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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ✲ ❯❋❇❆

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■▼

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P●▼❆❚

❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦

  

▼ét♦❞♦s ●❡♦♠étr✐❝♦s ♥♦ ❊st✉❞♦ ❡

■♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞♦ ❋❧✉①♦ ●❡♦❞és✐❝♦

❋❡❧✐♣❡ ▼♦s❝♦③♦ ❆r❛ú❥♦ ❞❛ ❈r✉③

  ❙❛❧✈❛❞♦r✲❇❛❤✐❛ ▼❛✐♦ ❞❡ ✷✵✶✷

  ▼ét♦❞♦s ●❡♦♠étr✐❝♦s ♥♦ ❊st✉❞♦ ❡ ■♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞♦ ❋❧✉①♦ ●❡♦❞és✐❝♦ ❋❡❧✐♣❡ ▼♦s❝♦③♦ ❆r❛ú❥♦ ❞❛ ❈r✉③

  ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❉✐❡❣♦ ❈❛t❛❧❛♥♦ ❋❡r✲ r❛✐♦❧✐✳

  ❙❛❧✈❛❞♦r✲❇❛❤✐❛ ▼❛✐♦ ❞❡ ✷✵✶✷

  ❞❛ ❈r✉③✱ ❋❡❧✐♣❡ ▼♦s❝♦③♦ ❆r❛ú❥♦✳ ▼ét♦❞♦s ●❡♦♠étr✐❝♦s ♥♦ ❊st✉❞♦ ❡ ■♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞♦ ❋❧✉①♦ ●❡♦❞és✐❝♦✳ ✴ ❋❡❧✐♣❡ ▼♦s❝♦③♦ ❆r❛ú❥♦ ❞❛ ❈r✉③✳ ✕ ❙❛❧✈❛❞♦r✿ ❯❋❇❆✱ ✷✵✶✷✳ ✽✾ ❢✳ ✿ ✐❧✳

  ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❉✐❡❣♦ ❈❛t❛❧❛♥♦ ❋❡rr❛✐♦❧✐✳ ❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ✷✵✶✷✳ ❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳

  ✶✳ ❋❧✉①♦ ●❡♦❞és✐❝♦✳ ✷✳ ❉✐str✐❜✉✐çõ❡s✳ ✸✳ ●❡♦♠❡tr✐❛ ❙✐♠♣❧ét✐❝❛✳ ✹✳❊s✲

tr✉t✉r❛s ❙♦❧ú✈❡✐s✳ ✺✳❙✐♠❡tr✐❛s✳ ✻✳■♥t❡❣r❛çã♦ ♣♦r ❙✐♠❡tr✐❛s✳ ■✳ ❋❡rr❛✐♦❧✐✱ ❉✐❡❣♦

❈❛t❛❧❛♥♦✳ ■■✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ■■■✳

❚ít✉❧♦✳

  ❈❉❯ ✿ ✺✶✹✳✼✼✹✳✽

  ▼ét♦❞♦s ●❡♦♠étr✐❝♦s ♥♦ ❊st✉❞♦ ❡ ■♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞♦ ❋❧✉①♦ ●❡♦❞és✐❝♦✳ ❋❡❧✐♣❡ ▼♦s❝♦③♦ ❆r❛ú❥♦ ❞❛ ❈r✉③

  ❉✐ss❡rt❛çã♦ ❞❡ ▼❡str❛❞♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❛♣r♦✈❛❞❛ ❡♠ ✷✽ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✷✳

  ❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

  Pr♦❢✳ ❉r✳ ❉✐❡❣♦ ❈❛t❛❧❛♥♦ ❋❡rr❛✐♦❧✐✭❖r✐❡♥t❛❞♦r✮ ❯❋❇❆

  Pr♦❢✳ ❉r✳ ❋❡❧✐❝✐❛♥♦ ▼❛r❝í❧✐♦ ❆❣✉✐❛r ❱✐tór✐♦ ❯❋❆▲

  Pr♦❢✳ ❉r✳ ❏♦sé ◆❡❧s♦♥ ❇❛st♦s ❇❛r❜♦s❛ ❯❋❇❆

  ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛ ❡ ❛♦s ♠❡✉s ♣❛✐s✳

  ✏❖ t❡♠♦r ❛ ❉❡✉s é ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝✐ê♥❝✐❛✳✑ Pr♦✈ér❜✐♦s ✶✳✼

  ❘❡s✉♠♦

  ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♦ ❡st✉❞♦ ❞❡ ♠ét♦❞♦s ❣❡♦♠étr✐❝♦s út❡✐s ♥❛ ✐♥t❡❣r❛çã♦ ♣♦r q✉❛❞r❛t✉r❛s ❞❡ ✢✉①♦s ❣❡♦❞és✐❝♦s✳ ❆❧é♠ ❞❡ ❞✐s❝✉t✐r ❛ ❛❜♦r❞❛❣❡♠ s✐♠♣❧ét✐❝❛✱ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ❛❞♦t❛❞❛ ♥❡st❡ t✐♣♦ ❞❡ ♣r♦❜✲ ❧❡♠❛✱ ❛♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ✉♠❛ ♥♦✈❛ ❛❜♦r❞❛❣❡♠ ❜❛s❡❛❞❛ ♥❛ ♥♦çã♦ ❞❡ ❡str✉t✉r❛ s♦❧ú✈❡❧✳ ❆ ❛♣❧✐❝❛çã♦ ❞❡st❡s ♠ét♦❞♦s é ✐❧✉str❛❞❛ ❛tr❛✈és ❞❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s✳

  P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❋❧✉①♦ ●❡♦❞és✐❝♦❀ ❉✐str✐❜✉✐çõ❡s❀ ●❊♦♠❡tr✐❛ ❙✐♠♣❧ét✐❝❛❀ ❊str✉✲ t✉r❛s ❙♦❧ú✈❡✐s❀ ❙✐♠❡tr✐❛s❀ ■♥t❡❣r❛çã♦ ♣♦r ❙✐♠❡tr✐❛s✳

  ❆❜str❛❝t

  ❚❤✐s ✇♦r❦ ❛✐♠s ❛t st✉❞② s♦♠❡ ❣❡♦♠❡tr✐❝ ♠❡t❤♦❞s ✇✐t❤ ❛r❡ ✉s❡❢✉❧ ✐♥ t❤❡ ✐♥t❡❣r❛✲ t✐♦♥ ❜② q✉❛❞r❛t✉r❡s ♦❢ ❣❡♦❞❡s✐❝ ✢♦✇s✳ ❆ ♥❡✇ ❛♣♣r♦❛❝❤✱ ❜❛s❡❞ ♦♥ t❤❡ ♥♦t✐♦♥ ♦❢ s♦❧✈❛❜❧❡ str✉❝t✉r❡✱ ✐s ♣r❡s❡♥t❡❞ t♦❣❡t❤❡r ✇✐t❤ t❤❡ s②♠♣❧❡❝t✐❝ ♦♥❡✱ tr❛❞✐t✐♦♥❛❧❧② ❢♦❧❧♦✇❡❞ ✐♥ t❤✐s ❦✐♥❞ ♦❢ ♣r♦❜❧❡♠✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❡t❤♦❞s ♦❢ ✐♥t❡❣r❛t✐♦♥ ❛r❡ ♣r❡s❡♥t❡❞ ❜② ♠❡❛♥s ♦❢ s♦♠❡ ❡①❛♠♣❧❡s✳

  ❑❡②✇♦r❞s✿ ●❡♦❞❡s✐❝ ❋❧♦✇❀ ❉✐str✐❜✉t✐♦♥s❀ ❙②♠♣❧❡❝t✐❝ ●❡♦♠❡tr②❀ ❙♦❧✈❛❜❧❡ ❙tr✉❝✲ t✉r❡s❀ ❙②♠♠❡tr✐❡s❀ ❙②♠♠❡tr②✲r❡❞✉❝t✐♦♥✳

  ❙✉♠ár✐♦

  ■♥tr♦❞✉çã♦ ①

  ✶ Pr❡❧✐♠✐♥❛r❡s ❞❡ ●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧ ✶

  ✶✳✶ ❱❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✷ ❊s♣❛ç♦ ❚❛♥❣❡♥t❡ ❡♠ ✉♠ ♣♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✸ ❉✐❢❡r❡♥❝✐❛❧ F ∗ ❞❡ ✉♠❛ ❆♣❧✐❝❛çã♦ F ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✹ ❋✐❜r❛❞♦s ❉✐❢❡r❡♥❝✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

  ✶✳✷ ❈❛♠♣♦s ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ❚❡♥s♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

  ✶✳✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸✳✷ ❋♦r♠❛s ❉✐❡❢❡r❡♥❝✐❛✐s ❡ ❋✐❜r❛❞♦ ❈♦t❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✸ ❉❡r✐✈❛❞❛ ❞❡ ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

  ✷ ❙✐♠❡tr✐❛s ❡ ❋♦r♠❛❧✐s♠♦ ❱❛r✐❛❝✐♦♥❛❧ ✷✷

  ✷✳✶ ❉✐str✐❜✉✐çõ❡s ❡ s✐♠❡tr✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✶ ❖ ❚❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶✳✷ ❙✐♠❡tr✐❛s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✸ ■♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

  ✷✳✷ ❊s♣❛ç♦s ❞❡ ❥❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

  

1

  (M, n) ✷✳✷✳✶ ❖ ♣r✐♠❡✐r♦ ❡s♣❛ç♦ ❞❡ ❥❛t♦s J ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷✳✷ Pr♦❧♦♥❣❛♠❡♥t♦s ❞❡ s✉❜✈❛r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✷✳✸ ❉✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✷✳✹ ❙✐♠❡tr✐❛s ✐♥✜♥✐t❡s✐♠❛✐s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✷✳✺ ❊s♣❛ç♦s ❞❡ ❥❛t♦s ❞❡ s❡çõ❡s ❞❡ ✉♠ ✜❜r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷✳✻ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ s♦❧✉çõ❡s ❡ s✐♠❡tr✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✷✳✼ ❊q✉❛çã♦ ❞❛s ❣❡♦❞és✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

  ✷✳✸ ❋♦r♠❛❧✐s♠♦ ❞❡ ❈❛rt❛♥ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✸✳✶ ❋♦r♠❛ ❞❡ ❈❛rt❛♥ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

  ✷✳✹ ❙✐♠❡tr✐❛s ❱❛r✐❛❝✐♦♥❛✐s ❡ ❚❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸ ■♥t❡❣r❛çã♦ ❞❡ ❋❧✉①♦s ●❡♦❞és✐❝♦s ❝♦♠ ▼ét♦❞♦s ❙✐♠♣❧ét✐❝♦s ✺✷

  ✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✷ ❱❛r✐❡❞❛❞❡s ❙✐♠♣❧ét✐❝❛s ❡ ❈❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✸ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✹ ❈❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s ❡ ❡str✉t✉r❛ ❞❡ P♦✐ss♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✺ ▼ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

  ✸✳✺✳✶ ❋✉♥çõ❡s ❣❡r❛❞♦r❛s ❞❡ tr❛♥s❢♦r♠❛çõ❡s s✐♠♣❧ét✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✺✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ❙✐♠♣❧ét✐❝❛s ❡ ♦ ▼ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥ ❏❛❝♦❜✐ ✳ ✳ ✳ ✳ ✻✵

  ✸✳✻ ❚❡♦r✐❛ ●❡♦♠étr✐❝❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✼ ❆♣❧✐❝❛çã♦ ❛ ❛❧❣✉♠❛s ▼étr✐❝❛s ❞❡ ❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

  ✸✳✼✳✶ ❈❛s♦ (a) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✸✳✼✳✷ ❈❛s♦ (b) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶

  ✹ ■♥t❡❣r❛çã♦ ❝♦♠ ❊str✉t✉r❛s ❙♦❧ú✈❡✐s ✼✸

  ✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✷ ➪❧❣❡❜r❛s ❙♦❧ú✈❡✐s ❞❡ ❙✐♠❡tr✐❛s ❡ ❊str✉t✉r❛s ❙♦❧ú✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✸ ■♥t❡❣r❛çã♦ ♣♦r q✉❛❞r❛t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✹✳✹ ❆♣❧✐❝❛çõ❡s ❛ ❊❉❖✬s ❞♦ ❚✐♣♦ ❱❛r✐❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✺ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

  2

  2

  2

  2

  • dx +2x dx dx +(1 + x ) dx

  1

  2

  3

  ✹✳✺✳✶ ❆♣❧✐❝❛çã♦ às ▼étr✐❝❛s ❞♦ ❚✐♣♦ g = dx

  1

  2

  1 3 ✼✾

  2 2 2x

  2

  2

  • φ (x ) (dx + e dx )

  1

  ✹✳✺✳✷ ❆♣❧✐❝❛çã♦ às ▼étr✐❝❛s ❞♦ t✐♣♦ g = dx

  1

  2 3 ✳ ✳ ✳ ✽✶

  ❘❡❢❡rê♥❝✐❛s ✽✺

  ■♥tr♦❞✉çã♦

  ❚❛♥t♦ ♥❛ ♠❛t❡♠át✐❝❛ q✉❛♥t♦ ♥❛ ❢ís✐❝❛✱ ❛s ❡q✉❛çõ❡s q✉❡ ❞❡s❝r❡✈❡♠ ❛s ❣❡♦❞és✐❝❛s ❞❡s❡♠♣❡♥❤❛♠ ✉♠ ♣❛♣❡❧ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✳ P♦r ✐ss♦✱ t♦r♥❛✲s❡ ♣❛rt✐❝✉❧❛r♠❡♥t❡ út✐❧ ♦ ❡st✉❞♦ ❞❡ ♠ét♦❞♦s ❞❡ ✐♥t❡❣r❛çã♦ ❡①❛t❛ ♣❛r❛ ❡st❡ t✐♣♦ ❞❡ ❡q✉❛çõ❡s✳

  ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ♦ ❡st✉❞♦ ❞❡ ♠ét♦❞♦s ❣❡♦♠étr✐❝♦s q✉❡ ♣❡r♠✐t❡♠ ❛ ✐♥t❡❣r❛çã♦ ♣♦r q✉❛❞r❛t✉r❛s ❞❡ ✢✉①♦s ❣❡♦❞és✐❝♦s✳ ❯♠❛ ê♥❢❛s❡ ♣❛rt✐❝✉❧❛r✱ ♥❡st❡ ❡st✉❞♦✱ é ❞❛❞❛ ❛♦s ❛s♣❡❝t♦s q✉❡ ❡♥✈♦❧✈❡♠ ❛ ♥♦çã♦ ❞❡ s✐♠❡tr✐❛✳ ❆ ♣r✐♥❝✐♣❛❧ ❥✉st✐✲ ✜❝❛t✐✈❛ ♣❛r❛ ❡st❡ ✐♥t❡r❡ss❡ ♥❛s s✐♠❡tr✐❛s é ❞❡✈✐❞♦✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❛♦ ❢❛t♦ q✉❡✱ ❥á ♥❛ ❛❜♦r❞❛❣❡♠ s✐♠♣❧ét✐❝❛✱ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ❛❞♦t❛❞❛ ♥♦ ❡st✉❞♦ ❞❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞❡ ✢✉①♦s ❣❡♦❞és✐❝♦s✱ ❛s s✐♠❡tr✐❛s ❍❛♠✐❧t♦♥✐❛♥❛s ❞❡s❡♥♣❡♥❤❛♠ ✉♠ ♣❛♣❡❧ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ✭✈✐❞❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✺✱ ✹✱ ✶✸✱ ✶❪ ❝♦♠♦ t❛♠❜é♠ ❬✸✽✱ ✸✾✱ ✶✹❪✮✳ ❊♠ s❡❣✉♥❞♦ ❧✉❣❛r✱ ❛ ✐♥t❡❣r❛❜✐❧✐✲ ❞❛❞❡ ❞♦s ✢✉①♦s ❣❡♦❞és✐❝♦s é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ✐♥t❡❣r❛çã♦ ❞❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s✱ ♣r♦❜❧❡♠❛ ♥♦ q✉❛❧ ♦ ♣❛♣❡❧ ❞❛s á❧❣❡❜r❛s ❞❡ s✐♠❡tr✐❛s ❛✐♥❞❛ é ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ✭✈✐❞❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✶✵✱ ✶✶✱ ✹✵✱ ✹✶✱ ✺✹✱ ✹✽❪ ❝♦♠♦ t❛♠❜é♠ ❬✷✱ ✷✾❪✮✳ ❯♠❛ s✐t✉❛çã♦ ❛♥❛❧♦❣❛✱ ❡♠❜♦r❛ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛✱ s❡ ❡♥❝♦♥tr❛ ♥♦ ❝❛s♦ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ✭✈✐❞❡✱ ♣♦r ❡①❡♠♣❧♦ ❬✷✱ ✶✵✱ ✶✶✱ ✸✸✱ ✹✵✱ ✹✶✱ ✹✾✱ ✺✹❪✮✳

  ❖ ❡st✉❞♦ ❞❡ ✢✉①♦s ❣❡♦❞és✐❝♦s é ✉♠ ❞♦s tó♣✐❝♦s ❞❡ ♣❡sq✉✐s❛ ♠❛✐s ✐♥t❡r❡ss❛♥t❡s ♣❛r❛ q✉❡♠ ❡st✉❞❛ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❝♦♠ ♠ét♦❞♦s s✐♠♣❧ét✐❝♦s✱ ❜❡♠ ❝♦♠♦ ❝♦♠ ♠ét♦❞♦s ❣❡♦♠étr✐❝♦s ♠❛✐s ❣❡r❛✐s✳ ❉❡ ❢❛t♦✱ ♥❛s ❛♣❧✐❝❛çõ❡s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♥❛ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✱ é ❢r❡q✉❡♥t❡ ❛ ♣r❡s❡♥ç❛ ❞❡ s✐♠❡tr✐❛s ♥❛ ♠étr✐❝❛ ✭✈✐❞❡✱ ♣♦r ❡①❡♠♣❧♦ ❬✹✾❪✮✳ ❊st❛s s✐♠❡tr✐❛s ❞♦ ❡s♣❛ç♦ s❡ r❡✢❡t❡♠ ❡♠ ♣r♦♣r✐❡❞❛❞❡s ❞❡ s✐♠❡tr✐❛s ❞❛s ❡q✉❛çõ❡s ❡✱ ♣♦rt❛♥t♦✱ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛ ✐♥t❡❣r❛çã♦ ❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✢✉①♦ ❣❡♦❞és✐❝♦✳

  ❚r❛❞✐❝✐♦♥❛❧♠❡♥t❡✱ ✉s❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ s✐♠♣❧ét✐❝♦✱ ❛ ✐♥t❡❣r❛çã♦ ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ❢❡✐t❛ ♣♦r ♠❡✐♦ ❞❡ ♠ét♦❞♦s ❝♦♠♦ ♦ ♠ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ♦✉ ✉s❛♥❞♦ t❡♦r❡♠❛s ❞❡ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❝♦♠♦ ♦ t❡♦r❡♠❛ ❞❡ ▲✐♦✉✈✐❧❧❡ ❝♦♠✉t❛t✐✈♦ ❡ ♥ã♦ ❝♦♠✉t❛t✐✈♦✳ ▼❛s✱ ♥❡♠ s❡♠✲ ♣r❡✱ ❡st❡s ♠ét♦❞♦s ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞♦s✳ P♦r ❡①❡♠♣❧♦✱ ♥❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✐♥t❡❣r❛✐s ❝♦♠♣❧❡t❛s ❞❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ❆ss✐♠ ❝♦♠♦ ♥❡♠ s❡♠♣r❡ ❛s ❤✐♣ót❡✲ s❡s ❞♦ t❡♦r❡♠❛ ❞❡ ▲✐♦✉✈✐❧❧❡ ✭❝♦♠✉t❛t✐✈♦ ❡ ♥ã♦ ❝♦♠✉t❛t✐✈♦✮ sã♦ s❛t✐s❢❡✐t❛s✳ P♦ré♠✱ ♥❡st❡s ❝❛s♦s✱ ✐ss♦ ♥ã♦ s✐❣♥✐✜❝❛ q✉❡ ❛s ❡q✉❛çõ❡s q✉❡ ❞❡s❝r❡✈❡♠ ♦ ✢✉①♦ ♥ã♦ ♣♦❞❡♠ s❡r ✐♥t❡❣r❛❞❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠ ❛❧❣✉♠❛ té❝♥✐❝❛ ♠❛✐s ❣❡r❛❧✳ ❊ss❡ t✐♣♦ ❞❡ s✐t✉❛çã♦ é ♠✉✐t♦ ❝♦♠✉♠ ♥♦

  ✵ ❡st✉❞♦ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ♦ q✉❡ t❡♠ ❡st✐♠✉❧❛❞♦✱ ♥♦s ú❧t✐♠♦s ❛♥♦s✱ ♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞❡ té❝♥✐❝❛s s❡♠♣r❡ ♠❛✐s ❣❡r❛✐s ✭✈✐❞❡ ❬✶✵✱ ✶✶✱ ✸✺❪ ❝♦♠♦ t❛♠❜é♠ ❬✷✷✱ ✷✵✱ ✷✶❪ ❡ s✉❛s r❡s♣❡❝t✐✈❛s r❡❢❡rê♥❝✐❛s✮✳ ❊♥tr❡ ❡ss❡s✱ ♦ ♠ét♦❞♦ ❞❛s ❡str✉t✉r❛s s♦❧ú✈❡✐s t❡♠ s❡ ♠♦str❛❞♦ ♣❛rt✐❝✉❧❛r♠❡♥t❡ út✐❧ ♥❛ ✐♥t❡❣r❛çã♦ ❞❡ ❊❉❖s✳

  ◆❡ss❛ ❞✐ss❡rt❛çã♦✱ ❛❧é♠ ❞❡ ❞✐s❝✉t✐r ❛ ❛❜♦r❞❛❣❡♠ s✐♠♣❧ét✐❝❛✱ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ❛❞♦t❛❞❛ ♥❡st❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✳ ❆♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠ ✉♠❛ ♥♦✈❛ ❛❜♦r❞❛❣❡♠ ❜❛s❡❛❞❛ ♥❛ ♥♦çã♦ ❞❡ ❡str✉t✉r❛s s♦❧ú✈❡✐s✳ ❆ ❛♣❧✐❝❛çã♦ ❞❡ss❡s ♠ét♦❞♦s s❡rá ✐❧✉str❛❞❛ ❛tr❛✈és ❞❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s✳

  ❆ ♣r❡s❡♥t❡ ❞✐ss❡rt❛çã♦ ❡stá ♦r❣❛♥✐③❛❞❛ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✳ ❖ ❈❛♣ít✉❧♦ ✶ ❧❡♠❜r❛ ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳ ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣ós ✉♠❛ ✐♥tr♦❞✉çã♦ à ♥♦çã♦ ❞❡ s✐♠❡tr✐❛ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦✱ é ❞✐s❝✉t✐❞❛ ❛ ❞❡s❝r✐çã♦ ✈❛r✐❛❝✐♦♥❛❧ ♣❛r❛ ❛s ❡sq✉❛çõ❡s ❞❛s ❣❡♦❞és✐❝❛s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥❡st❡ ❝❛♣ít✉❧♦✱ ✉s❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ✐♥✈❛r✐❛♥t❡ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥✱ é t❛♠❜é♠ ✐♥tr♦❞✉③✐❞❛ ❛ ♥♦çã♦ ❞❡ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ❥✉♥t♦ ❛♦ t❡♦r❡♠❛ ❞❡ ◆öt❤❡r q✉❡ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❡st❡ t✐♣♦ ❞❡ s✐♠❡tr✐❛ ❡ ❛s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✳ ◆♦ ❈❛♣ít✉❧♦ ✸✱ ✉s❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ s✐♠♣❧ét✐❝♦✱ tr❛t❛♠♦s ❛ ✐♥t❡❣r❛çã♦ ❞♦s ✢✉①♦s ❣❡♦❞és✐❝♦s ♣♦r ♠❡✐♦ ❞♦ ♠ét♦❞♦ ❝❧áss✐❝♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ◆❛ ♣❛rt❡ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✱ ♠♦str❛♠♦s ❛ ❛♣❧✐❝❛çã♦ ❞❡ss❡ ♠ét♦❞♦ à ✐♥t❡❣r❛çã♦ ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❞❡ ❛❧❣✉♠❛s ♠étr✐❝❛s ❞❡ ❊✐♥st❡✐♥✳ ◆♦ ❈❛♣ít✉❧♦ ✹✱ ❞✐s❝✉t✐♠♦s ♦ ✉s♦ ❞❡ s✐♠❡tr✐❛s ♥❛ ✐♥t❡❣r❛çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❛♣ós ✉♠❛ ✐♥tr♦❞✉çã♦ às ❡str✉t✉r❛s s♦❧ú✈❡✐s✱ ♣r♦✈❛♠♦s ♦ t❡♦r❡♠❛ s♦❜r❡ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ♣♦r q✉❛❞r❛t✉r❛s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠❛ t❛❧ ❡str✉t✉r❛✳ ◆❛ ♣❛rt❡ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦✱ ❛♣❧✐❝❛♠♦s ❡st❡s r❡s✉❧t❛❞♦s à ✐♥t❡❣r❛çã♦ ❞❡ ✢✉①♦s ❣❡♦❞és✐❝♦s✳

  ❈❛♣ít✉❧♦ ✶ Pr❡❧✐♠✐♥❛r❡s ❞❡ ●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧

  ◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ♥♦çõ❡s ❜ás✐❝❛s✱ ❛❧é♠ ❞❛s ♥♦t❛çõ❡s ♣r✐♥❝✐♣❛✐s✱ q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✳ ❈♦♠♦ ♥ã♦ é ♦❜❥❡t✐✈♦ ❞❡st❡ ❝❛♣ít✉❧♦ ❢♦r♥❡❝❡r t♦❞♦s ♦s ❞❡t❛❧❤❡s ❞♦ ♠❛t❡r✐❛❧ ♣r❡❧✐♠✐♥❛r ❛♣r❡s❡♥t❛❞♦ ❛q✉✐✱ ♦ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ❡♥❝♦♥tr❛rá ♠❛✐s ❞❡t❛❧❤❡s ♥❛s r❡❢❡rê♥❝✐❛s ❬✺✷❪✱ ❬✺✵❪✱ ❬✶✺❪✱ ❬✷✼❪✱ ❬✶✽❪✱ ❬✸✼❪✳

  ✶✳✶ ❱❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ✶✳✶✳✶ ■♥tr♦❞✉çã♦

  ❙❡rá ❛♣r❡s❡♥t❛❞❛ ❛q✉✐ ❛ ♥♦çã♦ ❞❡ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❛❧é♠ ❞❡ ❛❧❣✉♥s ❢❛t♦s út❡✐s ❛ r❡s♣❡✐t♦ ❞❡st❛✳ P❛r❛ ❡ss❡ ✜♠✱ ❝♦♠❡ç❛♠♦s ❝♦♠ ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ▼ ❞❡ ❍❛✉ss❞♦r❢ ❡ ❝♦♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧ é ✉♠❛

  k

  ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ ♥ ❡ ❝❧❛ss❡ C s❡ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ ✜♥✐t❛ ♦✉ ❡♥✉♠❡rá✈❡❧ ❞❡ ❤♦♠❡♦♠♦r✜s♠♦s

  

n

U : U U (U )

  A = {ϕ ⊂ R −→ ϕ ⊂ M} t❛✐s q✉❡ S

  ϕ U (U ) ✭✶✮ ▼ ❂ ❀

  U , ϕ

  V U (U ) V (V )

  ✭✷✮ ∀ϕ ∈ A t❛✐s q✉❡ ϕ ∩ ϕ 6= φ t❡♠♦s

  −1 −1 −1

  ϕ U : ϕ (ϕ U (U )

  V (V )) (ϕ U (U ) V (V )) V ◦ ϕ U ∩ ϕ −→ ϕ V ∩ ϕ k

  é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C ✳

  k

  ❆ ❢❛♠í❧✐❛ A s❛t✐s❢❛③❡♥❞♦ (1) ❡ (2) é ❝❤❛♠❛❞❛ ❛t❧❛s ❞❡ ❝❧❛ss❡ C ❡ s❡✉s ❡❧❡✲

  ✷

  ∞

  ❊♠ s❡❣✉✐❞❛✱ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r s♦♠❡♥t❡ ❛t❧❛s ❞❡ ❝❧❛ss❡ C ✳

  1

  2

  ❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❉♦✐s ❛t❧❛s A ❡ A s♦❜r❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M sã♦ ❡q✉✐✈❛❧❡♥t❡s

  ∞

  1

  2

  s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A ∪ A é ❛✐♥❞❛ ✉♠ ❛t❧❛s ❞❡ ❝❧❛ss❡ C ✭♥ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ t❛❧ r❡❧❛çã♦ ❡♥tr❡ ♦s ❛t❧❛s é✱ ❞❡ ❢❛t♦✱ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ♦✉ s❡❥❛✱ ✉♠❛ r❡❧❛çã♦ r❡✢❡①✐✈❛✱

  ∞

  s✐♠étr✐❝❛ ❡ tr❛♥s✐t✐✈❛✮✳ ❯♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧❡♥❝✐❛ ❞❡ ❛t❧❛s ❞❡ ❝❧❛ss❡ C ❞❡✜♥❡ ✉♠❛

  ∞

  ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ C s♦❜r❡ ♦ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M✳ ❙❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r q✉❡✱ ♣❛r❛ ♠✉♥✐r ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M

  ∞

  ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧✱ é s✉✜❝✐❡♥t❡ ✜①❛r ✉♠ ❛t❧❛s ❞❡ ❝❧❛ss❡ C ♣♦✐s t❛❧ ❡str✉t✉r❛

  ∞

  ♣♦❞❡ s❡r ♦❜t✐❞❛ ❝♦❧❡❝✐♦♥❛♥❞♦ t♦❞♦s ♦s ❛t❧❛s ❞❡ ❝❧❛ss❡ C q✉❡ sã♦ ❡sq✉✐✈❛❧❡♥t❡s ❛♦ ❛t❧❛s ✜①❛❞♦✳

  n i

  ❙❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡ x := R −→ R sã♦ ❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s ❡♠

  n U (U ) i : ϕ U (U )

  R ✱ ❝❛❞❛ ✈✐③✐♥❤❛♥ç❛ ϕ ♣♦❞❡ s❡r ❡q✉✐♣❛❞❛ ❝♦♠ n✲❢✉♥çõ❡s x → R ❞❡✜♥✐❞❛s

  −1 i := x i U (U )

  ♣♦r x ◦ ϕ U ❡ ❝❤❛♠❛❞❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❡♠ ϕ ✳ ❯s❛♥❞♦ ❛s ❢✉♥çõ❡s ❝♦♦r✲

  n

  ❞❡♥❛❞❛s✱ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣♦❞❡ s❡r ❧♦❝❛❧♠❡♥t❡ ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠ ✉♠ ❛❜❡rt♦ ❞♦ R ❡ ❝❛❞❛ ♣♦♥t♦ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣♦r ♠❡✐♦ ❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡s n ❝♦♦r❞❡♥❛❞❛s✳ P♦❞❡♠♦s✱ ♣♦r✲ t❛♥t♦✱ ❛♣❧✐❝❛r ❧♦❝❛❧♠❡♥t❡ ♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❛♦ ❡st✉❞♦ ❞❡ ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s✳ ▼❛s✱ t❡♥❞♦ ❡♠ ❝♦♥t❛ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✱ é ♣♦ssí✈❡❧ t❛♠❜é♠ ❝♦♥str✉✐r ✉♠ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡✜♥✐❞♦ ❣❧♦❜❛❧♠❡♥t❡ ❡♠ t♦❞❛ M✳

  ◆♦s ♣ró①✐♠♦s ♣❛rá❣r❛❢♦s s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ❛s♣❡❝t♦s ♣r✐♥❝✐♣❛✐s ❞❡st❛ ❝♦♥✲ str✉çã♦✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧

  U

  ❞❡✜♥✐❞❛ ♣❡❧♦ ❛t❧❛s A = {ϕ }✳ ❯♠❛ ❢✉♥çã♦ f : M → R é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ p✱

  ∗ ∞

  (f ) = f U

  U

  ❝♦♠ r❡s♣❡✐t♦ ❛ A✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ϕ ◦ ϕ é ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ C ❡♠

  −1

  ϕ (p) U U (U )

  

U ✱ ♣❛r❛ t♦❞❛ ϕ t❛❧ q✉❡ p ∈ ϕ ✳ ❆ ❢✉♥çã♦ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ M ❝♦♠ r❡s♣❡✐t♦

  ❛ A s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❢♦r ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ M ❝♦♠ r❡s♣❡✐t♦ ❛ A✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ M✱ ❝♦♠ r❡s♣❡✐t♦ ❛ A✱ t❡♠ ✉♠❛ ❡str✉t✉r❛

  ♥❛t✉r❛❧ ❞❡ R✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❛ s♦♠❛ ✉s✉❛❧ ❞❡ ❢✉♥çõ❡s ❡ ♦ ♣r♦❞✉t♦ ♣♦r ❡❧❡♠❡♥t♦s

  ∞ ∞

  (M ) (M ) ❞❡ R ❡ s❡rá ✐♥❞✐❝❛❞♦ ♣♦r C ✭♦✉ C A s❡ ❢♦r ♥❡❝❡ssár✐♦ ❡s♣❡❝✐✜❝❛r ❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧✮✳

  Pr❡❝✐s❛♠♦s t❛♠❜é♠ ❞❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✶✳✶✳✹✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ ❡str✉t✉r❛s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡✜♥✐❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣❡❧♦s ❛t❧❛s A

  1 ❡ A 2 ✳ ❯♠❛ ❢✉♥çã♦ F : M → N é ❞✐❢❡r❡♥❝✐á✈❡❧ ∗

  (f ) = f ♥♦ ♣♦♥t♦ p ∈ M s❡✱ ❡ s♦♠❡♥t❡ s❡✱ F ◦ F é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ p✱ ♣♦r ❝❛❞❛

  ∞ ∗ ∞

  f (N ) (C (N ))

  ∈ C A

  

2 ✳ ❆ ❛♣❧✐❝❛çã♦ F é ❞✐t❛ ❞✐❢❡r❡♥❝✐á✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ F A

2 ⊂ ∞

  C (M ) ✳

  A

  1

  ✸

  ✶✳✶✳✷ ❊s♣❛ç♦ ❚❛♥❣❡♥t❡ ❡♠ ✉♠ ♣♦♥t♦

  ◆❡st❛ s❡çã♦ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ♥♦çã♦ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t❛♥❣❡♥t❡ ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡♠ ✉♠ ♣♦♥t♦ ❛❧é♠ ❞❡ ♦✉tr❛s ❞❡✜♥✐çõ❡s q✉❡ ❞❡♣❡♥❞❡♠ ❞❡st❛ ♥♦çã♦✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ❛ ❉❡✜♥✐çã♦ ✶✳✶✳✺✳ ❙❡❥❛♠ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ p ∈ M✳ ❯♠ ✈❡t♦r t❛♥❣❡♥t❡ ❛ M ♥♦ ♣♦♥t♦ p é ✉♠❛ ❛♣❧✐❝❛çã♦

  ∞

  X : C (M ) −→ R q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

  ∞

  (M ), ✐✮ R ✲ ▲✐♥❡❛r✐❞❛❞❡✿ X(f + λg) = X(f) + λX(g), ∀f, g ∈ C ∀λ ∈ R

  ∞

  (M ) ✐✐✮ Pr♦♣r✐❡❞❛❞❡ ❞❡ ▲❡✐❜♥✐③✿ X(f · g) = X(f) · g(p) + f(p) · X(g), ∀f, g ∈ C

  P♦❞❡♠♦s✱ ❡♥tã♦✱ ❝♦♥s✐❞❡r❛r ✉♠ ✈❡t♦r t❛♥❣❡♥t❡ ❝♦♠♦ ✉♠❛ ✏❞❡r✐✈❛çã♦ ♥♦ ♣♦♥t♦ p✧✳

  p M

  ❖ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ M ❡♠ ✉♠ ♣♦♥t♦ p s❡rá ❞❡♥♦t❛❞♦ ♣♦r T ❡ ❞❡♥♦♠✐♥❛❞♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ M ♥♦ ♣♦♥t♦ p✳ ◆❡st❡ ❝♦♥❥✉♥t♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛s s❡❣✉✐♥t❡s ♦♣❡r❛çõ❡s✿

  (ξ + η)(f ) := ξ(f ) + η(f ) (c

  · ξ)(f) := c · [ξ(f)],

  ∞ p M (M )

  ∀ξ, η ∈ T ✱ ∀f ∈ C ❡ c ∈ R✳

  p M

  ❈♦♠ ❡st❛s ❞✉❛s ♦♣❡r❛çõ❡s✱ T é ✉♠ R✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❉❡st❛❝❛♠♦s✱ ♥❛ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡✱ ❞✉❛s ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s s❛t✐s❢❡✐t❛s

  ♣♦r t♦❞♦ ✈❡t♦r t❛♥❣❡♥t❡✿

  p M

  Pr♦♣♦s✐çã♦ ✶✳✶✳✻✳ ❙❡❥❛ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ P❛r❛ t♦❞♦ p ∈ M ❡ t♦❞♦ ξ ∈ T ✈❛❧❡♠✿

  ∞

  (M ) P✳✶✮ ξ(c) = 0✱ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ c ∈ C ✳

  ∞

  (M ) P✳✷✮ ξ(f) = ξ(g) ♣❛r❛ t♦❞❛s f, g ∈ C q✉❡ ❝♦✐♥❝✐❞❡♠ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p✳

  ✭▲♦❝❛❧✐③❛❜✐❧✐❞❛❞❡✮ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡ s❡rá ❜❛s❡❛❞❛ ♥♦ s❡❣✉✐♥t❡ ❧❡♠❛ ❡❧❡♠❡♥t❛r✿

  n

  ▲❡♠❛ ✶✳✶✳✼✳ ❙❡❥❛♠ U ❡ V ❜♦❧❛s ❢❡❝❤❛❞❛s ❞❡ R t❛✐s q✉❡ U ⊂ V ❡ V \U 6= ∅✳ ❊①✐st❡ ✉♠❛

  n

  ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ϕ : R −→ R✱ ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ❜♦ss❛✱ t❛❧ q✉❡

  U

  ✶✳ ϕ| ≡ 1❀

  n

  ✷✳ ϕ| R \V ≡ 0✳

  ✹ ❯♠❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✼❪ ♦✉ ❬✺✷❪✳

  n

  ❖ ♠❡s♠♦ r❡s✉❧t❛❞♦✱ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ✐❞❡♥t✐✜❝❛çã♦ ❧♦❝❛❧ ❞❛ ✈❛r✐❡❞❛❞❡ ❝♦♠ ♦ R ✈❛❧❡✱ ❧♦❝❛❧♠❡♥t❡✱ ♣❛r❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ q✉❛❧q✉❡r✳

  ❆ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✻ é ❛ s❡❣✉✐♥t❡✿ Pr♦✈❛✿ P❡❧❛ r❡❣r❛ ❞❡ ▲❡✐❜♥✐③✱ t❡♠♦s q✉❡ ξ(1) = ξ(1·1) = ξ(1)·1+1·ξ(1)✳ ▲♦❣♦✱ ξ(1) = 0✳ ❆ ♣r♦♣r✐❡❞❛❞❡ P✳✶✮ é ♦❜t✐❞❛ ♣❡❧❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❡ ξ✳

  ∞

  (M ) P❛r❛ ♣r♦✈❛r ❛ ♣r♦♣r✐❡❞❛❞❡ P✳✷✮✱ ❝♦♥s✐❞❡r❡ ❞✉❛s ❢✉♥çõ❡s f, g ∈ C q✉❡ ❝♦✲

  

1

  2

  1

  ✐♥❝✐❞❡♠ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ p✳ ❙❡ U ⊂ U ⊂ U sã♦ ❜♦❧❛s ❢❡❝❤❛❞❛s✱ ❝♦♠ p ∈ U ✱

  ∞

  (M ) = 1 M = 0 ❡♥tã♦✱ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❡①✐st❡ h ∈ C t❛❧ q✉❡ h| U

  1 ❡ h| \U 2 ✳ ❉❡ss❛ ❢♦r♠❛✱

  0 = (f − g).h ❡✱ ♣❡❧❛ r❡❣r❛ ❞❡ ▲❡✐❜♥✐③✱ ♦❜t❡♠♦s

  ξ((f − g) · h) = ξ(f − g) · h(p) + (f − g)(p) · ξ(h) = 0✳

  P♦rt❛♥t♦✱ ξ(f) = ξ(g)✳ ◗✉❡r❡♠♦s✱ ❛❣♦r❛✱ ❞❡s❝r❡✈❡r ❛ ❢♦r♠❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ ✉♠ ✈❡t♦r t❛♥❣❡♥t❡✳ P❛r❛ ✐ss♦✱

  n

  X

  n U

  1

  ∈ R | −

  , ..., ¯ x n ) (¯ x i s❡❥❛ ϕ ✉♠❛ ❝❛rt❛ ❝♦♠ ❞♦♠í♥✐♦ U✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❜♦❧❛ W = {(¯x

  i =1 2 −1

  p ¯ i ) < ǫ (p) } ⊂ U ❞❡ r❛✐♦ ǫ ❡ ❝❡♥tr♦ ¯p = ϕ U ✱ t❛❧ q✉❡ W ⊂ U✳

  ∞

  , ..., ¯ x n ) (M ) P❡❧♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❝á❧❝✉❧♦✱ ♣❛r❛ t♦❞♦ (¯x

  1 ❡ t♦❞❛ f ∈ C ✱

  ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿ Z

  1

  d

  ∗ ∗ ∗

  ϕ (f )(¯ x , ..., ¯ x n ) = ϕ (f )(¯ p , ..., ¯ p n ) + [ϕ (f )(¯ p + t(¯ x ), ..., ¯ p n + t(¯ x n

  U

  1 U

  1 U

  1

  1

  1

  − ¯p − dt p ¯ n ))]dt

  ✳ f = f ϕ f

  ▲♦❣♦✱ s❡ ¯ | (ω) ✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡♣r❡s❡♥t❛çã♦ ❝♦♦r❞❡♥❛❞❛ ❞❡ ¯

  

n

  X ¯ f (x) = f (p) + (x i i ) i (x),

  − p · g

  i

=1

  ♦♥❞❡ Z

  1 ∗

  ∂ϕ (f )

  U

  g i (x) = (¯ p + t(¯ x ), ..., ¯ p n + t(¯ x n n ))dt

  1

  1

  1 − ¯p − ¯p ✱ i ∈ {1, ..., n}.

  ∂ ¯ x i f f P❛r❛ ❛♣❧✐❝❛r X ❛ ¯ ✱ é ♥❡❝❡ssár✐♦ ✏❡st❡♥❞❡r✧ ¯ ♣♦✐s ✈❡t♦r❡s t❛♥❣❡♥t❡s s❡ ❛♣❧✐❝❛♠ ❛

  ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡✜♥✐❞❛s ❡♠ t♦❞❛ ❛ ✈❛r✐❡❞❛❞❡✳ P❛r❛ ✐ss♦✱ ✉s❛♥❞♦ ♦ ❧❡♠❛ ❛♥t❡r✐♦r✱

  ∞

  (M ) = 1 ❝♦♥s✐❞❡r❡ ✉♠❛ ❜♦❧❛ ❢❡❝❤❛❞❛ U ⊂ W t❛❧ q✉❡ p ∈ U ❡ ✉♠❛ h ∈ C t❛❧ q✉❡ h| U ❡ h M

  i i

  | \U ✳ P♦❞❡♠♦s✱ ❡♥tã♦✱✏❡st❡♥❞❡r✧❛s ❢✉♥çõ❡s x ❡ g à ✈❛r✐❡❞❛❞❡ M ❢❛③❡♥❞♦ (

  (x i · h)(q) ✱s❡ q ∈ W ; x i = e ✱s❡ q ∈ M\W .

  ✺ ❡ e g i =

  n

  1

  ∂ ∂x

  ∈ R✳ ❆❧é♠ ❞✐ss♦✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ ♦s ✈❡t♦r❡s

  i

  ✱ ♦♥❞❡ α

  p

  ∂x i |

  α i ∂

  i =1

  X

  t❡♠ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ❝♦♦r❞❡♥❛❞❛ ξ =

  p , ...,

  p M

  P♦rt❛♥t♦✱ t♦❞♦ ❡❧❡♠❡♥t♦ ξ ∈ T

  (f )) ✳

  ∗ U

  (ϕ

  1 U (p)

  −

  | ϕ

  ∂ ∂x

i

  (f ) =

  |

  ∂ ∂x n

  |

  ✶✳✶✳✸ ❉✐❢❡r❡♥❝✐❛❧ F ∗

  ∗

  (N ) ) p (ξ)] (f ) = ξ(F

  ∞

  N

  F (p)

  ) p : T p M −→ T

  ∗

  (F

  ❉❡✜♥✐❞❛ ❛ ♥♦çã♦ ❞❡ ✈❡t♦r ❡ ❡s♣❛ç♦ t❛♥❣❡♥t❡s✱ ✈❡❥❛♠♦s✱ ❛❣♦r❛✱ ❝♦♠♦ ❞❡✜♥✐r ❛ ❞✐❢❡r❡♥❝✐❛❧ ✭♦✉✏♣✉s❤✲❢♦✇❛r❞✧✮ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❋ ❡♥tr❡ ❞✉❛s ✈❛r✐❡❞❛❞❡s✳ ❉❡✜♥✐çã♦ ✶✳✶✳✽✳ ❙❡❥❛♠ M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ p ∈ M ❡ F : M −→ N ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ p✳ ❆ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ F ❡♠ p é ✉♠❛ ❛♣❧✐❝❛çã♦ R✲❧✐♥❡❛r

  ❞❡ ✉♠❛ ❆♣❧✐❝❛çã♦ F

  ❡ ❝♦♥❝❧✉✐♠♦s q✉❡ ❡st❡ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ ❞❡ ❞✐♠❡♥sã♦ n✳

  |

  p M

  ❢♦r♠❛ ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ T

  

p

  |

  ∂ ∂x n

  p , ...,

  |

  1

  ∂ ∂x

  sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡✲ ♣❡♥❞❡♥t❡s✳ ▲♦❣♦ ♦ ❝♦♥❥✉♥t♦

  p

  p

  ∂ ∂x i

  ( (g i

  p M

  ξ(x i ) ∂ϕ

  i =1

  X

  n

  ξ( e x i ) · e g i (p) =

  i =1

  X

  n

  ξ(f ) =

  ✱ ξ(f) = ξ( ¯ f ) ❆❣♦r❛✱ ✉s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s P✳✶✮ ❡ P✳✷✮✱ ♦❜t❡♠♦s✿

  | U ✳ P♦rt❛♥t♦✱ s❡ ξ ∈ T

  (f ) ∂x

  ✱ ♠❛s f| U = ˜ f

  ✳ ❈❧❛r❛♠❡♥t❡✱ f 6= ¯ f

  · e g i #

  i )

  ( e x i − p

  i

=1

  X

  

n

  "

  ❞❡ ¯ f ˜ f = f (p) +

  · h)(q) ✱s❡ q ∈ W ; ✱s❡ q ∈ M\W . ❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ s❡❣✉✐♥t❡ ❡①t❡♥sã♦ ˜ f

  ∗ U

  i

  (M ) −→ R é ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❞❡✜♥✐❞♦ ♣♦r

  (f )) = "

  p : C ∞

  ∂x i |

  ♦♥❞❡ ∂

  # (f ),

  p

  ∂x i |

  ξ( e x i ) ∂

  i =1

  X

  n

  ∗ U

  (ϕ

  # (ϕ

  (p)

  −

  ∂x i | ϕ

  ξ( e x i ) ∂

  i =1

  X

  n

  "

  (p)) =

  −1 U

  (f ))

  ✻ ) p

  ◗✉❛♥❞♦ ❛ ❛♣❧✐❝❛çã♦ (F ∗ é s♦❜r❡❥❡t✐✈❛✱ ♣❛r❛ t♦❞♦ p ∈ M✱ ❞✐③❡♠♦s q✉❡ F é ✉♠❛ ) p s✉❜♠❡rsã♦ ❞❡ M ❡♠ N✳ ◗✉❛♥❞♦ (F é ✐♥❥❡t✐✈❛ ♣❛r❛ t♦❞♦ p ∈ M ❞✐③❡♠♦s q✉❡ F é

  ∗

  ✉♠❛ ✐♠❡rsã♦✳ ❙❡✱ ❛❧é♠ ❞❡ F s❡r ✉♠❛ ✐♠❡rsã♦✱ ❢♦r t❛♠❜é♠ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ s✉❛ ✐♠❛❣❡♠ F (M)✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ F é ✉♠ ♠❡r❣✉❧❤♦ ❞❡ M ❡♠ N✳ ❉❡✜♥✐çã♦ ✶✳✶✳✾✳ ❙❡❥❛♠ ▼ ❡ ◆ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ M ⊂ N✳ ▼ é ✉♠❛ s✉❜✲ ✈❛r✐❡❞❛❞❡ ❞❡ ◆ s❡ ❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦ i : M −→ N é ✉♠ ♠❡r❣✉❧❤♦✳

  ❊♠ ❝♦♦r❞❡♥❛❞❛s✱ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ F ♥✉♠ ♣♦♥t♦ ❞❡ a ∈ M ♣♦❞❡ s❡r

  n

  , ..., x n U : U

  1

  ❡①♣r❡ss❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ s❡❥❛♠ x ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ϕ ⊂ R

  m U (U ) , ..., y n V : V

  t❛❧ q✉❡ a ∈ ϕ ❡ y

  1 ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ψ ⊂ R −→ N t❛❧

  (V ) , ..., x ) =

  V 1 n

  q✉❡ F (a) ∈ ψ ✳ ◆❡ss❛s ❝♦♦r❞❡♥❛❞❛s✱ F ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❝♦♠♦ F (x (y , ..., x n ), ..., y m , ..., x n )) =: (F (x , ..., x n ), ..., F m (x , ..., x n )) a a M

  1

  1

  1

  

1

  1

  1

  ◦ F (x ◦ F (x ❡ X ∈ T

  n

  X ∂

  ∞ a = α i a a M (N )

  ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ X ✳ P♦rt❛♥t♦✱ s❡ f ∈ C ❡♥tã♦ | ∈ T

  ∂x i

  i =1 n

  X ∂

  ∗

  [(F ) a (X a )] (f ) = α i a (F (f ))

  ∗ |

  ∂x i

  i =1

  !

  n m

  X X ∂f ∂F j

  = α i (F (a)) (a) ∂y j ∂x i

  i j =1 =1

  " ! #

  m n

  X X ∂F ∂

  j = α i (a) F (a) (f ).

  | ∂x i ∂y j

  j =1 i =1 n

  n o

  X ∂F j

  ∂

  ) a (X a ) F (a) α i (a) ▲♦❣♦✱ ❛ ❥✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡ ❞♦ ✈❡t♦r (F ∗ ❝♦♠ r❡s♣❡✐t♦ à ❜❛s❡ | é

  ∂y j

  ∂x i

  i =1

  ) a ❡ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❛ ❛♣❧✐❝❛çã♦ (F ∗ ♣❡❧❛ ♠❛tr✐③

    ∂F ∂F ∂F

  1

  1

  1

  (a) (a) (a) · · ·

    ∂x ∂x ∂x n

  1

  2

    ∂F ∂F ∂F

  2

  2

  2

    (a) (a) (a)

   · · ·   ∂x

  1 ∂x 2 ∂x n 

     

  ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

     

  ∂F ∂F ∂F

  m m m

  (a) (a) (a) · · ·

  ∂x ∂x ∂x n

  1

  2

  q✉❡ é ❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ ❞❛ F ✳

  ✶✳✶✳✹ ❋✐❜r❛❞♦s ❉✐❢❡r❡♥❝✐á✈❡✐s

  ❖ ♣r♦❞✉t♦ ❞✐r❡t♦ ✭♦✉ ❈❛rt❡s✐❛♥♦✮ M × N ❞❡ ❞✉❛s ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s é

  M = α , U α )

  ❡q✉✐♣❛❞♦✱ ❞❡ ♠❛♥❡✐r❛ ♥❛t✉r❛❧✱ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧✿ s❡ A {(ϕ } ❡

  ∞

  = , U )

  N β β

  A {(ψ } sã♦ ❛t❧❛s ❞✐❢❡r❡♥❝✐á✈❡✐s ✭❞❡ ❝❧❛ss❡ C ✮ s♦❜r❡ M ❡ N r❡s♣❡❝t✐✈❛♠❡♥t❡✱

  ∞ α β , U α β ) α β ) (u, v) =

  ✼ (ϕ α (u), ψ β (v)) α β

  ∈ M × N✱ u ∈ U ✱ v ∈ V ✳ ❯♠❛ ♣❡q✉❡♥❛ ❛❧t❡r❛çã♦ ❞❡st❛ ❝♦♥tr✉çã♦ ❣❡r❛ ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❝♦♥❝❡✐t♦s ❞❛ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧✿ ✉♠ ✜❜r❛❞♦ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❆ s❛❜❡r✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ π : E −→ M é ❝❤❛♠❛❞♦ ✜❜r❛❞♦ ❞✐❢❡r❡♥❝✐á✈❡❧ s♦❜r❡ M

  ❝♦♠ ✜❜r❛ N s❡ ✶✮ π é ✉♠❛ ❛♣❧✐❝❛çã♦ s♦❜r❡❥❡t♦r❛❀

  x

  ✷✮ ♣❛r❛ t♦❞♦ ♣♦♥t♦ x ∈ M✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❝♦♥t❡♥❞♦ ♦ ♣♦♥t♦ x t❛❧ q✉❡

  −1

  (U x ) x s✉❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ π é ❞✐❢❡♦♠♦r❢❛ ❛♦ ♣r♦❞✉t♦ U × N✱ ✐st♦ é✱ ❡①✐st❡ ✉♠

  −1 −1 U : U x (U x ) U (x, y) (x) x x

  ❞✐❢❡♦♠♦r✜s♠♦ f| × N −→ π t❛❧ q✉❡ f| ∈ π ✱ ♣❛r❛ t♦❞♦s x x ∈ U ❡ y ∈ N✳

  ❆ ♥♦t❛çã♦ ✉s✉❛❧ ❞❡ ✉♠ ✜❜r❛❞♦ é π : E −→ M ♦✉ (E, M, π)✳

  −1

  (x) ❆s s✉❜✈❛r✐❡❞❛❞❡s π ⊂ E✱ x ∈ M✱ sã♦ ❝❤❛♠❛❞❛s ✜❜r❛s ❞❡st❡ ✜❜r❛❞♦✳ ❊❧❛s sã♦ t♦❞❛s ❞✐❢❡♦♠♦r❢❛s ❛ N✳ ❖ ❡s♣❛ç♦ ❞♦ ✜❜r❛❞♦ E é ❛ ✉♥✐ã♦ ❞❡st❛s ✜❜r❛s q✉❡ ♥ã♦ s❡

  ✐♥t❡rs❡♣t❛♠ ❡♠ ♣❛r❡s✱ ✐st♦ é✱ ❡❧❡ é ✜❜r❛❞♦ ♣♦r ❡❧❛s✳ ❆ ✈❛r✐❡❞❛❞❡ M é ❝❤❛♠❛❞❛ ❜❛s❡ ❞♦ ✜❜r❛❞♦ (E, M, π)✳ ❊①❡♠♣❧♦ ✶✳✶✳✶✵✳ ❆ ♣r♦❥❡çã♦ π : M × N −→ M ❞❡ ✉♠ ♣r♦❞✉t♦ ❞✐r❡t♦ ♥♦ s❡✉ ♣r✐♠❡✐r♦ ❢❛t♦r é ✉♠ ✜❜r❛❞♦ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ♦ q✉❛❧ é ❝❤❛♠❛❞♦ ✜❜r❛❞♦ tr✐✈✐❛❧✳

  ❖ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ❞á ✉♠❛ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ q✉❛♥❞♦ ❞✐③❡♠♦s q✉❡✱ ❧♦❝❛❧♠❡♥t❡✱ ♦s ✜❜r❛❞♦s sã♦ ♣r♦❞✉t♦s ❞✐r❡t♦s✳

  −1 U α

  (V α ) ❙❡❥❛ A = {ϕ

  ✱ } ✉♠ ❛t❧❛s ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ M t❛❧ q✉❡ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ π

  −1

  V α = ϕ α (U α ) α = f

  V α : V α (V α )

  ✱ ❛❞♠✐t❡ ❛ ❛♣❧✐❝❛çã♦ tr✐✈✐❛❧✐③❛♥t❡ f | ×N −→ π s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ 2) ❞❡ ✉♠ ✜❜r❛❞♦✳ ❆ ❛♣❧✐❝❛çã♦

  −1

  f α : (V α β ) α β )

  β ◦ f ∩ V × N −→ (V ∩ V × N, α β ) α β

  ❛♣❧✐❝❛ ❞✐❢❡♦♠♦r✜❝❛♠❡♥t❡ t♦❞❛ ✜❜r❛ {x} × N ⊂ (V ✱ ♥❡❧❛ ♠❡s♠❛✳ ∩ V × N✱ x ∈ V ∩ V

  ❉❡♥♦t❛r❡♠♦s ❡st❡ ❞✐❢❡♦♠♦r✜s♠♦✱ ♣♦st❡r✐♦r♠❡♥t❡ ❛ ✐❞❡♥t✐✜❝❛çã♦ ó❜✈✐❛ ❞❡ N ❡ {x} × N✱ : N

  αβ

  ♣♦r h −→ N ❡ ❛ ❝❤❛♠❛r❡♠♦s ❢✉♥çã♦ ❞❡ tr❛♥s✐çã♦✳ ➱ ó❜✈✐♦ q✉❡

  −1

  h αα (x) = id, h βα (x) = h (x), h αβ (x) βγ (x) = h αγ (x)

  αβ ◦ h αβ (x) α β

  ❡ q✉❡ ♦s ❞✐❢❡♦♠♦r✜s♠♦s h ❞❡♣❡♥❞❡♠ ❞✐❢❡r❡♥❝✐❛✈❡❧♠❡♥t❡ ❞❛ ✈❛r✐á✈❡❧ x ∈ U ∩ U ✳

  U α

  ❆♦ ❝♦♥trár✐♦✱ ❡s♣❡❝✐✜❝❛♥❞♦ ✉♠ ❛t❧❛s A = {ϕ } ❡♠ M ❡ ✉♠ s✐st❡♠❛ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s h (x)

  αβ

  q✉❡ s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ✉♠ ✜❜r❛❞♦ π : E −→ M

  αβ (x) αβ

  ♣❛r❛ ♦ q✉❛❧ ❛s h ✬s sã♦ ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦✳ ■♥❢♦r♠❛❧♠❡♥t❡ ❢❛❧❛♥❞♦✱ ❛s ❢✉♥çõ❡s h

  α

  ♠♦str❛♠ ❝♦♠♦ ❛ ✈❛r✐❡❞❛❞❡ E é ✉♠❛ ❝♦❧❛❣❡♠ ❞❡ ❜❧♦❝♦s ❞❛ ❢♦r♠❛ U × N✳

  ✽ ❯♠❛ ❛♣❧✐❝❛çã♦ σ : M −→ E t❛❧ q✉❡ π ◦ σ = id é ❝❤❛♠❛❞❛ s❡çã♦ ❞♦ ✜❜r❛❞♦

  −1

  (E, M, π) (x)

  ✳ ❖❜✈✐❛♠❡♥t❡✱ σ é ✉♠❛ s❡çã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ σ(x) ∈ π ✳ ❆ ❝♦❧❡çã♦ ❞❡ t♦❞❛s ❛s s❡çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞♦ ✜❜r❛❞♦ ❡♠ q✉❡stã♦ é ❞❡♥♦t❛❞❛ ♣♦r Γ(π)✳

  i , M i , π i )

  1

  2

  ❙❡❥❛♠ (E ✱ i ∈ {1, 2}✱ ✜❜r❛❞♦s✳ ❖s ♣❛r❡s ❞❡ ❛♣❧✐❝❛çõ❡s F : E −→ E ✱ f : M

  1

  2

  2

  1

  −→ M sã♦ ❝❤❛♠❛❞♦s ♠♦r✜s♠♦s ❞♦ ♣r✐♠❡✐r♦ ✜❜r❛❞♦ ♥♦ s❡❣✉♥❞♦ s❡ π ◦F = f ◦π ✳

  1

  2

  ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠ ♠♦r✜s♠♦ ❛♣❧✐❝❛ ✜❜r❛s ❞❡ π ❡♠ ✜❜r❛s ❞❡ π ✳ ❖ ♠♦r✜s♠♦ ✐♥✈❡rs♦ é ❝❤❛♠❛❞♦ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ✭♦✉ ✐s♦♠♦r✜s♠♦✮ ❞♦s ✜❜r❛❞♦s ❡♠ q✉❡stã♦✳

  1

  ❖ ✜❜r❛❞♦ π é ❝❤❛♠❛❞♦ s✉❜✜❜r❛❞♦ s❡ ♦ ♠♦r✜s♠♦ (F, f) ❞❡t❡r♠✐♥❛ ♠❡r❣✉❧❤♦s ❞❡ ֒ ֒

  1

  2

  1

  2

  s✉❜✈❛r✐❡❞❛❞❡s F : E → E ✱ f : M → M ✳

  1

  ◗✉❛❧q✉❡r ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ f : M −→ M ♥❛ ❜❛s❡ ❞♦ ✜❜r❛❞♦ π : E −→ M

  ∗ ∗

  (x) : f (E) ❣❡r❛ ✉♠ ✜❜r❛❞♦ s♦❜r❡ M

  1 ✱ ❝❤❛♠❛❞♦ ✜❜r❛❞♦ ✐♥❞✉③✐❞♦✱ ❡ ❞❡♥♦t❛❞♦ ♣♦r f

  −→ M

  1

  ✳

  1

  ❯♠❛ ✜❜r❛✱ ❞♦ ✜❜r❛❞♦ ✐♥❞✉③✐❞♦✱ ❡♠ ✉♠ ♣♦♥t♦ y ∈ M ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ✜❜r❛ ❞♦

  ∗

  (E) ✜❜r❛❞♦ ♦r✐❣✐♥❛❧ ❡♠ ✉♠ ♣♦♥t♦ x = f(y)✳ ❋♦r♠❛❧♠❡♥t❡✱ ❛ ✈❛r✐❡❞❛❞❡ f é ❞❡✜♥✐❞❛ ❝♦♠♦

  1

  ❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ M × E ❝♦♥s✐st✐♥❞♦ ❞❡ ♣❛r❡s (y, e) t❛✐s q✉❡ f(y) = π(e) ❡ ❛ ♣r♦❥❡çã♦

  ∗

  f (π) é ❞❡✜♥✐❞❛ ❝♦♠♦ ❛ ❛♣❧✐❝❛çã♦ (y, e) 7→ y✳

  −1

  (x) ❯♠ ✜❜r❛❞♦ (E, M, π) é ❝❤❛♠❛❞♦ ✜❜r❛❞♦ ✈❡t♦r✐❛❧ s❡ s✉❛s ✜❜r❛s π sã♦

  ❡q✉✐♣❛❞❛s ❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ✏❞❡♣❡♥❞❡♥❞♦ ❞✐❢❡r❡♥❝✐❛✈❡❧♠❡♥t❡✧❞♦ ♣♦♥t♦ x ∈ M✳ ■ss♦ ♠♦str❛ q✉❡ ❡st❡ ✜❜r❛❞♦ ♣♦❞❡ s❡r ❡s♣❡❝✐✜❝❛❞♦ ♣♦r ❢✉♥çõ❡s ❞❡ tr❛♥✲

  αβ (x)

  s✐çã♦ ❧✐♥❡❛r❡s✱ ✐st♦ é✱ h sã♦ tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ❞❛ ✜❜r❛ ❝❛♥ô♥✐❝❛ N✱ ❛ q✉❛❧ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳

  ∞ i i (M )

  ❙❡❥❛♠ S ✱ i ∈ {1, ..., k}✱ s❡çõ❡s ❞♦ ✜❜r❛❞♦ ✈❡t♦r✐❛❧ π ❡ s❡❥❛♠ f ∈ C P f i s i

  ❢✉♥çõ❡s s♦❜r❡ ❛ ❜❛s❡ M✳ ❊♥tã♦ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ s❡çã♦ s = i ❞❛❞❛ ♣♦r P

  −1

  s(x) = f i (x)s i (x) (x)

  i ✱ ❥á q✉❡ ❛ ✜❜r❛ π é ✉♠ ❡s♣ç♦ ✈❡t♦r✐❛❧✳ ❉❡ss❛ ❢♦r♠❛✱ ❞✐③❡♠♦s ∞

  q✉❡ Γ(π) é ✉♠ C ✲♠ó❞✉❧♦✳ ❙✐♠✐❧❛r♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✜❜r❛❞♦s ❝✉❥❛s ✜❜r❛s t❡♠ ♦✉tr❛ ❡str✉t✉r❛✱ ♣♦r

  ❡①❡♠♣❧♦✱ ❛ ❡str✉t✉r❛ ❞❡ ✉♠ ❡s♣❛ç♦ ❛✜♠✱ ❛ ❡str✉t✉r❛ ❞❡ ✉♠ ❣r✉♣♦✱ ❡ ♦✉tr❛s ❛✐♥❞❛✳ ❚❛✐s

  αβ (x)

  ✜❜r❛❞♦s sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ♣❡❧♦ ❢❛t♦ q✉❡ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ h q✉❡ ♦ ❡s♣❡❝✐✜❝❛♠ sã♦ tr❛♥s❢♦r♠❛çõ❡s ❞❛ ✜❜r❛ ❝❛♥ô♥✐❝❛ N q✉❡ ♣r❡s❡r✈❛♠ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡str✉t✉r❛✳ ❉❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M ❞❡ ❞✐♠❡♥sã♦ n✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦♥❥✉♥t♦

  [ T M = T p M

  p ) = p

  ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ s♦❜r❡❥❡t♦r❛ π : T M −→ M ❞❡✜♥✐❞❛ ♣♦r π(ξ ❞❡✲

  p ∈M

  ♥♦♠✐♥❛❞❛ ♣r♦❥❡çã♦ ♥❛t✉r❛❧✳ ❖ ❝♦♥❥✉♥t♦ T M ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ♥❛t✉r❛❧ ❞❡ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ s❡❥❛♠ ♦s ❛❜❡rt♦s ❞❡ T M ❛s ✐♠❛❣❡♥s ✐♥✈❡rs❛s ❞♦s ❛❜❡rt♦s ❞❡ M ♣❡❧❛ ❛♣❧✐❝❛çã♦ π ❡ A ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛ ❛ ✈❛r✐❡❞❛❞❡ M✳ P❛r❛

  U , ..., x n U

  t♦❞❛ ϕ ∈ A ❝♦♠ x

  1 ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ϕ ✱ ❞❡✜♥❛ ❛s ❛♣❧✐❝❛çõ❡s 2n

  ϕ : e U e U e −→ R

  ✾ ϕ (ξ p ) = (x (p), ..., x n (p), ξ p (x ), ..., ξ p (x n )) e

  1

  1 U e

  ✱

  −1

  U = π (U ) ♦♥❞❡ ˜ ✳

  ❊st❛s ❛♣❧✐❝❛çõ❡s sã♦ ✐♥❥❡t♦r❛s ❡✱ ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ❡♠ T M✱ sã♦ ❤♦♠❡✲ ♦♠♦r✜s♠♦s✳ ❆❧é♠ ❞✐ss♦✱ T M é ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉ss❞♦r❢ ♣❛r❛❝♦♠♣❛❝t♦ ❝♦♠ ❡str✉t✉r❛

  2n −1 ∞

  ϕ : e U ϕ ϕ

  ❞✐❢❡r❡♥❝✐á✈❡❧ e A = { e U e −→ R } ❞❡ ❝❧❛ss❡ C ♣♦✐s ❛s ❛♣❧✐❝❛çõ❡s e U e ◦ e sã♦ ❞✐❢❡♦✲

  V e ∞

  ϕ ϕ ♠♦r✜s♠♦s ❞❡ ❝❧❛ss❡ C ♣❛r❛ t♦❞❛s e

  U e , e

V e ∈ e A✳

  ❆❣♦r❛ q✉❡ ❞♦t❛♠♦s T M ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧✱ é ❢á❝✐❧ ✈❡r q✉❡ π : T M

  −→ M

  −1 p M = π (p)

  é ✉♠ ✜❜r❛❞♦ ✈❡t♦r✐❛❧ q✉❡ é ❞❡♥♦♠✐♥❛❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❛ ▼✳ ❖s ❡s♣❛ç♦s T sã♦ ❛s ✜❜r❛s✳ ❯♠❛ s❡çã♦ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ é✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❛♣❧✐❝❛çã♦ Φ : M −→ T M t❛❧

  −1

  (x) q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ❝❛❞❛ ♣♦♥t♦ x ∈ M ❡stá ♥❛ ✜❜r❛ π s♦❜r❡ x✱ ♦✉ s❡❥❛✱ ❛ ✐♠❛❣❡♠ ❞❡

  p M

  ❝❛❞❛ ♣♦♥t♦ x ∈ M ♣❡rt❡♥❝❡ ❛♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ T ✳ P♦rt❛♥t♦✱ ✉♠❛ s❡çã♦ ❞❡ π ♣♦❞❡ s❡r ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s✳ ◆❛ ♣ró①✐♠❛ s❡çã♦✱ s❡rá ✐♥tr♦❞✉③✐❞❛✱ t❛♠❜é♠✱ ✉♠❛ ♦✉tr❛ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡ ❞❡ ❝♦♥s✐❞❡r❛r ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s✳

  ✶✳✷ ❈❛♠♣♦s ❞❡ ✈❡t♦r❡s

  ◆❡st❛ s❡çã♦ tr❛t❛r❡♠♦s ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡

  ∞

  (M ) ❝♦♠♦ ❞❡r✐✈❛çõ❡s ❞❛ á❧❣❡❜r❛ C ✳ ❉❡ss❛ ❢♦r♠❛✱ ✈❡r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❞❡ R✲á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❆❧é♠ ❞✐ss♦✱ ❡♠ ❞❡❝♦rrê♥❝✐❛ ❞♦s t❡♦r❡♠❛s ❞❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡✱ s❡rá ❞❡✜♥✐❞♦✱ t❛♠❜é♠✱ ♦ ✢✉①♦ ❞❡ ✉♠ ❝❛♠♣♦ ❥✉♥t♦ ❛ s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ❛ s❡❣✉✐♥t❡

  ∞

  ❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s C s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ▼ é ✉♠❛ ❛♣❧✐❝❛çã♦

  ∞ ∞

  X : C (M ) (M ) −→ C q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

  ∞

  (M ) ✭✐✮ R✲❧✐♥❡❛r✐❞❛❞❡ ✿ X(αf + βg) = αX(f) + βX(g)✱ ∀α, β ∈ R✱ ∀f, g ∈ C ✳

  ∞

  (M ) ✭✐✐✮ Pr♦♣r✐❡❞❛❞❡ ❞❡ ▲❡✐❜♥✐③✿ X(fg) = X(f)g + X(g)f✱ ✳

  ∀f, g ∈ C

  ∞

  (M ) P♦rt❛♥t♦✱ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s é ✉♠❛ ❞❡r✐✈❛çã♦ s♦❜r❡ C ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡st❛s

  ∞

  ❞❡r✐✈❛çõ❡s s❡rá ❞❡♥♦t❛❞♦ ♣♦r D(M)✳ ❈♦♥s✐❞❡r❛r❡♠♦s s❡♠♣r❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s C ✳ ❈♦♠♦ ♥♦ ❝❛s♦ ❞♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡♥tr❡

  ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❝♦♠♦ t❛♠❜é♠ ♣r♦❞✉t♦ ❞❡ ✉♠ ❝❛♠♣♦ ♣♦r ✉♠❛ ❢✉♥çã♦✿

  ✶✵

  ∞

  (X + X )(f ) := X (f ) + X (f ) , X (M )

  1

  2

  1

  2

  1

  2

  ✱ ∀X ∈ D(M)✱ ∀f ∈ C

  ∞

  (f X)(g) := f X(g) (M ) ✱ ✳

  ∀X ∈ D(M)✱ ∀f, g ∈ C

  ∞

  (M ) ❈♦♠ ❡st❛s ♦♣❡r❛çõ❡s✱ D(M) é ✉♠ C ✲♠ó❞✉❧♦✳ ❙❡❥❛♠ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ X ∈ D(M) ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✳ ❊♠ ❝❛❞❛

  p

  ♣♦♥t♦ p ∈ M✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ✈❡t♦r t❛♥❣❡♥t❡ X ♣♦r

  ∞

  X p (f ) := X(f )(p) (M ) ✱ ✳

  ∀f ∈ C

  X :

  ❉❡ss❛ ❢♦r♠❛✱ ❞❛❞♦ ✉♠ ❝❛♠♣♦ X ∈ D(M)✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❛♣❧✐❝❛çã♦ φ M

  X (p) = X p

  −→ T M ♣♦r φ ✱ ♦✉ s❡❥❛✱ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❞❡✜♥❡ ✉♠❛ s❡çã♦ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡✳ ■ss♦ ❥✉st✐✜❝❛ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ❝❛♠♣♦ ❞❛❞❛ ♥❛ s❡çã♦ ❛♥t❡r✐♦r ❡ ❛q✉❡❧❛ ❞❛❞❛ ♥❛ ♣r❡s❡♥t❡ s❡çã♦✳

  ◗✉❡r❡♠♦s ❡①♣r❡ss❛r ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❡♠ ❝♦♦r❞❡♥❛❞❛s✳ P❛r❛ ✐ss♦✱ s❡❥❛♠

  ∞

  x , ..., x (M )

  1 n U

  ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ϕ ✱ X ∈ D(M) ❡ f ∈ C ✳ P❡❧❛ ♦❜s❡r✈❛çã♦

  p p (f ) =

  ❛♥t❡r✐♦r✱ ❡♠ ❝❛❞❛ ♣♦♥t♦ p ∈ M✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ✉♠ ✈❡t♦r t❛♥❣❡♥t❡ X t❛❧ q✉❡ X X(f )(p)

  U (U ) p

  ✳ ❆❧é♠ ❞✐ss♦✱ ❥á ❢♦✐ ✈✐st♦ q✉❡ s❡ p ∈ ϕ ❡♥tã♦ ♣♦❞❡♠♦s ❡①♣r❡ss❛r X ❡♠

  n n

  X X ∂

  ∂

  p =

  X p (x i ) p X(x i ) ❝♦♦r❞❡♥❛❞❛s ♣♦r X | ✳ ▲♦❣♦✱ ♦❜t❡♠♦s q✉❡ X = ✳

  ∂x i ∂x i

  i i =1

  =1 ∞ ∞

  ❉❡✜♥✐çã♦ ✶✳✷✳✷✳ ❙❡ F : M −→ N é ✉♠❛ ❛♣❧✐❝❛çã♦ C ❡ ❳ ❡ ❨ sã♦ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s C (X p ) = Y F

  p (p)

  ❡♠ ▼ ❡ ◆✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞✐③❡♠♦s q✉❡ ❳ ❡ ❨ sã♦ ❋✲r❡❧❛❝✐♦♥❛❞♦s s❡ F ∗ ✱ ♣❛r❛ t♦❞♦ p ∈ M✳

  ❉❡❝♦rr❡ ❞❡st❛ ❞❡✜♥✐çã♦ q✉❡ s❡ X ❡ Y sã♦ F − relacionados ❡♥tã♦ Y F (g) = F (X p )(g) = X p (g

  (p) p ∗ ◦ F ) ✭✶✳✶✮

  ❡ t❡♠♦s t❛♠❜é♠ q✉❡

  ∗ ∗ F .

  ◦ Y = X ◦ F ✭✶✳✷✮ ❆ ✈♦❧t❛ t❛♠❜é♠ ✈❛❧❡✳

  P♦❞❡♠♦s ♠✉♥✐r ♦ ♠ó❞✉❧♦ D(M) ❝♦♠ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❡str✉t✉r❛ ♣❛r❛ ❛ q✉❛❧ é ♥❡❝❡ssár✐❛ ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✶✳✷✳✸✳ ❖ ♣r♦❞✉t♦ ❞❡ ▲✐❡ ✭♦✉ ❝♦♠✉t❛❞♦r✮ ❞❡ ❞♦✐s ❝❛♠♣♦s ✈❡t♦r❡s X, Y ∈ D(M) é ❞❡✜♥✐❞♦ ♣♦r

  [X, Y ](f ) := X ◦ Y − Y ◦ X✳

  ✶✶ ➱ ❢á❝✐❧ ✈❡r q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ▲✐❡ ❞❡ ❞♦✐s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s é t❛♠❜é♠ ✉♠ ❝❛♠♣♦

  ✈❡t♦r✐❛❧✳ ❱❡r❡♠♦s ❛❣♦r❛ ❝♦♠♦ ❡①♣r❡ss❛r ♦ ♣r♦❞✉t♦ ❞❡ ▲✐❡ ❞❡ ❞♦✐s ❝❛♠♣♦s ❡♠ ❝♦♦r❞❡♥❛❞❛s✳ P P

  ∂ ∂ 1 , ..., x n

  a i b i P❛r❛ ✐ss♦✱ s❡❥❛♠ x ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s t❛✐s q✉❡ X = i ❡ Y = i ✳

  ∂x ∂x i i

  ❊♥tã♦✱

  n

  X [X, Y ] = (X(b i ) i )) .

  − Y (a

  i =1

  Pr♦♣♦s✐çã♦ ✶✳✷✳✹✳ ❖ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ é ❛♥t✐ss✐♠étr✐❝♦✱ R✲❧✐♥❡❛r ❡ s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐

  [[X, Y ], Z] + [[Z, X], Y ] + [[Y, Z], X] = 0 ❆ ❞❡♠♦♥str❛çã♦ ❞❡❝♦rr❡ ❞❡ ✈❡r✐✜❝❛çõ❡s ❞✐r❡t❛s✳ ❯♠❛ á❧❣❡❜r❛ ❝♦♠ ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦ s❛t✐s❢❛③❡♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❝♦❧❝❤❡t❡ ❞❡

  ▲✐❡ ❛❝✐♠❛ é ❞❡♥♦♠✐♥❛❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❉❡ss❛ ❢♦r♠❛✱ D(M) é ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❞♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s s♦❜r❡ ❛ ✈❛r✐❡❞❛❞❡ M✳

  i i

  Pr♦♣♦s✐çã♦ ✶✳✷✳✺✳ ❙❡ X ∈ D(M) ❡ Y ∈ D(N) sã♦ ❋✲r❡❧❛❝✐♦♥❛❞♦s✱ ♣❛r❛ i ∈ {1, 2}✱ , X ] , Y ]

  1

  2

  1

  2

  ❡♥tã♦ [X ❡ [Y sã♦ ❋✲r❡❧❛❝✐♦♥❛❞♦s✳

  ∞

  Pr♦✈❛✿ ❙❡ g : N −→ R é C ✱ ❡♥tã♦✱ ♣♦r ✶✳✶✱ (Y i (g)) i (g

  ◦ f = X ◦ f), ∀i ∈ {1, 2}✳ ▲♦❣♦✱

  ([Y , Y ](g)) (Y (g)) (Y (g))

  1

  2

  1

  

2

  2

  1

  ◦ f = Y ◦ f − Y ◦ f = X (Y (g) (Y (g)

  1

  

2

  2

  1

  ◦ f) − X ◦ f) = X

  1 (X 2 (g 2 (X 1 (g

  ◦ f)) − X ◦ f)) = [X , X ](g

  1

  2 ◦ f).

  ∞

  ❉❡✜♥✐çã♦ ✶✳✷✳✻✳ ❙❡❥❛ X ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s C ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✳ ❯♠❛ tr❛✲

  ′

  (t) = X c

  (t)

  ❥❡tór✐❛ ❧♦❝❛❧ ❞❡ X é ✉♠❛ ❝✉r✈❛ c : (a, b) −→ M t❛❧ q✉❡ c ✱ ♦♥❞❡ t ❞❡♥♦t❛ ♦

  d ′

  (t) = c ♣❛râ♠❡tr♦ ❞❛ ❝✉r✈❛ ❡ c ∗ ✳ ❙❡ ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ c t❡♠ ❞♦♠í♥✐♦ R✱ ❞✐③❡♠♦s

  dt

  q✉❡ c é ✉♠❛ tr❛❥❡tór✐❛ ❣❧♦❜❛❧✳ ◗✉❡r❡♠♦s s❛❜❡r s♦❜r❡ q✉❛✐s ❝♦♥❞✐çõ❡s ✉♠❛ ❝✉r✈❛ c : I ⊂ R −→ M é ✉♠❛

  1 ❢✉♥çõ❡s U U (U )

  , ..., x n tr❛❥❡tór✐❛ ❞❡ ✉♠ ❝❛♠♣♦ X✳ P❛r❛ ✐ss♦ s❡❥❛♠ τ ∈ I✱ c(τ) = a ❡ s❡❥❛♠ x

  ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ϕ ❝♦♠ a ∈ ϕ ✳ ❈♦♥s✐❞❡r❡ t❛♠❜é♠ q✉❡ X é r❡♣r❡s❡♥t❛❞♦

  n

  X ∂

  U (U ) α i (t), ..., c n (t))

  1

  ❡♠ ϕ ♣♦r X = ❡ q✉❡ c(t) = (c ✳ ▲♦❣♦ ∂x i

  ✶✷

  

n

  X dc i ∂

  d

t =τ c(t) = (τ ) a

  | | ✳

  dt

  dt ∂x i

  

i =1

  P♦rt❛♥t♦✱ ♣❛r❛ q✉❡ c s❡❥❛ ✉♠❛ tr❛❥❡tór✐❛ ❞♦ ❝❛♠♣♦ X✱ ❞❡✈❡ s❡r s❛t✐s❢❡✐t♦ ♦ s❡❣✉✐♥t❡✿

  n n

  X X ∂ d dx i ∂

  α (a) = X = c(t) = (τ )

  i a t =τ a

  | | ✱ ∂x dt dt ∂x i

  i a i =1 i =1

  ♦✉ s❡❥❛✱

  dc i

  (t) = α i (c(t)) ✱ i ∈ {1, ..., n}✳

  dt

  ■ss♦ ❥✉st✐✜❝❛ ♦ s❡❣✉✐♥t❡

  1 , ..., x n

  ❚❡♦r❡♠❛ ✶✳✷✳✼✳ ❙❡❥❛♠ ▼ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ x ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛

  n

  X ∂

  n U : U

  α i ❛ ❝❛rt❛ ϕ ⊂ R −→ M ❡ X ∈ D(M) ❡①♣r❡ss♦ ❧♦❝❛❧♠❡♥t❡ ♣♦r ✳ ❯♠❛

  ∂x i

  i =1

  ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ✉♠❛ ❛♣❧✐❝❛çã♦ c : I ⊂ R −→ M✱ r❡♣r❡s❡♥t❛❞❛

  1 (t), ..., c n (t) i (t)

  ♣❡❧❛s ❢✉♥çõ❡s c ✱ s❡❥❛ ✉♠❛ tr❛❥❡tór✐❛ ❧♦❝❛❧ ❞♦ ❝❛♠♣♦ X é q✉❡ ❛s ❢✉♥çõ❡s x s❛t✐s❢❛çã♠ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s

  dc i

  (t) = α i (c(t))

  

dt ✱ i ∈ {1, ..., n}✳

  ❚❡♠♦s✱ ♣♦r t❡♦r❡♠❛s ❞❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ❧♦❝❛❧ ✭❬✺✷❪✱❬✹✼❪✱❬✺✵❪✱❬✷✻❪✮✱ q✉❡✱ ♣❛r❛ t♦❞♦ a ∈ M✱ ❡①✐st❡ ✉♠❛ tr❛❥❡tór✐❛ ❧♦❝❛❧ ♣❛ss❛♥❞♦ ♣♦r a✳ ❉❡✜♥✐çã♦ ✶✳✷✳✽✳ ❙❡ t♦❞❛ tr❛❥❡tór✐❛ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣♦❞❡ s❡r ❡①t❡♥❞✐❞❛ ❛ ✉♠❛ tr❛❥❡tór✐❛ ❣❧♦❜❛❧ ❡♥tã♦ X é ❞✐t♦ ❝♦♠♣❧❡t♦✳

  ❯♠❛ ❝♦♥s❡qê♥❝✐❛ s✐♠♣❧❡s✱ ♣♦ré♠ ✐♠♣♦rt❛♥t❡✱ ❞❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦

  ❧♦❝❛❧ é q✉❡✱ ❞❛❞❛s ❞✉❛s tr❛❥❡tór✐❛s c −→ M✱ i ∈ {1, 2}✱ ♣❛ss❛♥❞♦ ♣♦r ✉♠ ♣♦♥t♦ a ∈ M

  −1 i (a) (t) = c (t+τ )

  1

  2

  

2

  1

  ❡ τ ∈ c i ✱ i ∈ {1, 2}✱ ❡♥tã♦ c −τ ✱ ♣❛r❛ t♦❞♦ t s✉✜❝✐❡♥t❡♠❡♥t❡ ♣r♦ó①✐♠♦

  1

  ❞❡ τ ✭❬✺✷❪✱❬✹✼❪✱❬✺✵❪✮✳

  ∞ x

  ❉❡✜♥✐çã♦ ✶✳✷✳✾✳ ❙❡❥❛♠ X ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s C s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❡ I ⊂ R ♦ ✐♥t❡r✈❛❧♦ ♠á①✐♠♦ ♣❛r❛ ♦ q✉❛❧ ✉♠❛ tr❛❥❡tór✐❛ ♣❛ss❛♥❞♦ ♣♦r x ∈ M ❡stá ❞❡✜♥✐❞❛ ❡ D = e y

  y

  {(t, y) ∈ R × M; t ∈ I ∈ M}✳ ❖ ✢✉①♦ ❞♦ ❝❛♠♣♦ X é ❛ ❛♣❧✐❝❛çã♦ A : D

  −→ M

  −1

A(t, y) = c(c (y) + t)

  ✱ ♦♥❞❡ c é ✉♠❛ tr❛❥❡tór✐❛ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ y✳

  ✶✸ ❆ ❛♣❧✐❝❛çã♦ ❛❝✐♠❛✱ ❞❡ ❢❛t♦✱ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣♦✐s s❡ s é ♦✉tr❛ tr❛❥❡tór✐❛ ♣❛ss❛♥❞♦

  ♣♦r y ❡♥tã♦✱ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡

  −1 −1

  s(t) = c(t + c (y) (y)) − s ✳

  ▲♦❣♦✱

  

−1 −1 −1 −1 −1

  s(s (y) + t) = c(s (y) + t + c (y) (y)) = c(t + c (y)) − s ✳

  

x

  ❋✐①❛❞♦s ✉♠ ♣♦♥t♦ x ∈ M ❡ t ∈ I t❛✐s q✉❡ A(−t, x) ❡ A(t, x) ❡stã♦ ❞❡✜♥✐❞♦s✱

  s : U

  ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ⊂ M ❞♦ ♣♦♥t♦ x t❛❧ q✉❡ ❛ ❢❛♠í❧✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s {A −→ M ; A s (y) = A(s, y)

  } |s|<|t| ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ✭✈✳ ❬✺✷❪✱❬✹✼❪✱❬✺✵❪✮✳ ❚❛❧ ❢❛♠í❧✐❛ é ❞❡♥♦♠✐♥❛❞❛ ❣r✉♣♦ ❧♦❝❛❧ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▼✱ ♣♦✐s

  A s s = A s , s + s

  1

  2 1 +s 2 ✱ ♣❛r❛ t♦❞♦ s

  1 2 t❛✐s q✉❡ |s

  1

  2

  1

  2

  ◦ A |, |s |, |s | < |t|✳

  s : M s

  ❙❡ ♦ ❝❛♠♣♦ ❢♦r ❝♦♠♣❧❡t♦ ❡♥tã♦ ❛ ❢❛♠í❧✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s {A −→ M} ∈R ❝♦♥st✐t✉✐ ✉♠ ❣r✉♣♦ ❛ ✉♠ ♣❛râ♠❡tr♦ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♠♣♦s✐çã♦✳

  ❉❡❝♦rr❡ ❞♦ t❡♦r❡♠❛ ❞♦ ✢✉①♦ ❧♦❝❛❧ ✭✈✳ ❬✺✷❪✱❬✹✼❪✱❬✺✵❪✮ q✉❡ ❛ ❛♣❧✐❝❛çã♦ A : D −→ M é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ q✉❡✱ ♣❛r❛ t♦❞♦ (t, x) ∈ D✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ x t❛❧ q✉❡ ❛ ❛♣❧✐❝❛çã♦ A t : U t (y) = A(t, y)

  −→ M✱A ✱ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧✳ ❚❛❧ ❛♣❧✐❝❛çã♦ é ❞❡♥♦♠✐♥❛❞❛ ♦♣❡r❛❞♦r ❞❡ tr❛♥s❧❛çã♦ ❧♦❝❛❧ ❛♦ ❧♦♥❣♦ ❞❛s tr❛❥❡tór✐❛s ❞❡ X✳ ❉❡ss❛ ❢♦r♠❛✱ ♥ã♦ é ❞✐❢í❝✐❧

  ∗

  d(A )

  t t = X

  ✈❡r q✉❡ =0 ✭✈✳ ❬✺✷❪✮✳ | dt

  P♦rt❛♥t♦✱ ❞❛❞♦ ✉♠ ❝❛♠♣♦ X✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ❢❛♠í❧✐❛ ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s ❧♦❝❛✐s ❞❡ M✳ ❖ ❝❛♠✐♥❤♦ ❝♦♥trár✐♦ é ♣♦ssí✈❡❧ ♣❡❧♦ s❡❣✉✐♥t❡

  t

  ❚❡♦r❡♠❛ ✶✳✷✳✶✵✳ ❚♦❞❛ ❢❛♠í❧✐❛ ❛ ✉♠ ♣❛râ♠❡tr♦ {A } ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❧♦❝❛✐s ❞❡ ▼ q✉❡ ❞❡♣❡♥❞❡ ❞✐❢❡r❡♥❝✐❛✈❡❧♠❡♥t❡ ❞❡ t ❞❡✜♥❡ ✉♠ ♦♣❡r❛❞♦r ❞❡ tr❛♥s❧❛çã♦ ❛♦ ❧♦♥❣♦ ❞❛s tr❛❥❡tór✐❛s ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❞❡✜♥✐❞♦ ♣❡❧❛ ❢ór♠✉❧❛

  ∗ ∗

  A

  t − A

  X = lim ✳

  t →0

  t ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✺✷❪✳ ❯♠ ❝❛♠♣♦ é✱ ♣♦rt❛♥t♦✱ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ s❡✉ ✢✉①♦ ❡ ❡s❝r❡✈❡r❡♠♦s

  X ∼ = t t {A } ♣❛r❛ ✐♥❞✐❝❛r q✉❡ A é ❛ ❢❛♠í❧✐❛ ❞❡ ♦♣❡r❛❞♦r❡s ❞❡ tr❛♥s❧❛çã♦ ❛♦ ❧♦♥❣♦ ❞❛s tr❛❥❡tór✐❛s ❞❡ X✳ d

  ∗ ∗ ∗ t = A A

  = Pr♦♣♦s✐çã♦ ✶✳✷✳✶✶✳ ❙❡ X ∼ {A }✱ ❡♥tã♦ X ◦ A t t ◦ X = t ✳ dt

  Pr♦✈❛✿ ❙❛❜❡♠♦s q✉❡

  ∗ ∗ ∗ ∗

  A d t t A

  • s − A s − A ∗ ∗ ∗

  A = lim = A lim = A

t t ◦ t ◦ X.

  s s →0 →0

  dt s s

  ✶✹ P♦r ♦✉tr♦ ❧❛❞♦✱

  ∗ ∗ ∗ ∗

  A d t t A

  • s − A s − A ∗

  ∗ ∗

  A = lim = lim = X

  t ◦ A t ◦ A t s s →0 →0

  dt s s ❡ ♦ t❡♦r❡♠❛ s❡❣✉❡✳ ❱❡r❡♠♦s✱ ❛❣♦r❛✱ ❝♦♠♦ s❡ ❝♦♠♣♦rt❛♠ ♦s ❝❛♠♣♦s ❝♦♠ r❡❧❛çã♦ ❛ ❞✐❢❡♦♠♦r✜s♠♦s

  ❛tr❛✈és ❞❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✶✳✷✳✶✷✳ ❙❡❥❛♠ M, N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ F : M −→ N ❞✐❢❡♦♠♦r✜s♠♦ ❡

  X ∈ D(M)✳ ❖ ♣✉s❤✲❢♦✇❛r❞ ❞♦ ❝❛♠♣♦ X ♣❡❧❛ ❛♣❧✐❝❛çã♦ F é ❞❡✜♥✐❞♦ ♣♦r

  −1 ∗

  F (X) = (F )

  ∗ ◦ X ◦ F ∗ ✳

  ➱ ❢á❝✐❧ ✈❡r q✉❡ ♦ ♣✉s❤✲❢♦✇❛r❞ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ♣♦r ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ é t❛♠❜é♠ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✳

  ✶✳✸ ❚❡♥s♦r❡s ✶✳✸✳✶ ■♥tr♦❞✉çã♦

  ◆❡st❛ s❡çã♦ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ♥♦çã♦ ❞❡ t❡♥s♦r❡s ❡ ❛❧❣✉♥s ✐♠♣♦rt❛♥t❡s ♦♣❡r❛❞♦r❡s ❞❡ ❞❡r✐✈❛çã♦ s♦❜r❡ ❡st❡s✳ ❆ ♥♦çã♦ ❞❡ t❡♥s♦r❡s s❡rá ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ tr❛❜❛❧❤❛r♠♦s ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣✲❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s ❡ ❛♣❧✐❝á✲❧❛s ♣❛r❛ ❡st✉❞♦s ❣❧♦❜❛✐s ♣♦st❡r✐♦r❡s✳

  ∗

  = Hom ( ❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❙❡❥❛♠ A ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ✉♥✐tár✐♦✱ S ✉♠ A✲♠ó❞✉❧♦ ❡ S A S, A) = {ψ : S −→ A; ψ é ❤♦♠♦♠♦r✜s♠♦ A✲❧✐♥❡❛r}✳ ❯♠ t❡♥s♦r ❞♦ t✐♣♦ (p, q) s♦❜r❡ ♦ ♠ó❞✉❧♦ S é ✉♠❛ ❛♣❧✐❝❛çã♦ A✲♠✉❧t✐❧✐♥❡❛r

  ∗ ∗

  T : S × ... × S × S × ... × S −→ A✳ | {z } | {z }

  p vezes q vezes

  ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s t❡♥s♦r❡s ❞♦ t✐♣♦ ✭♣✱q✮ s♦❜r❡ ♦ ♠ó❞✉❧♦ S é ❞❡♥♦t❛❞♦

  q

  ( ♣♦r T p

  S) ❡ ❝♦♥st✐t✉✐ ✉♠ A✲♠ó❞✉❧♦ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ♥❛t✉r❛✐s ❞❡ s♦♠❛ ❡ ♣r♦❞✉t♦ ♣♦r ❡❧❡♠❡♥t♦s ❞❡ A ❞❡✜♥✐❞❛s ❛❜❛✐①♦✿

  q q q

  • : T ( ( (

  p S) × T p S) −→ T p S)

  (T + S)(s) := T (s) + S(s)

  

q q

  ( (

  

p p

  × : A × T S) −→ T S) (α

  × T )(s) := αT (s)

  

1

  ( ( ❈♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✱ t❡♠♦s q✉❡ T S) = S ❡ T

  1 S) = S s❡ ❝♦♥s✐❞❡r❛r♠♦s ∗∗

  = =

  ✶✺

  X

  q

  T ( ❉❡✜♥✐çã♦ ✶✳✸✳✷✳ ❙❡❥❛ T (S) ❂ p S)✱ ❞❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧

  p,q ≥0

  ⊗ : T (S) × T (S) −→ T (S)

  q s ∗ ∗ 1 ( 2 (

  1 2 :

  t❛❧ q✉❡✱ s❡ f ∈ T p S) ❡ f ∈ T r S)✱ f ⊗ f S × ... × S × S × ..., ×S é ❞❛❞❛ ♣♦r | {z }

  | {z }

  

p

  • r vezes q
  • l vezes ∗ ∗ ∗ ∗ ∗ ∗

  f (s , ..., s , s , ..., s q ) = f (s , ..., s , s , ..., s q ).f (s , ..., s , s q , ..., s q )

  1

  2 1 +l

  1

  

1

2 +1 +l

  ⊗ f

  1 p +r 1 p p +1 p +r ✳ q

  • l

  ( ▲♦❣♦✱ f

  1 2 p +r

  ⊗ f ∈ T S)✳

  ∗

  , ..., e

  1 n i

  ❙❡ ♦ ♠ó❞✉❧♦ S ♣♦ss✉✐ ✉♠❛ ❜❛s❡ ✜♥✐t❛ {e }✱ ❡♥tã♦ ♦s ❡❧❡♠❡♥t♦s ǫ ∈ S

  q

  X

  k ∗ i (e j ) = δ ij j = s e k

  ❞❡✜♥✐❞♦s ♣♦r ǫ ❝♦♥st✐t✉❡♠ ✉♠❛ ❜❛s❡ ♣❛r❛ S ✳ ▲♦❣♦✱ ❞❛❞♦s s j ✱

  k =1 p

  X ∗

  r q ∗

  s = s ǫ r (

  i i ❡ f ∈ T p

  S)✱ ♦❜t❡♠✲s❡ ♣❡❧❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❡ f q✉❡

  r =1 ∗ ∗

  X

  i i p j 1 j q

  1 ∗ ∗

  f (s , ..., s , s , ..., s q ) = f (ǫ i , ..., ǫ i , e j , ..., e j )s . ... .s p .s . ... .s

  1 p q 1 p

  1

  1

  1 1 q ✱ i ,...,i p ,j ,...,j q

  1

  1

  ♦✉ s❡❥❛✱ f é ú♥✐❝❛♠❡♥t❡ ❞❡✜♥✐❞♦ ♣❡❧♦s ✈❛❧♦r❡s

  j ,...,j q

  1

  f := f (ǫ i , ..., ǫ i , e j , ..., e j )

  p q i ,...,i p

  1

  1

  1

  1

  }✳ ❉❡ss❡ ♠♦❞♦✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r f ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  , ..., e n q✉❡ sã♦ ❝❤❛♠❛❞♦s ❝♦♠♣♦♥❡♥t❡s ❞♦ t❡♥s♦r f ❝♦♠ r❡s♣❡✐t♦ à ❜❛s❡ {e

  X

  j ,...,j 1 q

  f = f e i e i j j ,

  i ,...,i c p q p 1 ⊗ ... ⊗ c ⊗ ǫ 1 ⊗ ... ⊗ ǫ ✭✶✳✸✮

  1 i ,...,i ,j ,...,j p q

  1

  1

p

  X ∗ ∗

  r j ∗ ∗ ∗

  e j : e j (s e j ( s ǫ r ) = s (e j ) = s e i ♦♥❞❡ b S −→ A é ❞❡✜♥✐❞♦ ♣♦r b ) = b ✳ ▲♦❣♦✱ {c

  1 ⊗ ... ⊗ r

=1

q

  e i j j i ,...,i ,j ,...,j (

  p q p q p

  c ⊗ ǫ

  1 ⊗ ... ⊗ ǫ }

  1

1 ∈{1,...,n} ❝♦♥st✐t✉✐ ✉♠❛ ❜❛s❡ ♣❛r❛ T S)✳

  ❱❡r❡♠♦s ❛❣♦r❛ ❝♦♠♦ ❛♣❧✐❝❛r ❛ ♥♦çã♦ ❞❡ t❡♥s♦r❡s ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ p✲❢♦r♠❛s ❞✐❢❡r✲ ❡♥❝✐❛✐s✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ❛ s❡❣✉✐♥t❡

  p

  ( ❉❡✜♥✐çã♦ ✶✳✸✳✸✳ ❙❡❥❛♠ f ∈ T S) ❡ ∆(p) ♦ ❣r✉♣♦ ❞❛s ♣❡r♠✉t❛çõ❡s ❞❡ {1, 2, ..., p}✳ ❢ é ❞✐t❛ ♣✲❢♦r♠❛ s✐♠étr✐❝❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛♥t✐s✐♠étr✐❝❛✮ s❡✱ ♣❛r❛ t♦❞♦ δ ∈ ∆(p)✱

  δ δ δ

  f (s , ..., f p ) = f (s δ , ..., s δ ) = f (s , ..., s p ) (s , ..., s p ) = ( f (s , ..., f p )

  1 (1) (p)

  1

  1

  1

  ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ f −1) ✮✱

  δ δ

  = 1 = ♦♥❞❡ (−1) s❡ δ é ✉♠❛ ♣❡r♠✉t❛çã♦ ♣❛r ❡ (−1) −1 s❡ δ é ✉♠❛ ♣❡r♠✉t❛çã♦ í♠♣❛r✳

  ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣✲❢♦r♠❛s ❛♥t✐s✐♠étr✐❝❛s s♦❜r❡ S ♣♦ss✉✐ ✉♠❛

  

p

  ( ( ❡str✉t✉r❛ ❞❡ A✲♠ó❞✉❧♦ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r Λ S)✱ s❡ p > 0✱ ❡ Λ S) = A s❡ p = 0✳

  X

  p

  T ( ❚❛♠❜é♠ ❞❡✜♥✐♠♦s Λ(S) =

  S)✳

  p ≥0

  ❉❛❞♦ ✉♠ t❡♥s♦r f ∈ T (S) q✉❛❧q✉❡r✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ♦♣❡r❛çã♦ q✉❡ ♦ tr❛♥s✲

  ✶✻ alt : T ( S) −→ Λ(S)

  X

  δ δ

  alt(f ) := ( f ✳

  −1)

  δ

∈∆(p)

  ◆✉♠ ❝♦♥t❡①t❡ ♠❛✐s ❣❡r❛❧ ❞❡ t❡♥s♦r❡s✱ ❢♦✐ ♣♦ssí✈❡❧ ❞❡✜♥✐r ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❡ ❢♦✐ ✈✐st♦ q✉❡ T (S) ❡r❛ ❢❡❝❤❛❞♦ s♦❜r❡ ❡st❛ ♦♣❡r❛çã♦✳ P♦ré♠✱ ♦ ♠❡s♠♦ ♥ã♦ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ♥♦s r❡str✐♥❣✐♠♦s ❛ Λ(S) ♣♦✐s ♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ t❡♥s♦r❡s ❛♥t✐s✐♠étr✐❝♦s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ❛♥t✐s✐♠étr✐❝♦✳ ◆❡st❡ ❝❛s♦✱ ❞❡✜♥✐✲s❡ ✉♠ ♥♦✈♦ ♣r♦❞✉t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

  p q 1 (

2 (

  ❉❡✜♥✐çã♦ ✶✳✸✳✹✳ ❙❡❥❛♠ f ∈ Λ S) ❡ f ∈ Λ S)✳ ❉❡✜♥✐♠♦s ♦ ♣r♦❞✉t♦ ❡①t❡r✐♦r ∧ : Λ(S) × Λ(S) −→ Λ(S)

  1

  (f , f ) := alt(f )

  1

  2

  1

  

2

  1

  2

  7−→ f ∧ f p ⊗ f ✳

  !q!

  ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ q✉❡ ❢♦✐ ❢❡✐t♦ ♣❛r❛ ♦ ❝❛s♦ ❣❡r❛❧ ❞❡ t❡♥s♦r❡s✱ ♦❜t❡♠✲s❡ q✉❡

  p p i i

  ( (

  p i ,...,i p

  {ǫ

  1 ∧...∧ǫ } 1 ∈{1,2,...,n} ❝♦♥st✐t✉✐ ✉♠❛ ❜❛s❡ ♣❛r❛ Λ S)✳ ▲♦❣♦✱ t♦❞♦ ❡❧❡♠❡♥t♦ ω ∈ Λ S)

  X ω(e i , ..., e i )ǫ i i

  p p

  é ❡s❝r✐t♦ ❞❛ ❢♦r♠❛

  1 1 ∧ ... ∧ ǫ ✳ P♦rt❛♥t♦✱ s❡ ❛ ❞✐♠❡♥sã♦ ❞♦ A✲ i ,...,i

  1 p ∈{1,...,n}

p n

  ( ♠ó❞✉❧♦ S ❢♦r n✱ ❡♥tã♦ ❛ ❞✐♠❡♥sã♦ ❞❡ Λ S) é ✭ p ✮✳

  ✶✳✸✳✷ ❋♦r♠❛s ❉✐❡❢❡r❡♥❝✐❛✐s ❡ ❋✐❜r❛❞♦ ❈♦t❛♥❣❡♥t❡

  ❆✜♠ ❞❡ ❛♣❧✐❝❛r ❛ t❡♦r✐❛ ❞❡ ❚❡♥s♦r❡s✱ ❝♦♥s✐❞❡r❛r❡♠♦s ❞♦✐s ❝❛s♦s✿ ♣r✐♠❡✐r♦✱ ♦ ❛♥❡❧

  

p M

  t❛♥❣❡♥t❡ ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ♥✉♠ ♣♦♥t♦ A s❡rá R ❡ ♦ A✲♠ó❞✉❧♦ S s❡rá ♦ ❡s♣❛ç♦ T

  ∞

  p (M ) ∈ M❀ s❡❣✉♥❞♦✱ ♦ ❛♥❡❧ A s❡rá C ❡ ♦ A✲♠ó❞✉❧♦ S s❡rá D(M)✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ♦s

  q

  (T p M ) ❡❧❡♠❡♥t♦s ❞❡ Λ s❡rã♦ ❞❡♥♦♠✐♥❛❞♦s q✲❝♦✈❡t♦r❡s ✭♦✉ ❝♦✈❡t♦r ❞❡ ❣r❛✉ q✮ ♥♦ ♣♦♥t♦

  q

  p ( ✳ ◆♦ s❡❣✉♥❞♦✱ ♦s ❡❧❡♠❡♥t♦s ❞❡ Λ D(M)) s❡rã♦ ❞❡♥♦♠✐♥❛❞♦s q✲❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s

  1

  (T p M ) ✭♦✉ ❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❣r❛✉ q✮✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ q ❂ ✶✱ Λ é ❞❡♥♦♠✐♥❛❞♦

  ∗

  M ❡s♣❛ç♦ ❝♦t❛♥❣❡♥t❡ ❛ M ❡♠ p ❡ é s❡rá ❞❡♥♦t❛❞♦ ♣♦r T p ✳

  ∞

  (M ) ❉❡✜♥✐çã♦ ✶✳✸✳✺✳ ❙❡❥❛♠ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ f ∈ C ✳ ❆ ✶✲❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧ df s♦❜r❡ M ❞❡✜♥✐❞❛ ♣♦r

  ∞

  df : (M ) D(M) −→ C df (X) = X(f )

  é ❝❤❛♠❛❞❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❛ ❢✉♥çã♦ ❢✳ ❯♠❛ r❡str✐çã♦ ♥♦ ❞♦♠í♥✐♦ ❞❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ❣❡r❛ ❛ s❡❣✉✐♥t❡

  ∞

  (M ) ❉❡✜♥✐çã♦ ✶✳✸✳✻✳ ❙❡❥❛♠ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ f ∈ C ✳ ❖ ❞✐❢❡r❡♥❝✐❛❧ ✈❡r✲ t✐❝❛❧ ❞❡ f é ❞❡✜♥✐❞♦ ♣♦r

  V

  1

  d f : V (M ) (M ) −→ Λ

  ξ

  ✶✼ (η) = 0

  ♦♥❞❡ V (M) = {η ∈ T M : π ∗ }✳ ❉❡✜♥✐r❡♠♦s✱ ❛❣♦r❛✱ ♦ ♣✉❧❧✲❜❛❝❦ ❞❡ ✉♠❛ p✲❢♦r♠❛✿

  ❉❡✜♥✐çã♦ ✶✳✸✳✼✳ ❙❡❥❛♠ M, N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ F : M −→ N ❞✐❢❡r❡♥❝✐á✈❡❧✱

  p

  ω ( , ..., X p

  1

  ∈ T D(N))❡ X ∈ D(M)✳ ❖ ♣✉❧❧❜❛❝❦ ❞❡ ω ♣❡❧❛ ❛♣❧✐❝❛çã♦ F é ❞❡✜♥✐❞♦ ♣♦r

  ∗ ∗

  F (ω)(X , ..., X p )(a) = ω F ((F ) a (X ), ..., (F ) a (X p ))

  1 (a) 1 a a ∗ ✱ ∀a ∈ M✳ p

  ❯♠ q✲❝♦✈❡t♦r ω ♥✉♠ ♣♦♥t♦ p ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ✉♠❛ ❛♣❧✐❝❛çã♦ ω p : T p M p M

  × ... × T −→ R✳ ◗✉❡r❡♠♦s✱ ❛❣♦r❛✱ ❞❡s❝r❡✈❡r ❧♦❝❛❧♠❡♥t❡ ✉♠ q✲❝♦✈❡t♦r✳ P❛r❛ | {z }

  q vezes U

  , ..., x n

  1

  ✐ss♦✱ s✉♣♦♥❤❛ dim(M) = n✱ p ∈ M✱ ϕ ❝❛rt❛ ❝♦♦r❞❡♥❛❞❛ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛✱ x

  U

  ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❞❛ ❝❛rt❛ ϕ ✳ ❯s❛♥❞♦ ✶✳✸✱ ♦❜t❡♠♦s

  X ∂ ∂

  ω p = ω p p , ..., p dx i i q | |

  1 ∧ ... ∧ dx ✱

  ∂x i ∂x i q

  1 i ,...,i q 1 ∈{1,...,n}

  ∂

  i = δ kl

  ♦♥❞❡ dx k ✳ ∂x i

  l p

  ∗

  M = ❙❡❥❛ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ n✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦♥❥✉♥t♦ T

  [

  ∗ ∗

  T M M p ) = p

  p

  ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ eπ : T −→ M ♣♦r eπ(ω ✳ ❆♥❛❧♦❣❛♠❡♥t❡ ❛♦ ✜❜r❛❞♦

  p ∈M ∗

  M t❛♥❣❡♥t❡✱ ♦ ❝♦♥❥✉♥t♦ T ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ♥❛t✉r❛❧ ❞❡ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❛❞❛

  V

  ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ s❡❥❛ A ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á❧✈❡❧ ♣❛r❛ ❛ ✈❛r✐❡❞❛❞❡ M✳ P❛r❛ t♦❞❛ ϕ ∈ A , ..., x n

  V

  ❝♦♠ x

  

1 ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ϕ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛s ❛♣❧✐❝❛çõ❡s

2n −1 ∗

  e ψ (V ) = e

  V M

  

V e : eπ ⊂ T −→ R

  ∂ ∂ e ψ (ω ) = x (p), ..., x (p), ω , ..., ω

  

V e | | ✳

p 1 n p p

  ∂x

  1 ∂x n

  ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ ❡st❛s ❛♣❧✐❝❛çõ❡s sã♦ ✐♥❥❡t♦r❛s ❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ t♦♣♦❧♦❣✐❛

  2n ∗

  M ❡♠ T ♦♥❞❡ ♦s ❛❜❡rt♦s sã♦ ❛s ❝♦♥tr❛✐♠❛❣❡♥s ❞❡ ❛❜❡rt♦s ❞♦ R ✳ ❈♦♠ ❡st❛ t♦♣♦❧♦❣✐❛

  ∗

  M ψ ❡♠ T ❛s ❛♣❧✐❝❛çõ❡s e

  V e sã♦ ♥❛t✉r❛❧♠❡♥t❡ ❤♦♠❡♦♠♦r✜s♠♦s✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠ ❡st❛ ∗

  M t♦♣♦❧♦❣✐❛✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ T é ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉ss❞♦r❢ ❡ ♣❛r❛❝♦♠♣❛❝t♦ ❡ t❛♠❜é♠

  2n ∗

  ψ : e

  V e ⊂ T −→ e V e ⊂ R } ❞❡✜♥❡ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ −1 ∞

  V M ψ ( e V ) q✉❡ e A = { e

  ∞

  C ψ ψ ψ , e ψ

  ♣♦✐s e

  V e ◦ e sã♦ ❞✐❢❡♦♠♦r✜s♠♦s ❞❡ ❝❧❛ss❡ C ♣❛r❛ t♦❞❛s e V e U e ∈ e A✳ U e ∗

  M ❖ ❝♦♥❥✉♥t♦ T ♠✉♥✐❞♦ ❝♦♠ ❡st❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ é ❞❡♥♦♠✐♥❛❞♦ ✜❜r❛❞♦

  ∗ −1

  (p) ❝♦t❛♥❣❡♥t❡ ❡ ♦s ❡s♣❛ç♦s ❝♦t❛♥❣❡♥t❡s ❛ M ❡♠ p T p M = eπ sã♦ ❛s ✜❜r❛s s♦❜r❡ ❝❛❞❛

  −1

  (p) ♣♦♥t♦ p✳ ❈♦♠♦ eπ t❡♠ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♣❛r❛ t♦❞♦ p ∈ M✱ ♦ ✜❜r❛❞♦

  ∗

  M ❝♦t❛♥❣❡♥t❡ T t❡♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ✜❜r❛❞♦ ✈❡t♦r✐❛❧✳ ❆❧é♠ ❞✐ss♦✱ é ❢á❝✐❧ ✈❡r q✉❡ ❛

  ∗

  M ❛♣❧✐❝❛çã♦ eπ : T

  −→ M é ✉♠❛ s✉❜♠❡rsã♦✳

  ✶✽

  p p a : Λ ( (T a (M ))

  ❖❜s❡r✈❛çã♦ ✶✳✸✳✽✳ P♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❛♣❧✐❝❛çã♦ Θ D(M)) −→ Λ ♣♦r Θ a (ω) = ω a t❛❧ q✉❡

  ω a (X a , ..., X a p ) = ω(X

  1 , ..., X p )(a) 1 ✱ a , ..., X a

  , ..., X n

  p

  1

  t❛❧ q✉❡ X

  1 sã♦ ❞❡✜♥✐❞♦s ♣❡❧♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s X r❡s♣❡❝t✐✈❛♠❡♥t❡✳

  , ..., X p a

  1

  ❈♦♠♦ ❡st❛ ❞❡✜♥✐çã♦ é ♣♦♥t✉❛❧✱ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ X ✳ ▲♦❣♦✱ Θ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳

  p

  ( P♦rt❛♥t♦✱ ❞❛❞❛ ✉♠❛ p✲❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧ ω ∈ Λ D(M))✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❡♠ ❝❛❞❛

  a

  ♣♦♥t♦ a ∈ M ✉♠ p✲❝♦✈❡t♦r ω ✳ ❖ ❝♦♥trár✐♦ t❛♠❜é♠ é ♣♦ssí✈❡❧✳ ❉✐❢❡r❡♥❝✐❛❧ ❡①t❡r♥♦

  ❖ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢♦r♠❛ ❞✐❢❡r❡♥❝✐á✈❡❧ é ❞❡✜♥✐❞♦ ❞❡ ♠❛♥❡✐r❛ ❛①✐♦♠át✐❝❛ ♣❡❧♦ s❡❣✉✐♥t❡ ❚❡♦r❡♠❛ ✶✳✸✳✾✳ ❙❡❥❛ ▼ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❊①✐st❡ ✉♠❛✱ ❡ s♦♠❡♥t❡ ✉♠❛✱ ❛♣❧✐❝❛çã♦ d M : Λ(

  D(M)) −→ Λ(D(M)) t❛❧ q✉❡✿

  M (ω + ω ) = d M (ω ) + d M (ω ) , ω

  1

  2

  1

  2

  1

  

2

  ✶✳ d ✱ ω ∈ Λ(D(M))❀

  gr (ω

1 )

  M (ω ) = d M (ω ) + ( ω M (ω ) )

  1

  2

  1

  2

  1

  2

  1

  1

  ✷✳ d ∧ ω ∧ ω −1) ∧ d ✱ ♦♥❞❡ gr(ω é ♦ ❣r❛✉ ❞❡ ω ❀

  M M = 0

  ✸✳ d ◦ d ❀ ( M (f )

  ✹✳ s❡ f ∈ Λ D)✱ ❡♥tã♦ d é ♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❛ ❢✉♥çã♦ ❢ ❞❡✜♥✐❞♦ ❡♠ ✶✳✸✳✽✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✺✵❪ ❡ ❬✺✷❪✳

  M

  ◗✉❛♥❞♦ ♦ ❞♦♠í♥✐♦ ❞♦ ♦♣❡r❛❞♦r d ❡st✐✈❡r ❝❧❛r♦✱ ♦ ❞❡♥♦t❛r❡♠♦s ❛♣❡♥❛s ♣♦r d✳ ❉❡✜♥✐çã♦ ✶✳✸✳✶✵✳ ❖ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ♥♦ t❡♦r❡♠❛ ❛❝✐♠❛ é ❝❤❛♠❛❞♦ ❞✐❢❡r❡♥❝✐❛❧ ❡①t❡✲ r✐♦r✳

  ❖ t❡♦r❡♠❛ ❛❜❛✐①♦ ❡①✐❜❡ ✉♠❛ ❡①♣r❡ssã♦ ❣❧♦❜❛❧ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❡①♣❧í❝✐t♦ ❞♦ ❞✐❢❡r✲ ❡♥❝✐❛❧ ❡①t❡r✐♦r✳

  p

  ( , ..., X p

  1 +1

  ❚❡♦r❡♠❛ ✶✳✸✳✶✶✳ ❙❡ ω ∈ Λ D(M))✱ p ≥ 1✱ X ∈ D(m)✱ ❡♥tã♦ ✈❛❧❡✿ dω(X

  1 , ..., X p +1 ) = p +1

  X X

  i i

  • 1 +j

  ( X i (ω(X , ..., c X i , ..., X p )) + ( ω([X i , X j ], X , ..., c X i , ..., c X j , ..., X p )

  1 +1 1 +1 ✱

  −1) −1)

  i i<j =1

  ♦♥❞❡ ♦ sí♠❜♦❧♦ b. ✐♥❞✐❝❛ ❛ ❡①❝❧✉sã♦ ❞♦ ❛r❣✉♠❡♥t♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳

  ✶✾ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✺✷❪✳

  ∞

  (M ) ❖❜s❡r✈❡ q✉❡✱ ♥♦ t❡♦r❡♠❛ ❛❝✐♠❛✱ s❡ p = 0 ❡ ω = f ∈ C ✱ ❛ ❢ór♠✉❧❛ s❡ r❡❞✉③

  à ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦✳ ❖ ❞✐❢❡r❡♥❝✐❛❧ ❡①t❡r✐♦r s❛t✐s❢❛③ ✉♠❛ ✐♠♣♦rt❛♥t❡ ♣r♦♣r✐❡❞❛❞❡✱ ❞❡♥♦♠✐♥❛❞❛ ♥❛t✉✲ r❛❧✐❞❛❞❡✱ ❞❛❞❛ ♣❡❧❛ s❡❣✉✐♥t❡

  Pr♦♣♦s✐çã♦ ✶✳✸✳✶✷✳ ❉❛❞❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ F : M −→ N ❡♥tr❡ ❛s ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ▼ ❡ ◆✳ ❖ ❞✐❢❡r❡♥❝✐❛❧ ❡①t❡r✐♦r d s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡

  

∗ ∗

  d M = F N ◦ F ◦ d ✳

  ❆ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✺✷❪✳

  p

  ( ❉❡✜♥✐çã♦ ✶✳✸✳✶✸✳ ❙❡❥❛♠ ▼ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ω ∈ Λ

  D(M))✳ ω é ❞✐t❛ ❢❡❝❤❛❞❛

  p −1

  ( s❡ dω = 0 ❡ é ❞✐t❛ ❡①❛t❛ s❡ ❡①✐st❡ ✉♠❛ ❢♦r♠❛ η ∈ Λ D(M)) t❛❧ q✉❡ dη = ω✳ ❙❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡ t♦❞❛ ❢♦r♠❛ ❡①❛t❛ é ❢❡❝❤❛❞❛✳

  ❉❡✜♥✐çã♦ ✶✳✸✳✶✹✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ▼ é ❞✐t❛ ♦r✐❡♥tá✈❡❧ s❡ ❡①✐st❡ ✉♠❛ ❢♦r♠❛

  n

  ω ( a ∈ Λ D(M))✱ ♥ ❂ ❞✐♠▼✱ t❛❧ q✉❡ ω 6= 0, ∀a ∈ M✳ ❚❛❧ ❢♦r♠❛ s❡ ❝❤❛♠❛ ❢♦r♠❛ ❞❡

  ✈♦❧✉♠❡ s♦❜r❡ ▼✳

  n ′

  ( ❈♦♠♦ dim (Λ D(M))) = 1✱ s❡ ω é ✉♠❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ s♦❜r❡ M ❡ ω ∈

  n ′ ∞ ′

  Λ ( = f ω (M ) D(M))✱ ❡♥tã♦ ω ✱ ♦♥❞❡ f ∈ C ✳ ❙❡ ω t❛♠❜é♠ ❢♦r ✉♠❛ ❢♦r♠❛ ❞❡ ✈♦❧✲

  ✉♠❡ s♦❜r❡ M✱ ❡♥tã♦ f(a) 6= 0, ∀a ∈ M✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♥❡①❛

  ′

  ❡♥tã♦ só ❡①✐st❡♠ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ f > 0 ♦✉ f < 0✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ω ❡ ω sã♦ ❞✐t❛s

  ′

  ❝♦♠♣❛tí✈❡✐s✳ ◆♦ s❡❣✉♥❞♦✱ ω ❡ ω sã♦ ❞✐t❛s ✐♥❝♦♠♣❛tí✈❡✐s✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ ❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❡✜♥❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡ ✐ss♦ ♥♦s ♣♦ss✐❜✐❧✐t❛ ✐♥tr♦❞✉③✐r ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✶✳✸✳✶✺✳ ❯♠❛ ❝❧❛ss❡ ❞❡ ❢♦r♠❛s ❞❡ ✈♦❧✉♠❡ ❝♦♠♣❛tí✈❡✐s é ❞✐t❛ ❖r✐❡♥t❛çã♦ ♣❛r❛ ▼✳

  ✶✳✸✳✸ ❉❡r✐✈❛❞❛ ❞❡ ▲✐❡ t

  = ❉❡✜♥✐çã♦ ✶✳✸✳✶✻✳ ❙❡❥❛ X ∈ D(M)✱ X ∼ {A }✱ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡

  ∞

  (M ) ❞✐❢❡r❡♥❝✐á✈❡❧ ▼✳ ❆ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡ ❞❡ ✉♠❛ ❢✉♥çã♦ f ∈ C ♥❛ ❞✐r❡çã♦ ❞♦ ❝❛♠♣♦ X é ❞❡✜♥✐❞❛ ♣♦r

  (A t ) (f ) ) (f )

  ∗ − (A ∗

X (f ) := lim = X(f )

  L ✱

  t →0

  t

  ∞

  (M ) ♣❛r❛ t♦❞❛ f ∈ C ✳

  ✷✵ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡✜♥✐♠♦s ❛s ❞❡r✐✈❛❞❛s ❞❡ ▲✐❡ ♣❛r❛ ❝❛♠♣♦s ❡ t❡♥s♦r❡s✳ ❱❡❥❛♠♦s

  ♣r✐♠❡✐r♦ ❛ s❡❣✉✐♥t❡ = t

  ❉❡✜♥✐çã♦ ✶✳✸✳✶✼✳ ❙❡❥❛♠ X ∼ {A } ❡ Y ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r✲

  X Y

  ❡♥❝✐á✈❡❧ M✳ ❆ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡ L ❞♦ ❝❛♠♣♦ Y ❝♦♠ r❡s♣❡✐t♦ ❛♦ ❝❛♠♣♦ X é ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s

  Y A t (Y p )

  t (p) ∗

  − A (

  X Y ) := lim

  L p

  t →0

  t

  t ) (Y p ) p

  ✳ ♦♥❞❡ (A ∗ ✐♥❞✐❝❛ ♦ ♣✉s❤✲❢♦✇❛r❞ ❞♦ ✈❡t♦r Y ✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r ✭✈✳ ❬✺✷❪✮ q✉❡

  

X (Y ) = [X, Y ]

  L ✳ ❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ❝♦♠ ✉♠❛ ✐♠♣♦rt❛♥t❡ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛✱ é q✉❛♥❞♦

  X (Y ) = [Y, X] = 0

  L ✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ X(Y ) = Y (X)✳ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ♠♦str❛ q✉❡✱ ♥❡st❡ ❝❛s♦✱ ♦s ♦♣❡r❛❞♦r❡s ❞❡ tr❛♥s❧❛çã♦ ❛♦ ❧♦♥❣♦ ❞❛s tr❛❥❡tór✐❛s ❞❡ X ❝♦♠✉t❛♠ ❝♦♠ ♦s ♦♣❡r❛❞♦r❡s ❞❡ tr❛♥s❧❛çã♦ ❛♦ ❧♦♥❣♦ ❞❛s tr❛❥❡tór✐❛s ❞❡ Y ✳ ❚❡♦r❡♠❛ ✶✳✸✳✶✽✳ ❙❡❥❛♠ X ❡ Y ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M ❡

  t s

  {A } ❡ {B } ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❣r✉♣♦s ❧♦❝❛✐s ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡ X ❡ Y r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦

  [X, Y ] = 0 t s = B s t s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A ◦ B ◦ A ✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✺✷❪✱ ❬✺✵❪✳ ❱❡❥❛♠♦s ❛❣♦r❛ ❝♦♠♦ ❞❡✜♥✐r ❛ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡ t❡♥s♦r✐❛❧✳

  p

  ( t ❉❡✜♥✐çã♦ ✶✳✸✳✶✾✳ ❙❡❥❛♠ ▼ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ω ∈ T D(M))✱ {A } ❣r✉♣♦ ❧♦❝❛❧ ❛ ✉♠ ♣❛râ♠❡tr♦ ❞♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ∈ D(M)✳ ❆ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡ L(ω) ❞♦ t❡♥s♦r ω ❝♦♠ r❡s♣❡✐t♦ ❛♦ ❝❛♠♣♦ ❳ é ❞❡✜♥✐❞❛ ♣♦r

  ∗ ∗

  A (ω) (ω)

  

t

  − A L(ω) := lim ✳

  t →0

  t ➱ ♣♦ssí✈❡❧ ♠♦str❛r ✭✈✳ ❬✺✷❪✮ q✉❡ ✈❛❧❡ ♦ s❡❣✉✐♥t❡✿

  p

  X X (ω)(X , ..., X p ) = X(ω(X , ..., X p )) + ω(X , ..., [X i , X], ..., X p )

  1

  1

  ✭✶✳✹✮

  1 L

  i =1 p

  ( ❉❡✜♥✐çã♦ ✶✳✸✳✷✵✳ ❙❡❥❛♠ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ X ∈ D(M)✱ ω ∈ T D(M))✳ ❉❡✜♥✐✲

  p p −1 X : T ( (

  ♠♦s ♦ ♦♣❡r❛❞♦r ❞❡ ❝♦♥tr❛çã♦ ♣♦r ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X i D(M)) −→ T D(M))

  ♣♦r

  ✷✶ i

  

X (ω)(X , ..., X p ) := ω(X, X , ..., X p )

  1

  1 −1 −1 ✳

  ❈♦♠ ❡st❛ ❞❡✜♥✐çã♦✱ é ♣♦ssí✈❡❧ ♠♦str❛r ✉♠❛ ✐♠♣♦rt❛♥t❡ ❢ór♠✉❧❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡ ♣❛r❛ t❡♥s♦r❡s✱ ❛ s❛❜❡r

  X (ω) = d x (ω) + i

  X L ◦ i ◦ d(ω)✳

  ❊st❛ ❢ór♠✉❧❛ é ❞❡♥♦♠✐♥❛❞❛ ❢ór♠✉❧❛ ❞❡ ❈❛rt❛♥ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡ t❡♥s♦r✐❛❧ ❡ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞❛ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✶✳✸✳✶✸ ❡ ❛ ❢ór♠✉❧❛ ❛❧❣é❜r✐❝❛ ✶✳✹ ♣❛r❛ Lω✳

  ❈❛♣ít✉❧♦ ✷ ❙✐♠❡tr✐❛s ❡ ❋♦r♠❛❧✐s♠♦ ❱❛r✐❛❝✐♦♥❛❧

  ❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ ❝❛♣ít✉❧♦ é ❞❛r ✉♠❛ ❞❡s❝r✐çã♦ ✈❛r✐❛❝✐♦♥❛❧ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞❛s ❣❡♦❞és✐❝❛s ❡ ♠♦str❛r q✉❛✐s r❡❧❛çõ❡s ❡①✐st❡♠ ❡♥tr❡ ❛s s✐♠❡tr✐❛s ❡ ❛s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❡st❛s ❡q✉❛çõ❡s✳ P❛r❛ ❡ss❡ ✜♠✱ ✉s❛r❡♠♦s ♦ ❢♦r♠❛❧✐s♠♦ ✐♥✈❛r✐❛♥t❡ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥ q✉❡ ♣♦ss✐❜✐❧✐t❛rá ✉♠❛ ♠❡❧❤♦r ❞❡s❝r✐çã♦ ❞❛s s✐♠❡tr✐❛s ❡ ❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ s✐♠❡✲ tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s ❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r✳ ❆s ❞✉❛s ♣r✐♠❡✐r❛s s❡çõ❡s ❛❜♦r❞❛♠ ✉♠ ❝♦♥t❡ú❞♦ ♣r❡❧✐♠✐♥❛r à ♣❛rt❡ ❝❡♥tr❛❧ ❞❡st❡ ❝❛♣ít✉❧♦ ❜❡♠ ❝♦♠♦ ♣❛r❛ ❝♦♥t❡ú❞♦s ♣♦st❡r✐♦r❡s✳ ❉❡ ❢❛t♦✱ ♦ ❝❛♣ít✉❧♦ ❝♦♠❡ç❛ ❝♦♠ ❛ ✐♥tr♦❞✉çã♦ ❞❛s ♥♦çõ❡s ❞❡ ❞✐s✲ tr✐❜✉✐çã♦ ❡ s✐♠❡tr✐❛ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦✳ ❆ s❡❣✉♥❞❛ s❡çã♦ é ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦s ❡s♣❛ç♦s ❞❡ ❥❛t♦s q✉❡ r❡♣r❡s❡♥t❛♠ ♦ ❛♠❜✐❡♥t❡ ♥❛t✉r❛❧ ♥♦ q✉❛❧ ❛s ❡q✉❛çõ❡s ❞❛s ❣❡♦❞és✐❝❛s ♣♦❞❡♠ s❡r tr❛t❛❞❛s ❝♦♠♦ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❆ss✐♠✱ ❛s s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ♣♦❞❡♠ s❡r ❞❡s❝r✐t❛s ❝♦♠♦ ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥ ✐♥❞✉③✉❞❛ s♦❜r❡ ❡st❛ ✈❛r✲ ✐❡❞❛❞❡✳ ❆s s✐♠❡tr✐❛s ❞❡ss❛ ❞✐str✐❜✉✐çã♦ sã♦ ❛s s✐♠❡tr✐❛s ❞❛s ❡q✉❛çõ❡s ❞❛s ❣❡♦❞és✐❝❛s✳ ❆ t❡r❝❡✐r❛ s❡çã♦ tr❛t❛ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥ q✉❡ ♣❡r♠✐t❡ ✉♠❛ ❢♦r♠✉❧❛çã♦ ✈❛r✐❛✲ ❝✐♦♥❛❧ ✐♥✈❛r✐❛♥t❡ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡✲▲❛❣r❛♥❣❡ ❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞❛s ❣❡♦❞és✐❝❛s✳ ◆❛ q✉❛rt❛ s❡çã♦ é ✐♥tr♦❞✉③✐❞❛ ❛ ♥♦çã♦ ❞❡ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ❥✉♥t♦ ❛ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r✳

  ✷✳✶ ❉✐str✐❜✉✐çõ❡s ❡ s✐♠❡tr✐❛s

  ◆❡st❛ s❡çã♦ s❡rá ❛♣r❡s❡♥t❛❞❛ ❛ ♥♦çã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❛❧é♠ ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s ❛ r❡s♣❡✐t♦ ❞❡st❛s✳ ❉❡♥tr❡ ❡st❡s✱ ❞❡st❛❝❛♠♦s ♦ ❝❧áss✐❝♦ t❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s ❡ s✉❛s ❝♦♥s❡q✉ê♥❝✐❛s s♦❜r❡ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞❡ ❞✐str✐❜✉✐çõ❡s✳ ❚❛♠❜é♠ s❡rá ✈✐st❛ ❛ ♥♦çã♦ ❞❡ s✐♠❡tr✐❛ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❥✉♥t♦ ❛ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ❞❡❝♦rr❡♠ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ s✐♠❡tr✐❛s✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ D s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ✉♠❛

  ✷✸

  p p M

  ❛♣❧✐❝❛çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ p ∈ M ✉♠ s✉❜❡s♣❛ç♦ D ∈ T ✳ ❉✐③❡♠♦s q✉❡ ❛

  p ) = k

  ❞✐♠❡♥sã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ é k s❡ dim(D ♣❛r❛ t♦❞♦ p ∈ M✳ ❉❡✜♥✐çã♦ ✷✳✶✳✷✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ k✲❞✐♠❡♥s✐♦♥❛❧ D é ❞✐t❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥✉♠ ♣♦♥t♦

  ∞

  p ∈ M q✉❛♥❞♦✱ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ p ∈ M✱ é ♣♦ssí✈❡❧ ♦❜t❡r k ❝❛♠♣♦s C ▲✳■✳ t❛✐s q✉❡ D é ❣❡r❛❞❛ ♣♦r ❡st❡s ❝❛♠♣♦s ❡♠ U✳ ◗✉❛♥❞♦ ❛ ❞✐str✐❜✉✐çã♦ D é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ t♦❞♦

  ∞

  ♣♦♥t♦ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ D é ❞❡ ❝❧❛ss❡ C ❖❜s❡r✈❡ q✉❡✱ ❞❛❞♦ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✱ ♣♦❞❡♠♦s s❡♠♣r❡ ❞❡✜♥✐r ✉♠❛ ❞✐str✐❜✉✐çã♦ D =< X >✳ ▲♦❣♦✱ ❛ ♥♦çã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❣❡♥❡r✲

  ❛❧✐③❛ ♦ ❡st✉❞♦ ❞♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s✳ Pr❡❝✐s❛♠♦s✱ ❡♥tã♦✱ ❣❡♥❡r❛❧✐③❛r ❛ ♥♦çã♦ ❞❡ ❝✉r✈❛s ✐♥t❡❣r❛✐s✳ P❛r❛ ✐ss♦✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✷✳✶✳✸✳ ❙❡❥❛ D ✉♠❛ ❞✐str✐❜✉✐çã♦ ❦✲❞✐♠❡♥s✐♦♥❛❧ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ▼✳ ❯♠❛

  n m

  s✉❜✈❛r✐❡❞❛❞❡ N ❞❡ M é ❝❤❛♠❛❞❛ ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ D s❡ n ≤ k ❡✱ ♣❛r❛ t♦❞♦ p ∈ N✱ t❡♠♦s q✉❡ i p (T p N ) p

  

∗ ⊂ D ✱

n m

  ♦♥❞❡ i : N é ❛ ✐♥❝❧✉sã♦✳ −→ M

  ❉✐❢❡r❡♥t❡ ❞♦ ❝❛s♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧✱ s✉❜✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ♠❛①✐♠❛✐s ✭✐st♦ é✱ ❞❡ ❞✐♠❡♥sã♦ ♠á①✐♠❛✮ ♥❡♠ s❡♠♣r❡ ❡①✐st❡♠ ♠❡s♠♦ ❧♦❝❛❧♠❡♥t❡✳ ❱❡r❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡

  3

  ❡♠ R ∂ ∂ ∂

  ; = span + y ,

  (x,y,z) (x,y,z) (x,y,z) (x,y,z) (x,y,z)

  D = {D D | | | ∂x ∂z ∂y

  ♥ã♦ ❛❞♠✐t❡ s✉❜✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ ❞✐♠❡♥sã♦ ♠á①✐♠❛✳

  ✷✳✶✳✶ ❖ ❚❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s

  ❈♦♥s✐❞❡r❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ k✲❞✐♠❡♥s✐♦♥❛❧ D s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✳ ❉✐③❡♠♦s

  p p

  q✉❡ ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ♣❡rt❡♥❝❡ ❛ D s❡ X ∈ D ♣❛r❛ t♦❞♦ p✳ ❙✉♣♦♥❤❛ q✉❡ N é ✉♠❛ ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ D ❡ i : N −→ M é ❛ ✐♥❝❧✉sã♦✳ ❙❡ X ❡ Y sã♦ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s

  p , Y p p N

  ♣❡rt❡♥❝❡♥t❡s ❛ D ❡♥tã♦ ♣❛r❛ t♦❞♦ p ∈ N ❡①✐st❡♠ ú♥✐❝♦s ✈❡t♦r❡s X t❛✐s q✉❡ ∈ T

  X p = i (X p ) p = i (Y p )

  

p p

∗ ❡ Y ∗ ✱

  ♦✉ s❡❥❛✱ X ❡ X sã♦ i✲r❡❧❛❝✐♦♥❛❞♦s ❡ Y ❡ Y t❛♠❜é♠✳ ❈♦♠♦ [X, Y ] ∈ N ❡♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✷✳✺✱ [X, Y ] ∈ D✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ D ❛❞♠✐t❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ ❞✐♠❡♥sã♦ ♠á①✐♠❛ ✭D é ❝♦♠♣❧❡✲ t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✮✱ ❡♥tã♦ [X, Y ] ∈ D ♣❛r❛ t♦❞♦s X, Y ∈ D✳ ❊st❛ ❝♦♥❞✐çã♦ ❞❡ [X, Y ] ∈ D✱

  ✷✹

  ∞

  ❚❡♦r❡♠❛ ✷✳✶✳✹✳ ✭❞❡ ❋r♦❜❡♥✐✉s✮ ❯♠❛ ❞✐str✐❜✉✐çã♦ ❦✲❞✐♠❡♥s✐♦♥❛❧ C D ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ▼ é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱ ♦✉ s❡❥❛✱ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ❋r♦❜❡♥✐✉s s❡✱ ❡ s♦♠❡♥t❡

  −1 U (p) = 0

  s❡✱ ♣❛r❛ t♦❞♦ p ∈ M ❡①✐st❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ϕ ❝♦♠ ϕ ❡ U = (

  k , ..., a n i

  • 1

  −ǫ, ǫ) × ... × (−ǫ, ǫ) t❛❧ q✉❡✱ ♣❛r❛ ❝❛❞❛ a ❝♦♠ |a | < ǫ✱ ♦ ❝♦♥❥✉♥t♦

  U (U ); x k (q) = a k , ..., x n (q) = a n

  • +1 +1

  {q ∈ ϕ } é ✉♠❛ ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ D✳

  ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ t❡♠♦s q✉❡ q✉❛❧q✉❡r ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ D ❝♦♥❡①❛ r❡str✐t❛ ❛ U

  ❡stá ❝♦♥t✐❞❛ ❡♠ ✉♠ ❞❡st❡s ❝♦♥❥✉♥t♦s✳ Pr♦✈❛✿ ❆ ✈♦❧t❛ ❞❡st❡ t❡♦r❡♠❛ ❞❡❝♦rr❡ ❞❛ ♦❜s❡r✈❛çã♦ ❛♥t❡r✐♦r✳

  U U (0) = p p =

  ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ϕ t❛❧ q✉❡ ϕ ❡ D ∂ ∂ span , ...,

  { | | }✳ ∂x ∂x k

  1 n k

  : ❙❡❥❛ π : R −→ R ❛ ♣r♦❥❡çã♦ ♥❛s ♣r✐♠❡✐r❛s k✲❝♦♦r❞❡♥❛❞❛s✳ ❊♥tã♦✱ π ∗ D −→

  k r

  R é ✉♠ ✐s♦♠♦r✜s♠♦✳ P♦r ❝♦♥t✐♥✉✐❞❛❞❡✱ π ∗ é ✐♥❥❡t✐✈❛ ✉♠ D ♣❛r❛ r ♣ró①✐♠♦ ❞❡ 0✳ ❊♥tã♦✱

  ♣ró①✐♠♦ ❞❡ 0 ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ú♥✐❝♦s X (r), ..., X n (r) r

  

1 ∈ D

  t❛✐s q✉❡ ∂

  π (X i (r)) = π ,

  

r (r)

∗ |

  ∂x i ♣❛r❛ t♦❞♦ i ∈ {1, ..., k}.

  ∂

  n i

  ❊♥tã♦ ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s X ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ 0 ❡♠ R ❡ ❡♠ ✉♠❛ ∂x i

  k

  ✈✐③✐♥❤❛♥ç❛ ❞❡ 0 ❡♠ R sã♦ π✲r❡❧❛❝✐♦♥❛❞♦s✳ ❊♥tã♦✱ ♣❡❧❛ ✶✳✷✳✷✸✱ ∂ ∂ π ([X i , X j ] r ) = , = 0.

  ∗

  ∂x i ∂x j

  π (r) i , X j ] r r r i , X j ] = 0

  ▼❛s✱ ♣♦r ❤✐♣ót❡s❡✱ [X ∈ D ❡ ✈✐♠♦s q✉❡ π ∗ é ✐♥❥❡t✐✈❛ ❡♠ D ✳ ❊♥tã♦✱ [X ✳ ❊♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✷✳✶✺✱ ❡①✐st❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s x t❛❧ q✉❡

  ∂ X i = ,

  ∀i ∈ {1, ..., k}✳ ∂x i

  

U ; x k (q) = a k , ..., x n (q) = a n

  • 1 +1

  ❖ ❝♦♥❥✉♥t♦ {q ∈ ϕ } sã♦ ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ❞❡ ∂

  D ❥á q✉❡ s❡✉s ❡s♣❛ç♦s t❛♥❣❡♥t❡s sã♦ ❣❡r❛❞♦s ♣♦r ♣❛r❛ i ∈ {1, ..., k}✳ ∂x i

  U (U ) U (U )

  ❙❡❥❛ N ∪ ϕ ✉♠❛ ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ D ❝♦♥❡①❛ r❡str✐t❛ ❛ ϕ ✱ ❝♦♠ ❛ (U )

  U q

  ✐♥❝❧✉sã♦ i : N −→ ϕ ⊂ M✳ P❛r❛ q✉❛❧q✉❡r ✈❡t♦r X t❛♥❣❡♥t❡ ❛ N ❡♠ q t❡♠♦s

  ✷✺ d(x l q ) = X q (x l (X q )(x l ) = 0

  q

  ◦ i)(X ◦ i) = i ∗ ✱ ∂ ∂

  (X q ) q , ...,

  q

  ♣❛r❛ t♦❞♦ l ∈ {k + 1, ..., n} ❥á q✉❡ i ∗ ∈ D q✉❡ é ❣❡r❛❞❛ ♣♦r ✳ ❉❡ss❡ ∂x

  1 ∂x k l

  ♠♦❞♦✱ d(x ◦ i) é ❝♦♥st❛♥t❡✱ ♣❛r❛ t♦❞♦ l ∈ {k + 1, ..., n}✱ ♥❛ ✈❛r✐❡❞❛❞❡ ❝♦♥❡①❛ N✳ ❖ t❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s✱ ♣♦r ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡

  ♣❛r❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ s❡r ✐♥t❡❣rá✈❡❧✱ ♠♦t✐✈❛ ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✷✳✶✳✺✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ D é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡ ❡❧❛ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ ❋r♦❜❡♥✐✉s✳

  P♦❞❡♠♦s ❡♥✉♥❝✐❛r ♦✉tr❛ ✈❡rsã♦ ♣❛r❛ ♦ t❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s ♣♦r ♠❡✐♦ ❞❡ ❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s ❛ q✉❛❧ é ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡ ❡♠ ❛❧❣✉♥s ❝❛s♦s✳ ❆ s❛❜❡r

  i

  ❚❡♦r❡♠❛ ✷✳✶✳✻✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ D = Ann{ω } é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡✱ ❡ s♦✲

  ij

  ♠❡♥t❡ s❡✱ ❡①✐st❡♠ 1✲❢♦r♠❛s α t❛✐s q✉❡

  X dω i = α ij j , ∧ ω

  j

  ♣❛r❛ t♦❞♦ i✳ ❯♠❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✸✻❪ ❡ ❬✹✼❪✳

  ✷✳✶✳✷ ❙✐♠❡tr✐❛s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦

  ❉❡ ♠❛♥❡✐r❛ ❣❡r❛❧✱ ✉♠❛ s✐♠❡tr✐❛ ❡♠ ✉♠ ❡s♣❛ç♦ M ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ❝♦♠♦ ✉♠❛ ❛✉t♦tr❛♥s❢♦r♠❛çã♦ ❞❡ M q✉❡ ♣r❡s❡r✈❛ ❛❧❣✉♠ t✐♣♦ ❞❡ ❡str✉t✉r❛✳ ❆q✉✐✱ tr❛t❛r❡♠♦s s✐♠❡tr✐❛s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✷✳✶✳✼✳ ❯♠ ❞✐❢❡♦♠♦r✜s♠♦ ϕ : M −→ M é ❝❤❛♠❛❞♦ s✐♠❡tr✐❛ ✭✜♥✐t❛✮ ❞❡ ✉♠❛

  ( p ) = ϕ

  (p)

  ❞✐str✐❜✉✐çã♦ D s❡ ♣r❡s❡r✈❛ ❡st❛ ❞✐str✐❜✉✐çã♦✱ ✐st♦ é✱ ϕ ∗ D D ♣❛r❛ t♦❞♦ p ∈ M✳ ❉❡✜♥✐çã♦ ✷✳✶✳✽✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M é ✉♠❛ s✐♠❡tr✐❛

  t

  ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ D s❡ ♦ s❡✉ ✢✉①♦ {A } ❝♦♥s✐st❡ ❞❡ s✐♠❡tr✐❛s ✜♥✐t❛s✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s s✐♠❡tr✐❛s ✐♥✜♥✐t❡s✐♠❛✐s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ D é ❞❡♥♦t❛❞♦ ♣♦r Sym(D)✳

  ➱ ❢á❝✐❧ ✈❡r q✉❡ ❛s s✐♠❡tr✐❛s ✜♥✐t❛s ❢♦r♠❛♠ ✉♠ ❣r✉♣♦ ✭❣❡r❛❧♠❡♥t❡✱ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✮✳ ❖✉tr♦ ❢❛t♦ ✐♠♣♦rt❛♥t❡ é q✉❡ ❛s s✐♠❡tr✐❛s ✐♥✜♥✐t❡s✐♠❛✐s ❢♦r♠❛♠ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ✭❣❡r❛❧♠❡♥t❡✱ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✮✳ ❉❡ ❢❛t♦✱ ❜❛st❛ ❧❡♠❜r❛r q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡

  = [

  X , Y ] [X,Y ]

  ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ L L L ❡✱ ❞❡✜♥✐♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❛s s✐♠❡tr✐❛s ❞❡ ✉♠❛

  ✷✻ ❞✐str✐❜✉✐çã♦ D ♣♦r Sym(D)✱ ✈❡r✐✜❝❛♠♦s ❞✐r❡t❛♠❡♥t❡ q✉❡ (Sym(D), [, ]) é ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳

  ❊♠ ❣❡r❛❧✱ ❛ ❞❡✜♥✐çã♦ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❞❡ s✐♠❡tr✐❛ ✐♥✜♥✐t❡s✐♠❛❧ ♥ã♦ é ♠✉✐t♦ ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ❢❛③❡r ❝á❧❝✉❧♦s✳ ▼❛s ♣♦❞❡♠♦s ❝♦♥t♦r♥❛r ❡st❛ ❞✐✜❝✉❧❞❛❞❡ ❝♦♠ ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✿ ❚❡♦r❡♠❛ ✷✳✶✳✾✳ ❙❡❥❛ D ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ M✳ ❙ã♦ ❡q✉✐✈❛❧❡♥t❡s ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

  ✭✐✮ X é ✉♠❛ s✐♠❡tr✐❛ ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ D❀ , ..., X n

  ✭✐✐✮ ❙❡ X

  1 sã♦ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❣❡r❛❞♦r❡s ❞❡ D ❡♥tã♦ ❡①✐st❡♠ ❢✉♥çõ❡s ❞✐❢❡r❡♥✲

  X

  ij i ] = µ ij

  X j ❝✐á✈❡✐s µ t❛✐s q✉❡ [X, X ✱ ♣❛r❛ t♦❞♦ i ∈ {1, ..., n}❀

  j

  , ..., ω m , ..., ω m

  1

  1

  ✭✐✐✐✮ ❙❡ ω sã♦ 1✲❢♦r♠❛s t❛✐s q✉❡ D = Ann{ω }✱ ❡♥tã♦ ❡①✐st❡♠ ❢✉♥çõ❡s

  X

  ij X (ω i ) = ν ij ω j

  ❞✐❢❡r❡♥❝✐á✈❡✐s ν t❛✐s q✉❡ L ✱ ♣❛r❛ t♦❞♦ i ∈ {1, ..., m}✳

  

j

t t

  Pr♦✈❛✿ i ⇒ ii✳ ❙❡❥❛ X ✉♠❛ s✐♠❡tr✐❛ ❞❡ D ❡ {A } ♦ s❡✉ ✢✉①♦✳ ❈♦♠♦ A ♣r❡s❡r✈❛

  i

  ❛ ❞✐str✐❜✉✐çã♦✱ ♣❛r❛ t♦❞♦ t✱ ❡♥tã♦ ♦ ♣♦✉s❤❢♦✇❛r❞ ❞♦ ❝❛♠♣♦ X ∈ D ♣❡rt❡♥❝❡ ❛ D✿

  X (A t ) (X i ) = α ij (t)X j ,

  ∗

j

ij (t)

  ♦♥❞❡ α é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ M ❞❡♣❡♥❞❡♥❞♦ ❞✐❢❡r❡♥❝✐❛✈❡❧♠❡♥t❡ ❞♦ ♣❛râ♠❡tr♦ t✳ ❉✐❢❡r❡♥❝✐❛♥❞♦ ❝♦♠ r❡s♣❡✐t♦ ❛ t ❡ ❛✈❛❧✐❛♥❞♦ ❡♠ t = 0✱ ♦❜t❡♠♦s

  X [X, X i ] = µ ij X j ,

  

j

  dα ij

  ij := t =0

  ♦♥❞❡ µ | ✳ dt ii

  ⇒ iii✳ ❙✉♣♦♥❤❛ q✉❡ X s❛t✐s❢❛③ (ii)✳ P❡❧❛ ❡q✉❛çã♦ (1.4)✱ t❡♠♦s q✉❡

  X (ω j )(X i ) = X(ω j (X i )) + ω j ([X i , X]) = j ([X, X i ]) = 0,

  L −ω

  i ] , ..., X n

  ♣❛r❛ t♦❞♦s i ∈ {1, ..., n} ❡ j ∈ 1, ..., m✱ ❥á q✉❡ [X, X é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ X

  1 ✳ X (ω j ) 1 , ..., ω m

  ▲♦❣♦✱ L é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ω ✳

  ∗ ∗ ∗

  iii = A

  ⇒ i✳ Pr✐♠❡✐r♦ ♥♦t❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ A t +s t ◦ A s ✐♠♣❧✐❝❛ q✉❡ d

  ∗ ∗ t A (ω) = A ( X (ω)).

  | =s t s L dt

  ✷✼

  ∗ ∗

  A (ω) (ω) d

  t +s − A s ∗ t A (ω) = lim =s t

  |

  t

→0

  dt t

  ∗ ∗

  A (ω) (ω)

  t

  − A

  

  = A (lim )

  

s

t →0

  t

  

  = A ( X (ω)).

  

s

  L ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ❛ (m + 1)✲❢♦r♠❛ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ♣❛râ♠❡tr♦ t

  ∗ Ω i (t) := A (ω i ) m . t

  1

  ∧ ω ∧ ... ∧ ω

  ∗

  (ω i ) = ω i i (0) = 0 ❈♦♠♦ A ✱ t❡♠♦s q✉❡ Ω ✳ ❆❧é♠ ❞✐ss♦✱ dΩ i

  ∗

  (t) = A (

  X (ω i )) m

  1 t L ∧ ω ∧ ... ∧ ω ✭✷✳✶✮

  dt

  X

  ∗ = A (ν ij )Ω j . t j

  ▲♦❣♦✱ t❡♠♦s ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ❝♦♠♣♦st♦ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❧✐♥❡❛r

  i (0) = 0 i (t)

  ❤♦♠♦❣ê♥❡❛ ❝♦♠ ✈❛❧♦r ✐♥✐❝✐❛❧ Ω ✱ ♣❛r❛ t♦❞♦ i ∈ {1, ..., m}✳ ▲♦❣♦✱ Ω ≡ 0✳ ■ss♦

  ∗

  (ω i )

  1 , ..., ω m t

  ♠♦str❛ q✉❡ A t é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ω ✱ ♣❛r❛ t♦❞♦ t ❡ t♦❞♦ i✳ ▲♦❣♦✱ A é ✉♠❛ s✐♠❡tr✐❛ ❞❡ D✱ ♣❛r❛ t♦❞♦ t✱ ✐st♦ é✱ X é s✐♠❡tr✐❛ ❞❡ D✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✵✳ ❙❡ X é ✉♠❛ s✐♠❡tr✐❛ ✐♥✜♥✐t❡s✐♠❛❧ ❞❡ D t❛❧ q✉❡ X ∈ D ❡♥tã♦ X é ❞❡♥♦♠✐♥❛❞❛ s✐♠❡tr✐❛ ❝❛r❛❝t❡ríst✐❝❛ ✭♦✉ tr✐✈✐❛❧✮ ❡ ❞❡♥♦t❛r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡st❛s s✐♠❡tr✐❛s ♣♦r Char(D)✳

  ❚❡♠♦s ❛ s❡❣✉✐♥t❡

  ∞

  (M ) Pr♦♣♦s✐çã♦ ✷✳✶✳✶✶✳ Char(D) é ✉♠ C ✲♠ó❞✉❧♦ ✭❡♠ ♣❛rt✐❝✉❧❛r✱ ✉♠ R✲❡s♣❛ç♦ ✈❡t♦✲ r✐❛❧✮ ❡ ✉♠ ✐❞❡❛❧ ❞❛ á❧❣❡❜r❛ Sym(D)✳

  X , ..., X k > α i X i

  1 Pr♦✈❛✿ ❙❡ X ∈ Char(D) ❡ D =< X ✱ t❡♠♦s q✉❡ X = ❡ i ∞

  L ∈ D✳ P♦rt❛♥t♦✱ ♣❛r❛ t♦❞❛ f ∈ C ✱ t❡♠♦s q✉❡

  X X i (M )

  f X

  X i = [f X, X i ] = f [X, X i ] i (f )X L − X ∈ D.

  , Y + Y ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ Y

  1

  2

  1

  2

  ∈ Char(D)✱ Y ∈ D✳ ▲♦❣♦✱ Char(D) é ✉♠ s✉❜♠ó✲

  ∞

  (M ) ❞✉❧♦ ❞♦ C ✲♠ó❞✉❧♦ ❞♦s ❝❛♠♣♦s ♣❡rt❡♥❝❡♥t❡s ❛ D✳

  ✷✽ ❆❣♦r❛✱ s❡ X ∈ Char(D) ❡ Y ∈ Sym(D)✱ ❡♥tã♦

  " #

  X X

  X [X, Y ] = Y, α + i X i = Y (α i )X i α i [Y, X i ]

  i i i

  X X

  • = Y (α i )X i α i f ij X j ,

  i i

  X X α i X i i ] = f ij X j s❡ X = ❡ [Y, X ✳ ▼❛s [Y, X] ∈ D ❡✱ ♣♦r ❤✐♣ót❡s❡✱ X ❡ Y sã♦

  i j

  s✐♠❡tr✐❛s✳ ▲♦❣♦✱ [X, Y ] ❛✐♥❞❛ é ✉♠❛ s✐♠❡tr✐❛ ❡ ♣❡rt❡♥❝❡ ❛ D✳ ❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ é ❛ s❡❣✉✐♥t❡

  Pr♦♣♦s✐çã♦ ✷✳✶✳✶✷✳ Char(D) é ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳ Pr♦✈❛✿ ❈♦♥s❡q✉ê♥❝✐❛ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ❡ ❞♦ t❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s✳

  ❉❡✜♥✐çã♦ ✷✳✶✳✶✸✳ ❆ á❧❣❡❜r❛ q✉♦❝✐❡♥t❡ Sym(

  D) sym( D) :=

  Char(

  D) é ❛ á❧❣❡❜r❛ ❞❛s s✐♠❡tr✐❛s ♥ã♦ ❝❛r❛❝t❡ríst✐❝❛s ✭♦✉ ♣ró♣r✐❛s✮ ❞❡ D✳

  ✷✳✶✳✸ ■♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦

  ❉❡✜♥✐çã♦ ✷✳✶✳✶✹✳ ❙❡❥❛ D ✉♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ M✳ ❯♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❞❡ D é ✉♠❛

  ∞

  (M ) ❢✉♥çã♦ f ∈ C t❛❧ q✉❡

  X(f ) = 0, ♣❛r❛ t♦❞♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ∈ D✳

  ❊st❛ ❞❡✜♥✐çã♦ ❛❞♠✐t❡ ❛ s❡❣✉✐♥t❡ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛✿

  ∞

  (M ) p p (Γ c (f )) Pr♦♣♦s✐çã♦ ✷✳✶✳✶✺✳ ❙❡ f ∈ C é ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❞❡ D✱ ❡♥tã♦ D ⊂ T ✱

  c (f ) := c (f )

  ♣❛r❛ t♦❞♦ p ∈ Γ {a ∈ M : f(a) = c}✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ Γ ❝♦♥t❡♠ ❛s ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s q✉❡ ♣❛ss❛♠ ♣♦r s❡✉s ♣♦♥t♦s✳

  ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❢❛t♦ é ❞✐r❡t❛✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✻✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ D k✲❞✐♠❡♥s✐♦♥❛❧ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✱ ❝♦♠ dim(M ) = n

  ✱ ♥ã♦ ♣♦❞❡ ❛❞♠✐t✐r ♠❛✐s q✉❡ n − k ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡✲

  ✷✾ , ..., f n , g , ..., g h

  1

  1 Pr♦✈❛✿ ❙❡ D é k✲❞✐♠❡♥s✐♦♥❛❧ ❡ f −k sã♦ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❢✉♥✲

  ❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♥tã♦✱ ❞❡✜♥✐♥❞♦ D := Ann , ..., df n , dg , ..., dg h

  1

  1

  {df −k } ⊇ D, t❡♠♦s q✉❡ k = dim(D) ≤ dim(D) = n − (n − k + h) = k − h✳ ▲♦❣♦✱ h = 0✳ ❆❣♦r❛✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦

  Pr♦♣♦s✐çã♦ ✷✳✶✳✶✼✳ ❙❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ D k✲❞✐♠❡♥s✐♦♥❛❧ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❛❞♠✐t❡ dim(M ) − k ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♥tã♦ D é ❝♦♠♣❧❡t❛♠❡♥t❡

  ✐♥t❡❣rá✈❡❧✳

  ∞

  , ..., f h (M ) Pr♦✈❛✿ ❙❡ D ❛❞♠✐t❡ ❛s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s f

  1 ✱ ❝♦♠ h = dim(M)−

  ∈ C k ✱ ❡♥tã♦

    f = c

  1

  1

   

  ✳✳✳    f h = c h sã♦ ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ♠❛①✐♠❛✐s✳ ▲♦❣♦✱ D é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳

  ❆s ❞✐str✐❜✉✐çõ❡s k✲❞✐♠❡♥s✐♦♥❛✐s ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡✐s q✉❡ ❛❞♠✐t❡♠ dim(M)− k ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s sã♦ ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧✳ ❉❡ ❢❛t♦✱ s❡ ✉♠❛

  ❞✐str✐❜✉✐çã♦ é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ♥ã♦ é ❣❛r❛♥t✐❞♦ q✉❡ ❡①✐st❡♠ dim(M)−k ✐♥t❡❣r❛✐s , ..., f n

  1

  ♣r✐♠❡✐r❛s ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ P♦r ❡①❡♠♣❧♦✱ s❡ dim(M) = n ❡ f −k ∈

  ∞

  C (M ) sã♦ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❡ D ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡♥tã♦✱ ❡♠ ❝❛❞❛ ✈✐③✲

  i

  ✐♥❤❛♥ç❛ ❝♦♦r❞❡♥❛❞❛ {x }✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛s ❡q✉❛çõ❡s   f (x , ..., x ) = y

  1 1 n

  1

   

  ✳✳✳    f n (x , ..., x n ) = y n

  1 −k −k

  , ..., x n

  1

  ❡ t❡♥t❛r r❡s♦❧✈ê✲❧❛s ❝♦♠ r❡s♣❡✐t♦ ❛ x −k ✭✐ss♦ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❛ ♠❡♥♦s ❞❡ ✉♠❛ ♣❡r♠✉t❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s✮

    x = ϕ (y , ..., y n , x n , ..., x n )

  1

  1

  1 −k −k+1

   

  ✳✳✳    x n = ϕ n (y , ..., y n , x n , ..., x n )

  1 −k −k −k −k+1

  P , ..., x

  n n

  ❆ss✐♠✱ ✜①❛♥❞♦ x −k+1 ✱ ♦ s✐st❡♠❛ ❛❝✐♠❛ ❞❡s❝r❡✈❡ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡

  ✸✵ , ..., y n

  y (f )

  1

  ❆ss✐♠✱ {y −k } ♣❛r❛♠❡tr✐③❛♠ ♦s ♣❡❞❛ç♦s ❞❡ ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ Γ q✉❡

  y (f )

  ❡stã♦ ❝♦♥t✐❞♦s ❡♠ U✳ ▼❛s ♣♦❞❡ ❛❝♦♥t❡❝❡r q✉❡ ❛ ♠❡s♠❛ Γ t❡♥❤❛ ♦✉tr❛s ✐♥t❡rs❡çõ❡s ❝♦♠ P✳

  P❡❧♦ t❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s✱ t❡♠♦s q✉❡✱ ♣❡❧♦ ♠❡♥♦s ❧♦❝❛❧♠❡♥t❡✱ ✉♠❛ ❞✐str✐❜✉✐çã♦ k ✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ n✲❞✐♠❡♥s✐♦♥❛❧ ❛❞♠✐t❡ s❡♠✲

  ♣r❡ n ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡✜♥✐❞❛s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦✳

  ❙❡❥❛ D ✉♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✳ ❙❡ dim(D) = k ❡ dim(M) = n✱ ♣♦❞❡♠ ❡①✐st✐r✱ ♥♦ ♠á①✐♠♦✱ n − k s✐♠❡tr✐❛s ❞❡ D tr❛♥s✈❡rs❛✐s ❛ D✳ ■ss♦ ♠♦t✐✈❛ ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✷✳✶✳✶✽✳ ❙❡❥❛♠ D ✉♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✱ ♦♥❞❡ dim(D) = k ❡ dim(M) = n✳ ❯♠❛ á❧❣❡❜r❛ ❞❡ n−k s✐♠❡tr✐❛s ❞❡ D tr❛♥s✈❡rs❛✐s ❛ D s❡ ❝❤❛♠❛ ♠❛①✐♠❛❧✳

  ◆❛ s❡çã♦ 4.3 ♣r♦✈❛r❡♠♦s ✉♠ t❡♦r❡♠❛✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ t❡♦r❡♠❛ ❞❡ ❇✐❛♥❝❤✐✲▲✐❡✱ q✉❡ ❛✜r♠❛ q✉❡ s❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ D ❛❞♠✐t❡ ✉♠❛ á❧❣❡❜r❛ ♠❛①✐♠❛❧ ❡ s♦❧ú✈❡❧ ❞❡ s✐♠❡tr✐❛s tr❛♥s✈❡rs❛✐s ✭✈❡❥❛ s❡çã♦ 4.2✮ ❡♥tã♦ D é ✐♥t❡❣rá✈❡❧ ♣♦r q✉❛❞r❛t✉r❛s✳

  ◆♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ tr❛t❛r❡♠♦s ❛s ❡str✉t✉r❛s s♦❧ú✈❡✐s✱ q✉❡ sã♦ ❡str✉t✉r❛s ♠❛✐s ❣❡r❛✐s q✉❡ ❛s á❧❣❡❜r❛s s♦❧ú✈❡✐s✱ ❡ ✈❡r❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ♠❛✐s ❣❡r❛❧ q✉❡ ♦ t❡♦r❡♠❛ ❞❡ ❇✐❛♥❝❤✐✲▲✐❡✳

  ✷✳✷ ❊s♣❛ç♦s ❞❡ ❥❛t♦s

  ❯♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❛♥❛❧ít✐❝♦✱ é ✉♠❛ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧

  j i i ) (x)

  ❡♥tr❡ ✈❛r✐á✈❡✐s (x ✱ ❢✉♥çõ❡s u ❞❡ss❛s ✈❛r✐á✈❡✐s ❡ ❛s ❞❡r✐✈❛❞❛s u i ❞❡st❛s ❢✉♥çõ❡s✳ ❊ss❡ t✐♣♦ ❞❡ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦ ♣❡r♠✐t❡ ❝r✐❛r ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s ❞❡ ❢❡♥ô♠❡♥♦s ♥❛t✉r❛✐s ♦✉ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ✐♥t❡r❡ss❡ t❡ór✐❝♦ ♦✉ ♣rát✐❝♦✳

  ❖ t✐♣♦ ♠❛✐s s✐♠♣❧❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ✭♣❛r❝✐❛❧✮ é ✉♠❛ ❡q✉❛çã♦ ❞❡ , ..., x )

  1 n

  ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡♠ ✉♠❛ ❢✉♥çã♦ ✐♥❝ó❣♥✐t❛ u = u(x ❞❡s❝r✐t❛ ♣♦r ✉♠❛ r❡❧❛çã♦ ❞♦ t✐♣♦ F (x , ..., x n , u, u , ..., u n ) = 0.

  1

  

1

  ✭✷✳✷✮ Pr❡❝✐s❛♠♦s ❞❛r ✉♠❛ ❞❡s❝r✐çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ ❡st❡ t✐♣♦ ❞❡ ♦❜❥❡t♦ ♣❛r❛ ♣♦❞❡r♠♦s

  ❢❛③❡r ✉♠ ❡st✉❞♦ ✐♥✈❛r✐❛♥t❡✳ ❖s ❡s♣❛ç♦s ❞❡ ❥❛t♦s sã♦ ♦ q✉❡ ♣❡r♠✐t❡♠ tr❛t❛r ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❡q✉❛çõ❡s ❝♦♠♦

  ∂u

  ✷✳✷✳ ◆♦s ❡s♣❛ç♦s ❞❡ ❥❛t♦s✱ ❞❡ ❢❛t♦✱ ♣♦❞❡♠♦s tr❛t❛r ❛s ❢✉♥çõ❡s x, u, ❝♦♠♦ ❢✉♥çõ❡s

  ∂x

  ❝♦♦r❞❡♥❛❞❛s✳

  ✸✶

  1 (M, n) ✷✳✷✳✶ ❖ ♣r✐♠❡✐r♦ ❡s♣❛ç♦ ❞❡ ❥❛t♦s J

  ❙❡❥❛♠ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ a ∈ M✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ❛ s❡❣✉✐♥t❡

  1

  2

  ❉❡✜♥✐çã♦ ✷✳✷✳✶✳ ❉✉❛s s✉❜✈❛r✐❡❞❛❞❡s N ❡ N ❞❡ ❞✐♠❡♥sã♦ n sã♦ 1✲t❛♥❣❡♥t❡s ✭♦✉ t❛♥✲

  1

  2

  ❣❡♥t❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✮ ❡♠ a ∈ N ∩ N s❡✱ ❡ s♦♠❡♥t❡ s❡ T a N = T a N .

  1

  2

  1

  2

  ❊st❛ ❞❡✜♥✐çã♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡♠ N ❡ N ♥✉♠❛ ✈✐③✐♥✲

  a

  ❤❛♥ç❛ ❞♦ ♣♦♥t♦ a✳ ❊ss❡ ❢❛t♦ ♥♦s ♣❡r♠✐t❡ ✐♥tr♦❞✉③✐r ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ∼ s♦❜r❡

  a (M, n)

  ♦ ❝♦♥❥✉♥t♦ Γ ❞❡ t♦❞❛s ❛s s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s ❞❡ M q✉❡ ♣❛ss❛♠ ♣❡❧♦ ♣♦♥t♦ a ❞❛❞❛ ♣♦r

  def

  N

  1 a N

  1

  2

  1 ∼ ⇔ ❡ N ✲t❛♥❣❡♥t❡s ❡♠ a✳

  

a

  ➱ ✐♠❡❞✐❛t♦ ✈❡r✐✜❝❛r q✉❡✱ ❞❡ ❢❛t♦✱ ∼ ❡st❛❜❡❧❡❝❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♠ Γ a (M, n)

  ✳ ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s ❛ s❡❣✉✐♥t❡

  1

  (M, n) ❉❡✜♥✐çã♦ ✷✳✷✳✷✳ ❖ ♣r✐♠❡✐r♦ ❡s♣❛ç♦ ❞❡ ❥❛t♦s J a ❞❛s s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s

  a (M, n)/ a

  ❞❡ M q✉❡ ♣❛ss❛♠ ♣❡❧♦ ♣♦♥t♦ a é ♦ q✉♦❝✐❡♥t❡ Γ ∼ ✱ ✐st♦ é✱

  a

  ∈ Γ

  a ∼

  ◆♦t❛çã♦ ✷✳✷✳✸✳ P❛r❛ s✐♠♣❧✐✜❝❛r ❛s ♥♦t❛çõ❡s✱ ❡s❝r❡✈❡r❡♠♦s [N] a ❡♠ ✈❡③ ❞❡ [N] a ✳ ❆❧é♠

  ∼ k 1 k

  (M, n)

  a

  ❞✐ss♦✱ ♠✉✐t❛s ✈❡③❡s ✉s❛✲s❡ ❛ ♥♦t❛çã♦ [N] a ♣❛r❛ ❞✐st✐♥❣✉✐r ❞❡ [N] a ∈ J ✳ ❉❡ ❢❛t♦✱

  1 k

  (M, n) (M, n) ♠♦str❛r❡♠♦s q✉❡ é ♣♦ssí✈❡❧ ❣❡♥❡r❛❧✐③❛r ❛ ❝♦♥str✉çã♦ ❞❡ J a ❡ ❞❡✜♥✐r J a ✳

  1

  (M, n) a

  a

  P♦r ❞❡✜♥✐çã♦✱ ✉♠ ❡❧❡♠❡♥t♦ θ ∈ J é ✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ θ = [N] ❞❡ s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s✱ t♦❞❛s ❝♦♠ ♦ ♠❡s♠♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❡♠ a✳ P♦rt❛♥t♦✱ θ

  a a M a ) = n

  ❞❡t❡r♠✐♥❛ ✉♠ ♣♦♥t♦ a ∈ M ❡ ✉♠ ❡s♣❛ç♦ t❛♥❣❡♥t❡ V ✱ dim(V ✳ ⊂ T

  a N

  ❙❡ N ∈ θ✱ T ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❝♦♠♦ s❡❣✉❡✿ ❝♦♥s✐❞❡r❡

  i

  {v } ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ a ∈ M✳ N ♣♦❞❡ s❡r ❞❡s❝r✐t❛✱ ♣❡❧♦ ♠❡♥♦s ❧♦❝❛❧♠❡♥t❡✱ ❝♦♠♦ ✉♠ ❣rá✜❝♦✳ ❉❡ ❢❛t♦✱ s❡❥❛ dim(M) = m+n✱ ❡♥tã♦ N✱ ❝♦♠♦ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ M✱ ❛❞♠✐t❡ ✉♠❛ ❞❡s❝r✐çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❞❛❞❛ ♣♦r

    v β = v β (v α , ..., v α n )

  1

  1

  1

   

  ✳✳✳    v β = v β (v α , ..., v α ).

  m m n

  1 i i := v α = v β i i

  ❉❡✜♥✐♥❞♦ x ❡ u ✱ ♦❜t❡♠♦s 

  1

  1

   u = u (x , ..., x n )

  

1

   

  ✭✷✳✸✮ ✳✳✳

   

  m m

   u = u (x

  1 , ..., x n ).

  ✸✷

  i i , u

  ■ss♦ ♠♦str❛ q✉❡ ❡①✐st❡♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s {x } ❡♠ M t❛✐s q✉❡ N é r❡♣r❡s❡♥✲

  a

  t❛❞❛ ♣♦r ✉♠ s✐st❡♠❛ ❞❛ ❢♦r♠❛ ✷✳✸✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ❡s♣❛ç♦ V t❛♥❣❡♥t❡ ❡♠ a à ✈❛r✐❡❞❛❞❡

  ∞

  N (M ) ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❛ss✐♠✿ s❡ f ∈ C ✱ ❡♥tã♦✱ ❝❛❧❝✉❧❛♥❞♦ s✉❛ r❡str✐çã♦ ❛ N✱ ♦❜t❡✲

  j

∂f P

∂ ∂u ∂

  (x (a) , u (x (a))) = + j (f ) (x (a) , u (x (a))) ♠♦s f(x, u(x)) ❡✱ ♣♦rt❛♥t♦✱ j ✳

  ∂x i ∂x i ∂x i ∂u

  ❊♥tã♦✱

  j j

  X X ∂ ∂u ∂ ∂ ∂u ∂

  V = < + a , ..., > ✭✷✳✹✮

  • j j

  ∂x

  1 ∂x n ∂u ∂x n ∂x n ∂u j j

  ∂ ∂ ∂ ∂

  j j

  = < + p , ..., + p >,

  

1 n

j j

  ∂x

  1 ∂u ∂x n ∂u j j ∂u

  := ♦♥❞❡ p i ❡✱ ♥❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ ✉s❛♠♦s ❛ ❝♦♥✈❡♥çã♦ ❞❡ ❊✐♥st❡✐♥ s♦❜r❡ ❛ s♦♠❛

  ∂x i

  r❡❧❛t✐✈❛♠❡♥t❡ ❛ í♥❞✐❝❡s r❡♣❡t✐❞♦s✳

  a a M

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ t❡♠♦s ✉♠ s✉❜❡s♣❛ç♦ V ⊂ T ❣❡r❛❞♦ ♣♦r n ✈❡t♦r❡s

  n n

  • m +m

  X X ∂ ∂

  h h

  ξ = A , ..., ξ = A ,

  1 n 1 n

  ∂v h ∂v h

  h h =1 =1

  ❡♥tã♦ ♣♦❞❡♠♦s s❡♠♣r❡ ♣❛ss❛r ❛ ♦✉tr♦ s✐st❡♠❛ ❞❡ ❣❡r❛❞♦r❡s ❞❛ ❢♦r♠❛ 

  m

  X ∂ j ∂

     + η = p

  1

  1

   

  ∂v α ∂v β

  1 j

j

=1 m

  X  ∂ ∂

  j

    η + n = p

  n

   

  ∂v α ∂v β

  n j

j

=1

  P

  

j ∂ j ∂

  • i := v α := v β i = p j

  i j

  ♦♥❞❡✱ ❞❡✜♥✐♥❞♦ x ❡ u ✱ ♦❜t❡♠♦s η n ✱ ♣❛r❛ t♦❞♦s i ∈ {1, ..., n}

  ∂x ∂u i

  α

  1 i

  ❡ j ∈ {1, ..., m}✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ ♦s ξ ✬s sã♦ ▲✳■✳✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ A

  1 6= 0 ❡ ❞❡✜♥✐r

  P

  1 ∂ k ∂

  η = = + α1 B

  i

  1 ∂v 1 ∂v

  ✳ ❖ ♠❡s♠♦ ♣♦❞❡ s❡r ❢❡✐t♦ ♣❛r❛ ♦s ❞❡♠❛✐s ξ ✬s✳

  A k α1

  1

  ❋❡✐t♦ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❛❝✐♠❛✱ ♣♦❞❡♠♦s s❡♠♣r❡ ❞❡✜♥✐r ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ N q✉❡✱

  j a = T a N i , u

  ❡♠ a ∈ M✱ ❛❞♠✐t❡ V ✳ ❉❡ ❢❛t♦✱ ♥❛s ❝♦♦r❞❡♥❛❞❛s ❝♦♥str✉✐❞❛s {x }✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r

  ( )

  n

  X

  j j N = u = p x i , j . i

  ∈ {1, ..., m}

  i =1

  1

  (M, n)

  a

  ❊st❛s ❝♦♥s✐❞❡r❛çõ❡s ♠♦str❛♠ q✉❡ θ ∈ J ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠ ✉♠ ♣❛r (a, V a ) a a M a ) = n a (M, n)

  ✱ ♦♥❞❡ V ⊂ T ❡ dim(V ✳ ❆❧é♠ ❞✐ss♦✱ t♦❞❛ N ∈ Γ ♣♦❞❡ s❡r ❞❡s❝r✐t❛

  j i , u

  ❧♦❝❛❧♠❡♥t❡ ❝♦♠♦ ❣rá✜❝♦ ❞❡ ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s {x }✳ P♦rt❛♥t♦✱ ❛ ♣❛rt✐r ❞❡ q✉❛❧q✉❡r

  j

  , u

  h i

  ❝❛rt❛ {v }✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ✉♠❛ ❝❛rt❛ ❡s♣❡❝✐❛❧ {x } ❝❤❛♠❛❞❛ s❡♣❛r❛❞❛✳ ❆ r❡✉♥✐ã♦

  ✸✸ ❞❡✜♥✐❞❛ ❡♠ M✱ ♣♦r ❡ss❡ ❛t❧❛s✱ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛q✉❡❧❛ ✐♥✐❝✐❛❧✳ ❈❛❞❛ ❝❛rt❛ s❡♣❛r❛❞❛ ❞❡ ✉♠❛

  [

  j j 1 i , u , p J (M, n)

  ✈✐③✐♥❤❛♥ç❛ U ❞❡ M ❞❡✜♥❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s {x i a ✳ } s♦❜r❡ ♦ ❡s♣❛ç♦

  a ∈M

  ❊st❛s ❢✉♥çõ❡s sã♦ ❞❡✜♥✐❞❛s ❝♦♠♦ s❡❣✉❡✿ [

  j j j

  1

  , u , p J (M, n) , u

  i i

  ✭✶✮ {x i } s♦❜r❡ a sã♦ ✉♠❛ ❡①t❡♥sã♦ ❞❛s {x } ❞❡✜♥✐❞❛s s♦❜r❡ M✳ ❙❡

  a ∈M

  [

  1 j j

  θ J (M, n) i (θ) = x i (a) (θ) = u (a) ∈ a ✱ t❡♠♦s q✉❡ x ❡ u ✳

  a ∈M j a ) a

  ✭✷✮ ❆s ❢✉♥çõ❡s p i sã♦ ❞❡✜♥✐❞❛s ❛ss✐♠✿ s❡ θ = (a, V ❡ V é ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ {u = u(x)}✱ ♥♦ ♣♦♥t♦ a✱ ❡♥tã♦

  j j ∂u

  p (θ) = (x(a)).

  i

  ∂x i

  j

  ◆♦t❛çã♦ ✷✳✷✳✹✳ ❊♠ ✈✐rt✉❞❡ ❞❡ss❛ ❞❡✜♥✐çã♦✱ ♣❛r❛ ❞❡♥♦t❛r ❛s ❝♦♦r❞❡♥❛❞❛s p i ✱ ✉s❛✲s❡ ❛

  j j

  ♥♦t❛çã♦ u i ✳ P♦❞❡♠♦s ❝❤❛♠❛r ❡st❛s ❝♦♦r❞❡♥❛❞❛s ❞❡r✐✈❛❞❛s ❢♦r♠❛✐s ❞❛s ❝♦♦r❞❡♥❛❞❛s u

  i

  ❝♦♠ r❡s♣❡✐t♦ às ❝♦♦r❞❡♥❛❞❛s x ✳

  1

  (M, n) ❉❡✜♥✐çã♦ ✷✳✷✳✺✳ ❉❡✜♥✐♠♦s J ✱ ♦ ❡s♣❛ç♦ ❞♦s 1✲❥❛t♦s ❞❡ s✉❜✈❛r✐❡❞❛❞❡s n✲ ❞✐♠❡♥s✐♦♥❛✐s ❞❡ M✱ ♣♦r

  [

  1

  1 J (M, n) = J (M, n). a a

  

∈M

  ❆❧é♠ ❞✐ss♦✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦❥❡çã♦ ♥❛t✉r❛❧

  1

  π : J (M, n)

  1,0

  −→ M θ = (a, V a )

  7−→ a

  1

  π (M, n)

  1,0

  ✐♥❞✉③ ✉♠❛ t♦♣♦❧♦❣✐❛ ♥❛t✉r❛❧ s♦❜r❡ J ✳ ❈♦♠ ❡st❛ t♦♣♦❧♦❣✐❛ s♦❜r❡

  (M, n) = M ✱ π 1,0 é ❝♦♥tí♥✉❛✳ ▼✉✐t❛s ✈❡③❡s é ❝♦♥✈❡♥✐❡♥t❡ ✐♥tr♦❞✉③✐r ❛ ♥♦t❛çã♦ J ✳

  ◆❡st❡ ❝❛s♦✱ s❡ ❞✐③ q✉❡ M é ♦ ❡s♣❛ç♦ ❞♦s 0✲❥❛t♦s ❞❡ s✉❜✈❛r✐❡❞❛❞❡s✳ ◆♦t❛çã♦ ✷✳✷✳✻✳ ◗✉❛♥❞♦ ♥ã♦ ❤á ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❝♦♥❢✉sã♦✱ ♣♦❞❡ s❡r ❝♦♥✈❡♥✐❡♥t❡ ♦♠✐t✐r í♥❞✐❝❡s ♥♦s ❝á❧❝✉❧♦s ❡ ♥❛s ❢ór♠✉❧❛s✳ ❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ ❡s❝r❡✈❡♠♦s {x, u, p} ❡♠ ✈❡③ ❞❡

  j j j j i , u , p = p dx i i i ✳

  {x } ♦✉ du = pdx ❡♠ ✈❡③ ❞❡ du

  m

  X ∂ ∂

  j

  := p

  • j

  i

  ❆❧é♠ ❞✐ss♦✱ ✉s❛r❡♠♦s ❛❣♦r❛ ❛ ♥♦t❛çã♦ D i ✳

  ∂x i ∂u

  j =1

  ✸✹

  1

  1

  (M, n) (M, n)

  a

  ▼♦str❛r❡♠♦s q✉❡ J ❡ J sã♦ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ P❛r❛ ❡ss❡ ✜♠✱ ♣r❡❝✐s❛♠♦s ❞♦ ❧❡♠❛ s❡❣✉✐♥t❡ q✉❡ é út✐❧ t❛♠❜é♠ ❡♠ ❝♦♠♣✉t❛çõ❡s ❝♦♠ ❛s ❢✉♥çõ❡s

  j j i , u , p

  {x i }✳

  a

  ▲❡♠❛ ✷✳✷✳✼✳ ❙❡ θ = [N] ❡♥tã♦✱ s♦❜ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s {¯x = ¯x(x, u), ¯u = u(x, u) ¯ (a)

  1,0

  } ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ a = π ✱ ¯p(θ) é ❞❡t❡r♠✐♥❛❞♦ ❡♠ ❢✉♥çã♦ ❞❡ x(θ), u(θ) ❡ p(θ)} ♣❡❧❛ ❢ór♠✉❧❛

       

  j j

D (¯ x ) . . . D (¯ x n ) p ¯ D (¯ u )

  1

  1

  1

  1

  1

             

  ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

      =   ,

  j j

  D n (¯ x ) . . . D n (¯ x n ) p ¯ D n (¯ u )

  1 n

  ♦♥❞❡ j ∈ {1, ..., m}✱ ❛♣❧✐❝❛♥❞♦✲❛ ♥♦ ♣♦♥t♦ θ s❡♥❞♦ q✉❡✱ s♦❜r❡ N✱  

  D

  1 (¯ x 1 ) . . . D 1 (¯ x n )

     

  ✳✳✳ ✳✳✳ ✳✳✳   6= 0.

  D n (¯ x ) . . . D n (¯ x n )

  1 j j i , u i , ¯ u

  Pr♦✈❛✿ ❈♦♠♦✱ ♣♦r ❤✐♣ót❡s❡✱ t❡♠♦s ❞✉❛s ❝❛rt❛s s❡♣❛r❛❞❛s {x } ❡ {¯x } ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ a ∈ M ❡♥tã♦ t❡♠♦s ❞✉❛s ♣♦ssí✈❡✐s r❡♣r❡s❡♥t❛çõ❡s ❞❡ N ❝♦♠♦ ❣rá✜❝♦

  N = {u = f(x)}

  = f (¯ x) {¯u = ¯ }. P♦rt❛♥t♦✱ ✉s❛♥❞♦ ❡st❛s r❡♣r❡s❡♥t❛çõ❡s✱ ♣♦❞♠❡♦s ❡st❛❜❡❧❡❝❡r ❛ r❡❧❛çã♦ ❡①✐st❡♥t❡

  

∂ ¯ f ∂f

  (¯ x(θ)) x(θ) ❡♥tr❡ ¯p(θ) ❡ p(θ)✳ ❉❡ ❢❛t♦✱ ¯p(θ) = ✱ p(θ) = ❡✱ t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡

  

∂ x ¯ ∂x

i

  X X ∂ ¯ x i ∂ ¯ x

  β

  x + d¯ i = dx α du

  β

  ∂x α ∂u

  α β

  X ∂ ¯ f

  j

  d¯ u θ = d¯ x i , |

  ∂ ¯ x i

  i

  ♦❜t❡♠♦s

  X

  j j

  d¯ u θ = p (θ)d¯ x i θ

  i

  | |

  i

  " #

  i

  X X

  X

  j ∂ ¯ x i ∂ ¯ x β

  = p (θ) + (x(θ), u(θ))dx α θ (x(θ), u(θ))du θ

  i | | β

  ∂x α ∂u

  i α β

  " #

  i

  X X

  X ∂ ¯ x i ∂ ¯ x

  j β

  = p (θ) (x(θ), u(θ)) + (x(θ), u(θ))p (θ) dx ,

  α θ i α | β

  ∂x α ∂u

  i α β

P

  β β θ = p (θ)dx α θ α

  ✸✺ P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s t❛♠❜é♠ q✉❡

  j j

  X X ∂u ∂u

  j β

  d¯ u + θ = dx α θ du θ | | |

  ∂x α ∂u β

  α β

  " #

  j j

  X X ∂u ∂u

  β = (θ) + p dx α θ . α |

  ∂x α ∂u β

  α β j θ

  ▲♦❣♦✱ ❝♦♠♣❛r❛♥❞♦ ❛s ❞✉❛s ❡①♣r❡ssõ❡s ❝❛❧❝✉❧❛❞❛s ♣❛r❛ d¯u | ✱ ♦❜t❡♠♦s ♦ r❡s✉❧t❛❞♦✳

  α (¯ x i ))

  ❖❜s❡r✈❡ q✉❡ det (D 6= 0 ♣♦✐s✱ s♦❜r❡ N✱ t❡♠♦s q✉❡

  X d¯ x i N = D α (¯ x i ) N dx α N | | |

  α j j i , ¯ u i , u

  ❡ {¯x }✱ {x } sã♦ ❞♦✐s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣♦ssí✈❡✐s s♦❜r❡ N✳ ❚❡♦r❡♠❛ ✷✳✷✳✽✳ ❙❡ ▼ é ❡q✉✐♣❛❞♦ ❝♦♠ ✉♠ ❛t❧❛s s❡♣❛r❛❞♦✱ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s {x, u}✱

  1

  (M, n) ❡♥tã♦ J é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s {x, u, p}✳ ❆❧é♠

  1

  (M, n)

  a

  ❞✐ss♦✱ ♣❛r❛ t♦❞♦ a ∈ M✱ J t❛♠❜é♠ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s {p}✳

  1

  (M, n) Pr♦✈❛✿ ❆s ❢✉♥çõ❡s x, u, p ❞❡t❡r♠✐♥❛♠ ✉♠ ❛t❧❛s ♣❛r❛ J ❡ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ sã♦ ❞✐❢❡♦♠♦r✜s♠♦s✳ ❉❡ ❢❛t♦✱ s❡ {¯x, ¯u, ¯p} sã♦ ♦✉tr❛s ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❡♠ ✉♠❛

  ✈✐③✐♥❤❛♥ç❛ ❝♦♠✉♠✱ ❡♥tã♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ss❛s ❝♦♦r❞❡♥❛❞❛s✱ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡✲ ✈❡♠ t❡r ❛ ❢♦r♠❛ {¯x = ¯x(x, u), ¯u = ¯u(x, u), ¯p = ¯p(x, u, p)}✱ ♦♥❞❡ (x, u) 7→ (¯x, ¯u) é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ✭♣♦✐s é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s s♦❜r❡ M✮ ❡ ¯p = ¯p(x, u, v) sã♦ ❞❡t❡r♠✐♥❛❞❛s ♣❡❧❛s ❢ór♠✉❧❛s ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r✳ ▲♦❣♦✱ ❝♦♠ ❛s ❝❛rt❛s ❞❡s❝r✐t❛s ♣❡❧❛s ❝♦♦r✲

  1

  (M, n) ❞❡♥❛❞❛s ❧♦❝❛✐s {x, u, p}✱ J é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣r♦✈❛✲s❡

  1

  (M, n) q✉❡✱ ♣❛r❛ t♦❞♦ a ∈ M✱ J a é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳

  1

  (M, n) 1,0 : ❖❜s❡r✈❡ q✉❡✱ ❝♦♠ r❡s♣❡✐t♦ ❛ ❡st❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛ J ✱ π

  1

  1 J (M, n)

  (M, n) −→ M é ✉♠❛ s✉❜♠❡rsã♦✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ t♦❞♦ θ ∈ J ✱ ❡①✐st❡ ✉♠❛

  (θ)

  1,0

  ✈✐③✐♥❤❛♥ç❛ U ⊂ M✱ a = π ∈ U ❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦

  1 −1

  π (U ) (M, n).

  1,0 ≃ U × J a −1

  (U ) ❉❡ ❢❛t♦✱ s❡ U é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ π 1,0 ♣♦❞❡ s❡r ❝♦❜❡rt❛

  ✸✻

  −1

  (U ) ❞♦ t✐♣♦ {x, u, p}✳ ❉❡ss❛ ❢♦r♠❛✱ ❝❛❞❛ ♣♦♥t♦ θ ∈ π 1,0 ✱ ❝♦♠ s✉❛s ❝♦♦r❞❡♥❛❞❛s p(θ)✱ ❞❡✲

  1 ′

  (M, n) t❡r♠✐♥❛ ✉♠ ú♥✐❝♦ ♣♦♥t♦ θ a ✭❝♦♠ ♠❡s♠❛s ❝♦♦r❞❡♥❛❞❛s p(θ)✮✳ P♦❞❡✲s❡ ♠♦str❛r ∈ J q✉❡ ❡ss❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❡✱ ♣♦rt❛♥t♦✱ é ❜❡♠ ❞❡✜♥✐❞❛ ❛

  ❛♣❧✐❝❛çã♦

  1 −1

  π (U ) (M, n)

  a 1,0 −→ U × J ′

  θ 1,0 (θ), θ ) 7−→ (π

  ❛ q✉❛❧✱ é ♣♦ssí✈❡❧ ♠♦str❛r✱ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✳ ❚❡♠♦s✱ ♣♦rt❛♥t♦✱ ♦ s❡❣✉✐♥t❡

  1

  : J (M, n)

  1,0

  ❚❡♦r❡♠❛ ✷✳✷✳✾✳ π −→ M é ✉♠ ✜❜r❛❞♦ ❞✐❢❡r❡♥❝✐❛✈❡❧✳ ❙❡ F : M −→ M é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✱ t♦❞❛ s✉❜✈❛r✐❡❞❛❞❡ n✲❞✐♠❡♥s✐♦♥❛❧ N ⊂ M

  é tr❛♥s❢♦r♠❛❞❛ ❡♠ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ F (N) ⊂ M✳ ❙❡ a ∈ M✱ F ✐♥❞✉③ ♥❛t✉r❛❧♠❡♥t❡ ✉♠❛

  1 (1)

  1

  1 (1)

  (M, n) (M, n) (θ) := ❛♣❧✐❝❛çã♦ F ❡♥tr❡ J a ❡ J F ✳ ❉❡ ❢❛t♦✱ s❡ θ = [N] a ✱ ❞❡✜♥✐♠♦s F

  (a)

  1 (1)

  1

  1

  [F (N )] : J (M, n) (M, n)

  F ✳ ❊st❛ ❛♣❧✐❝❛çã♦ ❞❡t❡r♠✐♥❛ ✉♠ ❛✉t♦♠♦r✜s♠♦ F −→ J ❞♦ (a)

  1

  1

  : J (M, n)

  1,0

  ✜❜r❛❞♦ π −→ M✳ F é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✳ ❖ ❧❡♠❛ ❛♥t❡r✐♦r ❢♦r♥❡❝❡ ✉♠❛

  (1)

  ❞❡s❝r✐çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❞❡ F ✳

  ✷✳✷✳✷ Pr♦❧♦♥❣❛♠❡♥t♦s ❞❡ s✉❜✈❛r✐❡❞❛❞❡s

  ❉❡♥♦t❡ ♣♦r Γ(M, n) ♦ ❡s♣❛ç♦ ❞❛s s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s ❞❡ M✳ ❙❡❥❛ N ∈ Γ(M, n)

  a N

  ✳ ❈♦♥s✐❞❡r❛♥❞♦✱ ❡♠ t♦❞♦ ♣♦♥t♦ a ∈ N✱ ♦ ❡s♣❛ç♦ T ✱ ❞❡✜♥✐♠♦s ❛ s✉❜✈❛r✐❡❞❛❞❡

  1 j (N ) := a N ) : a (M, n).

  1

  {(a, T ∈ N} ⊂ J ❊st❛ s✉❜✈❛r✐❡❞❛❞❡ é ❝❤❛♠❛❞❛ ♣r✐♠❡✐r♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ N✳

  (j (N )) = N

  1,0

  1

  ➱ ❢á❝✐❧ ✈❡r q✉❡ π ♣❛r❛ t♦❞♦ N ∈ Γ(M, n)✳

  n

  ❊♠ ❝♦♦r❞❡♥❛❞❛s s❡♣❛r❛❞❛s✱ s❡ N = {(x, f(x)) : x ∈ U ⊂ R } ❡♥tã♦ ∂f j (N ) = (x), x

  1 {(x, u, p) : u = f(x), p = ∈ U}.

  ∂x P♦rt❛♥t♦✱ ♦s ♣r✐♠❡✐r♦s ♣r♦❧♦♥❣❛♠❡♥t♦s ❞❡ s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s ❞❡ M sã♦

  1

  (M, n) s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s ❞❡ J q✉❡ t❛♠❜é♠ sã♦ ❣rá✜❝♦s ❞❛ ❢♦r♠❛ {(x, u, p) :

  n

  u = u(x), p = p(x), x ∈ U ⊂ R }✳ ❆s ❝♦♦r❞❡♥❛❞❛s {x} sã♦ ❝♦♦r❞❡♥❛❞❛s ✐♥t❡r♥❛s s♦❜r❡

  1 ∞

  (J (M, n)) ❡ss❛s ✈❛r✐❡❞❛❞❡s✳ ▲♦❣♦✱ s❡ F ∈ C ✱ s✉❛ r❡str✐çã♦ ❛ ✉♠ ♣r✐♠❡✐r♦ ♣r♦❧♦♥❣❛♠❡♥t♦ j (N )

  i θ (j (N ))

  1

  1

  t❡♠ ❛ ❢♦r♠❛ G(x) = F (x, u(x), p(x))✳ P♦rt❛♥t♦✱ ♦ ✈❡t♦r ξ ∈ T ✱ ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ♦♣❡r❛❞♦r ❞❡ ❞❡r✐✈❛çã♦ ❞✐r❡❝✐♦♥❛❧ ❛♦ ❧♦♥❣♦ ❞❛ i✲és✐♠❛ ❝✉r✈❛ ❝♦♦r❞❡♥❛❞❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ θ(x) = (x, u(x), p(x)) é t❛❧ q✉❡

  ✸✼ " ! #

  j

  X X ∂ ∂ ∂p ∂

  j s (G) = + ξ p (x) + i θ θ (F ). i (x) (x)

  | j |

  j

  ∂x i ∂u ∂x i

  s

  ∂p

  j s,j

  1

  1

  (M, n) θ (j (N )) θ (J (M, n))

  1

  ▲♦❣♦✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ J ✱ T ⊂ T é ❣❡r❛❞♦ ♣❡❧♦s ✈❡t♦r❡s

  j j

  X ∂ ∂f ∂ ∂ f ∂ + ξ i = + (x) (x) .

  2 X

  s

j

  ∂x i ∂x i ∂u ∂x i ∂x s ∂p

  j j j,s ✷✳✷✳✸ ❉✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥

  θ (j (N ))

  1

  ❯s❛♥❞♦ ❛ ❞❡s❝r✐çã♦ ❞❡ T ✱ ❞❛❞❛ ♥♦ ✜♥❛❧ ❞❛ s❡çã♦ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ❞❡✲

  ′ ′ a θ (j (N ))

  1

  ❞✉③✐r q✉❡✱ ❛♦ ✈❛r✐❛r ❞❡ N ∈ [N] ✱ ♦s ❡s♣❛ç♦s T ✈❛r✐❛♠ ❡ ❞❡s❝r❡✈❡♠ ✉♠ s✉❜❡s♣❛ç♦

  1

  1

  1

  1 θ (J (M, n))

  (M, n) C θ ⊂ T ✳ ■st♦ ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ C s♦❜r❡ J q✉❡ é ❝❤❛♠❛❞❛ ❞✐s✲ tr✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s✿

  X X ∂ ∂ ∂

  j j

  1 j

  • =< p , >= Ann p dx i C i {du − i }.

  j j

  ∂x ∂u

  i

  ∂p

  i j i

  ❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s✱ é ✐♠❡❞✐❛t♦ ✈❡r✐✜❝❛r q✉❡ Pr♦♣♦s✐çã♦ ✷✳✷✳✶✵✳ ❆ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥ ♥ã♦ é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳

  1

  1

  (M, n) ❆ ❞✐♠❡♥sã♦ ❞❡ J é n + m + n · m = n + m · (n + 1) ❡ ❛ ❞✐♠❡♥sã♦ ❞❡ C é n + m

  · n✳ ❊st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r ❛ ❢♦r♠❛ ❞❛s s✐♠❡tr✐❛s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛r✲ t❛♥✳ P♦r ❡♥q✉❛♥t♦✱ ♦❜s❡r✈❡♠♦s q✉❡ ❥á ❝♦♥❤❡❝❡♠♦s ✉♠❛ ❝❧❛ss❡ ❞❡ s✐♠❡tr✐❛s ✜♥✐t❛s✳ ❉❡

  (1)

  1

  : J (M, n) ❢❛t♦✱ ✈✐♠♦s q✉❡ ❝❛❞❛ ❞✐❢❡♦♠♦r✜s♠♦ F ❞❡t❡r♠✐♥❛ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ F −→

  (1)

  J (M, n) ✳

  (1)

  P❡❧❛ ♥❛t✉r❡③❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s {x, u, p}✱ ✁❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ ❡st❡ F é ✉♠❛ s✐♠❡tr✐❛

  1

  1

  1

  1

  (M, n) ❞❡ C ✳ P♦rt❛♥t♦✱ s❡ N ⊂ J é ✉♠❛ ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ C ✭❡✈✐❞❡♥t❡♠❡♥t❡

  1 1 (1)

  ) (N )

  1

  ❞❡ ❞✐♠❡♥sã♦ ♠❡♥♦r q✉❡ dim (C ✱ ♣♦rq✉❡ C é ♥ã♦ ✐♥t❡❣rá✈❡❧✮✱ F ❛✐♥❞❛ é ✉♠❛

  t

  ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧✳ ❯♠❛ ❛♥á❧♦❣❛ ❝♦♥s✐❞❡r❛çã♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ❛♦ ✢✉①♦ {A } ❞❡ ✉♠

  (1) (1)

  (M, n) ❝❛♠♣♦ X ∈ D(M)✳ ❊st❡ ✢✉①♦ ❞❡t❡r♠✐♥❛ ✉♠ ✢✉①♦ {A t ✳ ❊st❡ ✢✉①♦

  } s♦❜r❡ J

  (1)

  1

  (M, n) ❞❡✜♥❡ ✉♠ ❝❛♠♣♦ X s♦❜r❡ J q✉❡ s❡rá ❝❤❛♠❛❞♦ ♣r✐♠❡✐r♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞♦ ❝❛♠♣♦ X✳

  (1) 1 (1)

  ❈♦♠♦ ♦ ✢✉①♦ ❞♦ ❝❛♠♣♦ X ♣r❡s❡r✈❛ C ✱ ♦ ❝❛♠♣♦ X é ✉♠❛ s✐♠❡tr✐❛ ✐♥✜♥✐t❡s✲

  (1) 1 (1) 1 t

  ✐♠❛❧ ❞❡ C ✳ P♦rt❛♥t♦✱ ♦ ✢✉①♦ {A } ❞❡ X tr❛♥s❢♦r♠❛ ✈❛r✐❡❞❛❞❡s ❞❡ C ❡♠ ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s✳

  =

  1

  ❯♠ ❡①❡♠♣❧♦ ✐♠♣♦rt❛♥t❡ ❞❡ ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s é ❛q✉❡❧❡ ❞❛s ✈❛r✐❡❞❛❞❡s N j (N )

  1

  ✸✽

  ✷✳✷✳✹ ❙✐♠❡tr✐❛s ✐♥✜♥✐t❡s✐♠❛✐s ❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥

  1

  ◆❡st❛ s❡çã♦ ♠♦str❛r❡♠♦s ❛ ❡str✉t✉r❛ ❞❛s s✐♠❡tr✐❛s ✐♥✜♥✐t❡s✐♠❛✐s ❞❡ C ✳ ❊♠

  (1)

  ♣❛rt✐❝✉❧❛r✱ ♠♦str❛r❡♠♦s ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ♣r✐♠❡✐r♦ ♣r♦❧♦♥❣❛♠❡♥t♦ X ❛ ♣❛rt✐r ❞❡ X✳

  1

  (M, n)) ❈♦♥s✐❞❡r❡ Y ∈ D(J ❡ ❛ss✉♠❛ q✉❡✱ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s {x, u, p}✱ Y t❡♠ ❛ ❢♦r♠❛

  X X

  X ∂ ∂ j ∂ + ξ + Y = i η j µ .

  i j j

  ∂x i ∂u ∂p

  i i j i,j

  P P

  m m

  1

  1

  1

  = Ann = du p dx i , ..., ω = du p dx i ❊♥tã♦✱ s❡♥❞♦ q✉❡ C

  1 i 1 i

  {ω − i − i }✱ ❛ P

  1

  1 Y Y ω j = α js ω s js

  ❝♦♥❞✐çã♦ q✉❡ L C ⊆ C ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ L s ✱ ♦♥❞❡ ❛s α sã♦

  1

  (M, n) ❢✉♥çõ❡s s♦❜r❡ J ✳ ❊st❛ ❝♦♥❞✐çã♦ t❛♠❜é♠ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛

  Y ω j , ..., ω m

  1 L ≡ 0 mod{ω }. j = Y yω j

  ❆❣♦r❛✱ ♣♦♥❞♦ φ ✱ t❡♠♦s q✉❡

  Y ω j = d(Y yω j ) + Y ydω j

  L !

  X X

  j j

  • = dφ j ξ i dp µ dx i

  i − i i i

  X X

  X X

  X ∂φ j ∂φ j ∂φ j j j

  s s + = dx + i du dp ξ µ + i dp dx i . i i − i s

  ∂x i ∂u s ∂p

  i j s i,s i i

  P

  s s

  • = ω s p dx i ▼❛s✱ ❡s❝r❡✈❡♥❞♦ du i i ✱ ♦❜t❡♠♦s

  !

  X X

  X X

  X ∂φ j ∂φ j j ∂φ j j ∂φ j

  s s + + ω p + + Y j = dx i dp ξdp ω s . i i i i

  L − µ s

  s

  ∂x i ∂u ∂p ∂u s

  i i s i,s i s Y ω j 1 , ..., ω m

  ▲♦❣♦✱ L ≡ 0 mod{ω } s❡✱ ❡ s♦♠❡♥t❡ s❡✿

  j P

∂ s ∂

  µ = D + i (φ j ) i := p s

  i ✱ ❝♦♠ D s i ❀

∂x i ∂u

∂φ j s

  = 0

  ∂p s❡ s 6= j ✭q✉❛♥❞♦ m > 1✮❀ i

  ∂φ ∂φ j j ∂φ α j j α

  = i i = = ✭∀j ∈ {1, ..., m} ❡✱ ♣♦rt❛♥t♦✱ −ξ ✱ i, α ∈ {1, ..., m}✮✳

  −ξ

  ∂p ∂p ∂p i i i

  P

  j j := Y yω j = η j ξ s p

  ❉❡ss❛ ❢♦r♠❛✱ q✉❛♥❞♦ m > 1✱ t❡♠♦s q✉❡ ❛s ❢✉♥çõ❡s φ − s s

  j

  sã♦ ❧✐♥❡❛r❡s ♥❛s p i ✬s✳ P❛r❛ ❡st❡ ✜♠✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡✱ ♣❡❧❛ (iii)✱ ❛s s❡❣✉♥❞❛s ❞❡r✐✈❛❞❛s

  j s = ξ s (x, u)

  ❝♦♠ r❡❧❛çã♦ às p i ✬s sã♦ ♥✉❧❛s✳ P♦rt❛♥t♦✱ ♥♦ ❝❛s♦ m > 1✱ ξ ✳ ▼❛s t❛♠❜é♠

  ∂φ j s

  η j = η j (x, u) = 0 j ✱ q✉❛♥❞♦ m > 1✳ ❉❡ ❢❛t♦✱ ∂p ✱ s❡ s 6= j✱ ✐♠♣❧✐❝❛ q✉❡ φ só ♣♦❞❡ ❞❡♣❡♥❞❡r

  i j ∂φ ∂η P ∂η P j j ξ s j ξ s ∂ξ s j j j j j j j j

  = i = p i p = 0 = 0 ❞❛s p i ✬s✳ ▼❛s −ξ − s − ξ ✳ ▲♦❣♦✱ − s ❡✱ s❡♥❞♦ ✱

  s s ∂p ∂p p ∂p p ∂p i i i i i i

  ∂η j j

  = 0 ♦❜t❡♠♦s q✉❡ ✳

  ∂p i

  ❊st❡s ❝á❧❝✉❧♦s ♠♦str❛♠ q✉❡✱ ✐♥tr♦❞✉③✐♥❞♦ ♦s ♦♣❡r❛❞♦r❡s

  X ∂ ∂

  s

  D + i := p ,

  i s

  ∂x ∂u

  i

s

  ✸✾

  1

  ❚❡♦r❡♠❛ ✷✳✷✳✶✶✳ ❙❡ m = 1✱ ❛s s✐♠❡tr✐❛s ❞❡ C tê♠ ❛ ❢♦r♠❛ ❧♦❝❛❧ !

  X X

  X ∂φ ∂ ∂φ ∂ ∂

  Y = φ p (φ) , + + i D i − −

  ∂p i ∂x i ∂p i ∂u ∂p i

  i i i

  P p i dx i ❝♦♠ φ := Y yω✱ ω = du − ✳

  i

  ❙❡ m > 1✱ ❛s s✐♠❡tr✐❛s tê♠ ❛ ❢♦r♠❛ ❧♦❝❛❧

  X ∂

  Y = X + D i (φ j ) ,

  j

  ∂p

  i i,j

  ❝♦♠ X ❝❛♠♣♦ ❡♠ M ❞❛ ❢♦r♠❛

  X X ∂ ∂

  X = ξ + i η j

  j

  ∂x i ∂u

  i j

  , ..., φ m ) ❡ φ = (φ

  1 ❞❡✜♥✐❞❛ ♣♦r ♠❡✐♦ ❞❛s ❡q✉❛çõ❡s

  X

  j j φ j = Y yω j , ω j = du p dx i . i

  −

  i

  ❆ ❢✉♥çã♦ φ s❡ ❝❤❛♠❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❛ s✐♠❡tr✐❛✳ ◆♦ ❝❛s♦ m > 1✱ ❛s s✐♠❡tr✐❛s Y sã♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛s ♣♦r ✉♠ ❝❛♠♣

  (1)

  1 X

  (M, n) ❡♠ M✳ P♦rt❛♥t♦✱ ♥❡st❡ ❝❛s♦✱ Y ❝♦✐♥❝✐❞❡ s❡♠♣r❡ ❝♦♠ ♦ ❝❛♠♣♦ X s♦❜r❡ J

  (1)

  1

  (M, n) t

  t

  ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ ✢✉①♦ {A } q✉❡ ♣r♦❧♦♥❣❛ ❛ J ♦ ✢✉①♦ {A } ❞❡ X✳ ❚❡♠♦s✱ ♣♦rt❛♥t♦✱ ♦ s❡❣✉✐♥t❡

  P P

  ∂ ∂ (1)

  ξ i η j + j ❈♦r♦❧ár✐♦ ✷✳✷✳✶✷✳ ❖ ♣r✐♠❡✐r♦ ♣r♦❧♦♥❣❛♠❡♥t♦ X ❞❡ ✉♠ ❝❛♠♣♦ X =

  i j ∂x i ∂u

  s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M é

  X ∂

  (1)

  X = X + D x (φ j ) ,

  i j ✭✷✳✺✮

  ∂p

  i i,j j = Xyω j

  ♦♥❞❡ φ ✳

  ✷✳✷✳✺ ❊s♣❛ç♦s ❞❡ ❥❛t♦s ❞❡ s❡çõ❡s ❞❡ ✉♠ ✜❜r❛❞♦

  ❆ t❡♦r✐❛ ❞♦s ❡s♣❛ç♦s ❞❡ ❥❛t♦s ❞❡ s✉❜✈❛r✐❡❞❛❞❡s é ❛ ♠❛✐s ❣❡r❛❧ ❡ ♣❡r♠✐t❡ tr❛t❛r ❛s♣❡❝t♦s ❣❧♦❜❛✐s q✉❡ s❡r✐❛♠ ♠❛✐s ❝♦♠♣❧✐❝❛❞♦s ❝♦♠ ♦✉tr♦s t✐♣♦s ❞❡ ❡s♣❛ç♦s ❞❡ ❥❛t♦s✳

  ❯♠ ♦✉tr♦ t✐♣♦ ❞❡ ✜❜r❛❞♦ ❞❡ ❥❛t♦s✱ q✉❡ ✉s❛r❡♠♦s ❡♠ s❡❣✉✐❞❛✱ é ❛ ❞♦s ✜❜r❛❞♦s ❞❡ ❥❛t♦s ❞❡ s❡çõ❡s ❞❡ ✉♠ ✜❜r❛❞♦ π : M −→ B ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ dim(B) = n ❡ dim(M) =

  1

  m + n (π) ✳ ❊st❡s ❡s♣❛ç♦s sã♦ ❞❡♥♦t❛❞♦s ♣♦r J ❡✱ ♣♦r ❞❡✜♥✐çã♦✱ ❡st❡ é ♦ ❡s♣❛ç♦ ❞❛s ❝❧❛ss❡s

  ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ s❡çõ❡s ❝✉❥♦s ❣rá✜❝♦s tê♠ ♦ ♠❡s♠♦ ❡s♣❛ç♦ t❡♥❣❡♥t❡ ❡♠ ❛❧❣✉♠ ♣♦♥t♦✳

  ✹✵ ❞✐ss♦✱ é ❢✉♥❞❛♠❡♥t❛❧ ♦❜s❡r✈❛r q✉❡✱ ♥❡st❡ ❝❛s♦✱ ❛s ❝♦♦r❞❡♥❛❞❛s s❡♣❛r❛❞❛s sã♦ s✐♠♣❧❡s♠❡♥t❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ ✉♠❛ tr✐✈✐❛❧✐③❛çã♦ ❧♦❝❛❧ ❞♦ ✜❜r❛❞♦ π : M −→ B✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s

  1 , ..., x n

  ♣❡♥s❛r q✉❡ ❛s {x } sã♦ ❝♦♦r❞❡♥❛❞❛s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ ✉♠ ♣♦♥t♦ b ∈ B

  m

  1 −1

  , ..., u (U )

  ❡ ❛s ❝♦♦r❞❡♥❛❞❛s {u } sã♦ ❝♦♦r❞❡♥❛❞❛s s♦❜r❡ ❛s ✜❜r❛s✱ ❝♦♥t✐❞❛s ❡♠ π ✱ ❡♠

  −1

  U (U ) ✉♠❛ ✈✐③✐♥❤❛♥ç❛ e ⊂ π ❞❡ ✉♠ ♣♦♥t♦ b t❛❧ q✉❡ π(b) = b✳ ❈♦♠ ❡st❛s ❝♦♦r❞❡♥❛❞❛s✱ ❛s

  U s❡çõ❡s tê♠ ❣rá✜❝♦s ❞❛ ❢♦r♠❛ {(x, u) : u = f(x), x ∈ U} ⊂ e ✳

  1

  (π) ❈♦♠♦ J é ❝♦♥str✉í❞♦ ❝♦♥s✐❞❡r❛♥❞♦ ❛♣❡♥❛s ❛s s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s

  1

  (M, n) q✉❡ sã♦ ❣rá✜❝♦s ❞❡ s❡çõ❡s✱ t❡♠♦s ✉♠❛ ✐♠♣♦rt❛♥t❡ ❞✐❢❡r❡♥ç❛ ❝♦♠ J ✳ ❉❡ ❢❛t♦✱ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ♥❡♠ s❡♠♣r❡ tr❛♥s❢♦r♠❛ ✉♠❛ s❡çã♦ ❡♠ ♦✉tr❛ s❡çã♦✳ P♦rt❛♥t♦✱ ❛♣❡♥❛s

  (1)

  ❧♦❝❛❧♠❡♥t❡ ✭❡✱ ♣♦ss✐✈❡❧♠❡♥t❡✱ ❡①❝❧✉✐♥❞♦ ❛❧❣✉♥s ♣♦♥t♦s✮✱ ♣♦❞❡♠♦s ❞❡✜♥✐r F ✳

  1

  (M, n) ❆ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥ é ❞❡✜♥✐❞❛ ❞♦ ♠❡s♠♦ ♠♦❞♦ q✉❡ ❡♠ J ❡ t❡♠ ❛

  ♠❡s♠❛ ❢♦r♠❛ ❝♦♦r❞❡♥❛❞❛✳ ❆❧é♠ ❞✐ss♦✱ ♦s ❝á❧❝✉❧♦s s♦❜r❡ ❛s s✐♠❡tr✐❛s ❛✐♥❞❛ ✜❝❛♠ ✈á❧✐❞♦s✳

  ✷✳✷✳✻ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ s♦❧✉çõ❡s ❡ s✐♠❡tr✐❛s i

  ❯♠ s✐st❡♠❛ ❞❡ h ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡♠ n ✈❛r✐á✈❡✐s ✐♥❞❡♣❡♥❞❡♥t❡s {x } ❡♠ m

  ❢✉♥çõ❡s ❞❡♣❡♥❞❡♥t❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠

  j j

  ∂u ∂u

  j h j

  1 F (x i , u (x), ) = 0, ..., F (x i , u (x), ) = 0

  ✭✷✳✻✮ ∂x i ∂x i

  h

  1

  , ..., F ◗✉❛♥❞♦ ❛s ❢✉♥çõ❡s F s❛t✐s❢❛③❡♠ ❤✐♣ót❡s❡s ♥❡❝❡ssár✐❛s ❞❡ r❡❣✉❧❛r✐❞❛❞❡✱

  ♣♦❞❡✲s❡ tr❛t❛r ✷✳✻ ❝♦♠♦ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ E (n + m + m · n − h)✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ ✉♠

  1

  (E, n) ❡s♣❛ç♦ ❞❡ ❥❛t♦s J ❝♦♠ ❝♦♦r❞❡♥❛❞❛s {x, u, p}✳

  1

  1

  ❉❡♥♦t❛r❡♠♦s ❝♦♠ C ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥ ✐♥❞✉③✐❞❛ ❡♠ E ♣♦r C ✱ ✐st♦ é✱ ❛

  E

  1

  1

  ) = θ

  θ

  ❞✐str✐❜✉✐çã♦ t❛❧ q✉❡ (C θ C ∩ T E✱ ♣❛r❛ t♦❞♦ θ ∈ E✳

  E

  ❆s s♦❧✉çõ❡s ❞❡ ✷✳✻ sã♦ s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s ❞❡ E q✉❡✱ ♣♦rt❛♥t♦✱ sã♦

  1 (N )

  ❧♦❝❛❧♠❡♥t❡ ❞❛ ❢♦r♠❛ N = {(x, u) : u = f(x)} ❡ t❛✐s q✉❡ j ⊂ E✳ ❈♦♠♦ ❡st❡s

  1

  (N ) (N )

  1

  1

  ♣r♦❧♦♥❣❛♠❡♥t♦s j sã♦ ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ❞❡ C ✱ ❡♥tã♦ ❛ ❝♦♥❞✐çã♦ j ⊂ E ✐♠♣❧✐❝❛

  

1

  (N )

  1

  q✉❡ j é t❛♠❜é♠ ✈❛r✐❡❞❛❞❡ ✐♥t❡❣r❛❧ ❞❡ C ✳ ❊st❡ é ♦ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞❛s s♦❧✉çõ❡s

  

E

  ❞❡ ✉♠ s✐st❡♠❛ ❞♦ t✐♣♦ ✷✳✻✳

  1

  ❆s s✐♠❡tr✐❛s ✐♥✜♥✐t❡s✐♠❛✐s ❞❡ ✷✳✻ sã♦ ❝❛♠♣♦s ✈❡t♦r✐❛✐s ❡♠ E q✉❡ ♣r❡s❡r✈❛♠ C ✳

  E

  ❉❡ss❛ ❢♦r♠❛✱ ❛s s✐♠❡tr✐❛s ♣♦❞❡♠ s❡r ❞❡ ❞♦✐s t✐♣♦s ♣r✐♥❝✐♣❛✐s✿ ❡①t❡r♥❛s ✭♦✉ ❞❡ ▲✐❡✮ ❡ ✐♥t❡r♥❛s ✭♦✉ ❞❡ ❈❛rt❛♥✮✳ ❆s s✐♠❡tr✐❛s ❡①t❡r♥❛s sã♦ ❛q✉❡❧❛s q✉❡ s❡ ♦❜t❡♠ ❞❡ s✐♠❡tr✐❛s

  1

  ❞❡ C t❛♥❣❡♥t❡s ❛ E✳ ◆❡ss❡ ❝❛s♦✱ ❛ r❡str✐çã♦ ❛ E ❞❡ ✉♠❛ t❛❧ s✐♠❡tr✐❛ é ♦ q✉❡ s❡ ❝❤❛♠❛ s✐♠❡tr✐❛ ❡①t❡r♥❛✳ ❆s s✐♠❡tr✐❛s ✐♥t❡r♥❛s sã♦ s✐♠❡tr✐❛s q✉❡ ♥ã♦ s❡ ♦❜t❡♠ r❡str✐♥❣✐♥❞♦

  1

  ❛ E ✉♠❛ s✐♠❡tr✐❛ ❞❡ C t❛♥❣❡♥t❡ ❛ E✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ m > 1✱ s❛❜❡♠♦s q✉❡ t♦❞❛s ❛s s✐♠❡tr✐❛s ❡①t❡r♥❛s sã♦ ❞❛ ❢♦r♠❛

  (1)

  X

  ✹✶ ❊st❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ✷✳✻ ✈❛❧❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ ❛s s✉❜✈❛r✐❡❞❛❞❡s q✉❡ q✉❡r✲

  ❡♠♦s ❞❡s❝r❡✈❡r ♣♦r ♠❡✐♦ ❞❡ ✷✳✻ ✭✐st♦ é✱ ❛q✉❡❧❛s q✉❡ sã♦ ❝❛r❛❝t❡r✐③❛❞❛s ♣❡❧❛ ❝♦♥❞✐çã♦ ✷✳✻✮

  1

  (π) sã♦ s❡çõ❡s ❞❡ ✉♠ ✜❜r❛❞♦ π : E −→ B✳ ◆❡st❡ ❝❛s♦✱ E ⊂ J ✳

  ✷✳✷✳✼ ❊q✉❛çã♦ ❞❛s ❣❡♦❞és✐❝❛s

  ❙❡ (M, g) é ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣s❡✉❞♦✲❘✐❡♠❛♥♥✐❛♥❛ n✲❞✐♠❡♥s✐♦♥❛❧✱ ✉♠❛ ❝✉r✈❛ γ(t)

  i i (t))

  q✉❡✱ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s {x }✱ t❡♠ ❛ ❝❛rt❛ γ(t) = (x é ✉♠❛ ❣❡♦❞és✐❝❛ s❡ s✉❛s ❝♦♠♣♦♥❡♥t❡s s❛t✐s❢❛③❡♠ ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s

  n

  d x i dx h dx k

  i

  (t) + Γ (t) (t) i

  hk

  ∈ {1, ..., n} ✭✷✳✼✮

  2

  dt dt dt

  h,k =1

  ♦♥❞❡

  n

  X 1 ∂g ∂g ∂g

  hs ks hk i is

  Γ = g +

  hk −

  2 ∂x k ∂x h ∂x s

  s =1

  sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❈❤r✐st♦✛❡❧ ❞❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ▲❡✈✐✲❈✐✈✐t❛✳ ❆s ❢✉♥çõ❡s

  ij −1

  g )

  ij

  sã♦ ♦s ❡❧❡♠❡♥t♦s ❞❛ ♠❛tr✐③ (g ✳ ❆s ❡q✉❛çõ❡s ❡♠ ✷✳✼ s❡ ❝❤❛♠❛♠ ❡q✉❛çõ❡s ❞❛s ❣❡♦❞és✐❝❛s ✭✈✳ ❬✶✽✱ ✸✼❪ ♣❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s✮✳

  i (t))

  ❯♠❛ ❝✉r✈❛ γ(t) = (x s♦❜r❡ M ❞❡✜♥✐❞❛ ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ I ⊂ R ♣♦❞❡ s❡r

  i (t))

  ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠ ✉♠❛ s❡çã♦ ❧♦❝❛❧ t 7−→ (t, x ❞♦ ✜❜r❛❞♦ tr✐✈✐❛❧ π : R × M −→ R✳ ❆s ❣❡♦❞és✐❝❛s sã♦ s❡çõ❡s ♣❛rt✐❝✉❧❛r❡s ❞❡st❡ ✜❜r❛❞♦✳

  ❙❡♥❞♦ ✷✳✼ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈ê✲❧♦ ❝♦♠♦

  2

  (π) ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞♦ ❡s♣❛ç♦ J ❞♦s 2✲❥❛t♦s ❞❡ s❡çõ❡s ❞♦ ✜❜r❛❞♦ π✳ ❊ss❡ ✜❜r❛❞♦ ♣♦❞❡

  1

  1

  (π ) : J (π )

  1

  

1

  1

  s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❡J ✱ s❡♥❞♦ π −→ R✳

  1 i , ˙x i (π 1 )

  ❙❡✱ ♥❡st❡ ♠♦♠❡♥t♦✱ ❞❡♥♦t❛r♠♦s ❝♦♠ {t, x } ❛s ❝♦♦r❞❡♥❛❞❛s ♥❛t✉r❛✐s ❡♠ J

  2

  1

  , ˙x , ¨ x (π ) = J (π )

  i i i

  2

  1

  ❡ ❝♦♠ {t, x } ❛s ❝♦♦r❞❡♥❛❞❛s ♥❛t✉r❛✐s ❡♠ J ✳ ❈♦♠ ❡ss❛s ❝♦♦r❞❡♥❛❞❛s✱

  1

  (π )

  1

  ✷✳✼ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ❛ s✉❜✈❛r✐❡❞❛❞❡ E ⊂ J ✱ ❧♦❝❛❧♠❡♥t❡ ❞❡s❝r✐t❛ ♣❡❧♦ s✐st❡♠❛

  i i + Γ ˙x h ˙x k = 0 hk

  E := {(t, x, ˙x, ¨x) : ¨x }. ✭✷✳✽✮

  1 i (t))

  ▲♦❣♦✱ ✉♠❛ s❡çã♦ ❞❡ π ✱ ❧♦❝❛❧♠❡♥t❡ ❞❡s❝r✐t❛ ❝♦♠♦ t 7−→ (t, x ✱ é ✉♠❛ s♦❧✉çã♦ (t), ˙x (t), ¨ x(t))

  i i

  ❞❡ ✷✳✻ s❡ ♦ s❡✉ s❡❣✉♥❞♦ ♣r♦❧♦♥❣❛♠❡♥t♦ t 7−→ (t, x ❡stá ❝♦♥t✐❞♦ ❡♠ E✳ ◆❛ ♣ró①✐♠❛ s❡çã♦ ❞❡s❝r❡✈r❡♠♦s ❛s ❡q✉❛çõ❡s ✷✳✻ ❝♦♠♦ ❡q✉❛çõ❡s ✈❛r✐❛❝✐♦♥❛✐s ✉s✲

  ❛♥❞♦ ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥✳ P❛r❛ ❡ss❡ ✜♠✱ ♥ã♦ s❡rá ♥❡❝❡ssár✐♦ ❝✐t❛r ♦ ✜❜r❛❞♦

  ✳

  ✹✷

  ✷✳✸ ❋♦r♠❛❧✐s♠♦ ❞❡ ❈❛rt❛♥ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧

  ◆❡st❛ s❡çã♦ ❞❛r❡♠♦s ✉♠❛ ❢♦r♠✉❧❛çã♦ ✐♥✈❛r✐❛♥t❡ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ❝♦♠ ❡①tr❡♠♦s ✜①♦s ❡♠ ✉♠❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳ P❛r❛ ❡ss❡ ✜♠✱ ✉s❛r❡♠♦s ♦ ❢♦r♠❛❧✐s♠♦ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥ q✉❡ é✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡✱ ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ♦s ♥♦ss♦s ♦❜❥❡t✐✈♦s ♣♦st❡r✐♦r❡s✳ ❊♥tr❡t❛♥t♦✱ ♣❛r❛ t♦r♥❛r ♠❛✐s ❝❧❛r❛s ❛s ❞✐❢❡r❡♥ç❛s ❡♥tr❡ ❡st❛ ❛❜♦r❞❛❣❡♠ ✐♥✈❛r✐❛♥t❡ ❡ ❛ ❝❧áss✐❝❛✱ ❝♦♠❡ç❛♠♦s ❞❡❞✉③✐♥❞♦ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲ ▲❛❣r❛♥❣❡ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❡❧❡♠❡♥t❛r ❜❛s❡❛❞❛ ♥♦ ✉s♦ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✳ P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ❛ r❡s♣❡✐t♦ ❞♦ ❢♦r♠❛❧✐s♠♦ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❣❡r❛❧✱ ✈✐❞❡ ❬✸✵❪✳

  ◆❡st❛ s❡çã♦✱ π ❞❡♥♦t❛rá ✉♠ ✜❜r❛❞♦ π : E

  −→ I ♦♥❞❡ E é ✉♠❛ ✈❛r✐❡❞❛❞❡ (n + 1)✲❞✐♠❡♥s✐♦♥❛❧ ❡ I ⊂ R é ✉♠ ✐♥t❡r✈❛❧♦✳ ❈♦♠ t ❞❡♥♦t❛r❡♠♦s

  i

  ✉♠❛ ❝♦♦r❞❡♥❛❞❛ ❡♠ I ❡ {x } ❞❡♥♦t❛rã♦ ❝♦♦r❞❡♥❛❞❛s ♥❛s ✜❜r❛s ❞❡ π✳ ❆s ❝♦♦r❞❡♥❛❞❛s

  1 1 ∞

  (π) i , v i (J (π)) ♥❛t✉r❛✐s ❡♠ J s❡rã♦ ❞❡♥♦t❛❞❛s ♣♦r {t, x }✳ ❯♠❛ ❢✉♥çã♦ L ∈ C s❡rá ❝❤❛♠❛❞❛ ▲❛❣r❛♥❣❡❛♥❛✳

  ❙❡❥❛ ω ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ s♦❜r❡ I✳ P♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ❝❛❞❛ ❧❛❣r❛♥❣❡❛♥❛ L ✉♠ ❢✉♥❝✐♦♥❛❧ s♦❜r❡ ♦ ❡s♣❛ç♦ Γ(π) ❞❛s s❡çõ❡s ❞♦ ✜❜r❛❞♦ π

  Z

  ∗ A (I) = j (s) (L)ω. s

  

1

i τ

  ❯♠❛ ✈❛r✐❛çã♦ s ❞❡ ✉♠❛ s❡çã♦ s✱ ❝♦♠ ❡①tr❡♠✐❞❛❞❡s ✜①❛s✱ é ✉♠❛ ❢❛♠í❧✐❛ ❛ 1✲

  t

  ♣❛râ♠❡tr♦ τ ❞❡ s❡çõ❡s ❞❡ π t❛❧ q✉❡ s = s ❡ s ❝♦✐♥❝✐❞❡ ❝♦♠ s ♥♦s ❡①tr❡♠♦s ❞♦ ✐♥t❡r✈❛❧♦ ❞❡ ❞❡✜♥✐çã♦ ❞❡ s✳

  d τ A L [s τ ] = 0

  ❯♠❛ s❡çã♦ s é L✲❝rít✐❝❛ ✭♦✉ L✲❡st❛❝✐♦♥ár✐❛✮ s❡ =0 ✱ ♣❛r❛ t♦❞❛

  dτ | τ

  ✈❛r✐❛çã♦ s ❞❡ s ❝♦♠ ❡①tr❡♠✐❞❛❞❡s ✜①❛s✳ ❆s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ r❡♣r❡s❡♥t❛♠ ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s✱ ❡ s✉✜❝✐❡♥t❡s✱

  ♣❛r❛ s❡çõ❡s L✲❝rít✐❝❛s✳ ❙❡ s ∈ Γ(π)✱ s : [a, b] −→ E✱ ♣♦❞❡ s❡r ❞❡s❝r✐t❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ♥❛s ❝♦♦r❞❡♥❛❞❛s

  i

  ❧♦❝❛✐s {x }✱ ❡♥tã♦✱ ❡s❝♦❧❤❡♥❞♦ ω = dt✱ t❡♠♦s q✉❡ Z b A L [s] := L(t, x(t), v(t))dt.

  a

  ◆❡ss❡ ❝❛s♦✱ ✉♠❛ ✈❛r✐❛çã♦ ❞❡ s✱ ♣❛r❛ ✈❛❧♦r❡s ♣❡q✉❡♥♦s ❞❡ τ✱ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❡♠ ❝♦♦r❞❡♥❛❞❛s ♥❛ ❢♦r♠❛ s τ (t) = (t, x i (t) + τ h i (t)) ,

  • P

  1,0

  J

  , ∀a ∈

  1,0 ∗a

  ) = Kern π

  a (π 1,0

  (π) −→ I✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ V

  1

  : J

  ) ✱ ♦♥❞❡ π

  (π) ❀

  1,0 ∗

  ) = Kern (π

  1,0

  · V (π

  (π) ❀

  1

  ∀p ∈ J

  

p (π) = Kern π

∗ p ,

  ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ V

  1

  · V

  ❘❡❧❡♠❜r❛♠♦s q✉❡ π : E −→ I é ✉♠ ✜❜r❛❞♦ ♦♥❞❡ I ⊂ R✳ ❚❛♠❜é♠✱ ❡st❛❜❡❧❡❝❡♠♦s ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿

  ✳ ❚❡♠♦s ❛ s❡❣✉✐♥t❡ Pr♦♣♦s✐çã♦ ✷✳✸✳✶✳ ❊①✐st❡ ✉♠❛ ú♥✐❝❛ ✶✲❢♦r♠❛ ω

  1

  w

  ∗

  (γ)

  1

  (π) t❛❧ q✉❡✿

  1

  ❡♠ J

  

1

  1 ) = (V (π 1 )) ∗

  ∗

  (π

  

  ❀ V

  ∗

  (π 1,0 ) = (V (π 1,0 ))

  ∗

  ❀ V

  ∗

  (π) = (V (π))

  · V (π) = Kern (π ∗ )

  ❖ ❢♦r♠❛❧✐s♠♦ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥ é ❜❛s❡❛❞♦ ❡♠ ✉♠❛ ✶✲❢♦r♠❛ Θ ❝❤❛♠❛❞❛ ❢♦r♠❛ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥✳ P❛r❛ ❞❡✜♥✐r ❡st❛ ❢♦r♠❛✱ ♣r❡❝✐s❛♠♦s ❞❡ ❛❧❣✉♠❛s ❝♦♥str✉çõ❡s ♣r❡❧✐♠✲ ✐♥❛r❡s✳ ❈♦♠❡ç❛♠♦s ✜①❛♥❞♦ ❛s ♥♦t❛çõ❡s ♣r✐♥❝✐♣❛✐s✳

  ✹✸ ❝♦♠ h

  a

  ∂L ∂x i

  a

  = Z b

  b a

  (t, x(t), v(t))h i (t) |

  (t, x(t), v(t)) h i (t)dt + ∂L ∂v i

  ∂L ∂v i

  (t, x(t), v(t)) − d dt

  ∂L ∂x i

  = Z b

  ∂L ∂v i

  ∂v i (t, x(t), v(t)) ˙h(t) dt

  (t, x(t), v(t))h i (t) + ∂L

  ∂L ∂x i

  a

  A L [s τ ] = Z b

  τ =0

  |

  ✳ P♦rt❛♥t♦✱ d dτ

  i (a) = h i (b) = 0

  (t, x(t), v(t)) − d dt

  (t, x(t), v(t)) h i (t)dt ∀h(t). ▲♦❣♦✱ s(t, x(t)) é L✲❝rít✐❝❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ s❛t✐s❢❛③ ❛s ❡q✉❛çõ❡s

  ✷✳✸✳✶ ❋♦r♠❛ ❞❡ ❈❛rt❛♥ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡

  ✱ ❛s ❡q✉❛çõ❡s ✷✳✾ ♣♦❞❡♠ s❡r ❡s❝r✐t❛s ♥❛ ❢♦r♠❛

  g ij v i v j ✳

  2

  1

  ❆s ❡q✉❛çõ❡s ❞❛s ❣❡♦❞és✐❝❛s s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ (M, g) sã♦ ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ s♦❜r❡ ✉♠ ✜❜r❛❞♦ π : I × M −→ I ❡ ❝♦♠ ❢✉♥çã♦ ❧❛❣r❛♥❣❡❛♥❛ L =

  = 0, ∀i ∈ {1, ..., n}. ✭✷✳✶✵✮

  ∂L ∂v i

  t

  − D

  ∂L ∂x i

  i ∂ ∂x i

  ∂L ∂x i

  ∂ ∂t

  =

  t

  (π) ✱ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s (t, x, v) ❡ ✉s❛♥❞♦ D

  1

  ❊st❛s sã♦ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✳ ❊♠ J

  (t, x(t), v(t)) = 0, ∀i ∈ {1, ..., n}. ✭✷✳✾✮

  ∂L ∂v i

  (t, x(t), v(t)) − d dt

  = 0

  ✹✹ (ξ) = (π ) (ξ) )

  1 1,0

  1

  ✭✐✐✮ w ∗ ✱ ♣❛r❛ t♦❞♦ ξ ∈ V (π ✳

  1

  1 Pr♦✈❛✿ ✭✐✮❯♥✐❝✐❞❛❞❡✳ ❙✉♣♦♥❤❛ q✉❡ ω ❡ ω sã♦ ❞✉❛s ✶✲❢♦r♠❛s s❛t✐s❢❛③❡♥❞♦ ❛s

  1

p (J (π)) (s) π (ξ)

1 1 ∗

  ❝♦♥❞✐çõ❡s ❛❝✐♠❛✳ ❙❡ ξ ∈ T ✱ ❡♥tã♦ ξ − j ∗ ∈ V (π)✱ ♣❛r❛ t♦❞❛ s❡çã♦ s ∈ Γ(π)✳ ▲♦❣♦✱ ✉t✐❧✐③❛♥❞♦ ❛s ❤✐♣ót❡s❡s ✐✮ ❡ ✐✐✮✱

  (ω )(ξ) = (ω )(ξ (s) π ∗ (ξ)) + (ω )(j (s) π ∗ (ξ))

  1

  1

  1

  1

  1

  1

  1

  1

  1

  1

  − ω − ω − j ∗ − ω ∗ = 0.

  = ω ▲♦❣♦✱ ω

  1 1 ✳ 1 (π) 1 (s)(t)

  1

  ✭✐✐✮❊①✐stê♥❝✐❛✳ ❙❡❥❛ p ∈ j t❛❧ q✉❡ p = j ✱ s ∈ Γ(π)✳ ❉❡✜♥❛ ω ♣♦r

  1

  ω

  1 (ξ) = π 1,0 ∗ (ξ) 1 (s) π 1 ∗ (ξ), p J (π).

  − j ∗ ∀ξ ∈ T ✭✷✳✶✶✮

  1 (ξ) = π 1,0 ∗ (ξ) 1 (j 1 (s) (ν)) = 0

  ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ ω ✱ ♣❛r❛ t♦❞♦ ξ ∈ V (π)✱ ❡ ω ∗ ✱ ♣❛r❛ t♦❞❛ s❡çã♦ s ❞❡ π ❥á q✉❡ ω (j (s) (ν)) = π (j (s) (ν)) (s) π (j (s) (ν))

  1 1 1,0 ∗

  1

  1 1 ∗

  1 ∗ ∗ − j ∗ ∗

  = π (j (s) (ν)) (j (s) (ν))

  1,0 ∗ 1 1,0 ∗

  1 ∗ − π ∗

  = 0,

  ∗ (s) = id

  

1

  1

  ✉s❛♥❞♦✱ ♥❛ s❡❣✉♥❞❛ ✐❣✉❛❧❞❛❞❡✱ ♦ ❢❛t♦ q✉❡ π ◦ j ∗ ✳

  1

  (π) ❙❡❥❛♠ (t, x, v) ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ♣❛r❛ J ✱ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ω

  1 ❡♠ ❝♦♦r❞❡✲

  ♥❛❞❛s ♣♦r

  X ∂

  i ω = i dt).

  1

  ⊗ (dx − v ✭✷✳✶✷✮ ∂x

  i i n n

  X X ∂ ∂ ∂

  i j

  1

  1

  (s)(t) (π) p = τ ξ η p J (π) + +

  1

  ❉❡ ❢❛t♦✱ s❡❥❛♠ p = j ∈ J ❡ X ∈ T ✱ ∂t ∂x i ∂v i

  i j =1 =1

  t❡♠♦s✿

  n

  X ∂ ∂ ∂

  i

  • ω p (X p ) = τ ξ (τ )

  − σ ∗ ∂t ∂x i ∂t

  i =1 n n

  X X ∂ ∂ ∂ ∂

  i

  = τ ξ v + i (p) ) + − τ(

  ∂t ∂x i ∂t ∂x i

  i i =1 =1 n

  X ∂

  = (ξ i i (p)) − τv

  ∂x i

  i =1

  ∂ = i i (p)dt) p (X p ).

  ⊗ (dx − v

  ✹✺

  −1

  (a) ❆❣♦r❛✱ ♦❜s❡r✈❡ q✉❡ ❝❛❞❛ ✜❜r❛ π 1,0 é ✉♠ ❡s♣❛ç♦ ❛✜♠ ❛ss♦❝✐❛❞♦ ❛♦ ❡s♣❛ç♦ ✈❡t♦✲

  −1 a (π) , p (a) i = j (s i (t))

  r✐❛❧ V ✳ ❉❡ ❢❛t♦✱ s❡ p

  1 2 sã♦ t❛✐s q✉❡ p 1 ✱ ❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ❛✜♠

  ∈ π 1,0

  −1 −1

  (a) (a) a (π)

  1 , p 2 ) =

  ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ ♣♦r ♠❡✐♦ ❞❛ ❛♣❧✐❝❛çã♦ ϕ : π ×π −→ V ❞❛❞❛ ♣♦r ϕ(p

  1,0 1,0 ∂

  (s )( ) π a (π)

  a (π) 1 ∗ 2 ∗ (a)

  − s | ∈ V ✳ ❊st❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ❛✜♠ ♣❡r♠✐t❡ ✐❞❡♥t✐✜❝❛r ❝♦♠ V

  ∂t −1 ∗

  (a) (π )

  1,0

  ❝❛❞❛ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ π 1,0 ✳ P♦rt❛♥t♦✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ α ∈ V ♣♦❞❡ s❡r ✐❞❡♥t✐✜✲ (s)(t ) )

  1

  ❝❛❞♦ ❝♦♠ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ Hom(V (π), R)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ p = j ❡ ˙s(t ❞❡♥♦t❛ ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❛ s❡çã♦ s ❡♠ t ✱ ❡♥tã♦ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ ❛♣❧✐❝❛çã♦

  ∗

  ψ : V (π ) ) = π ( ˙s(t )) (π), T I)

  1,0 p p p π (p)

  −→ Hom(V (π), T I) t❛❧ q✉❡ ψ(α ∗ ⊗ α ∈ Hom(V

  1 ✱ ∂

  1

  (π) i ) = i ♣❛r❛ t♦❞♦ p ∈ J ✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s✱ ψ(dv ⊗ du ✳

  ∂t

  1

  (π) ❆❣♦r❛✱ ❞❡✜♥✐♠♦s ✉♠❛ 1✲❢♦r♠❛ θ s♦❜r❡ J ♣♦r

  θ j (ξ) = L(j (γ)(t)).π ∗ (ξ) + µ L (j (γ)(t))(ω (ξ)),

  1 (γ)(t)

  1

  1

  1 1 p

  ♦♥❞❡

  1

  µ L : J (π) (T ransf ormada de Legendre) −→ Hom(V (E), T I)

  V µ L (p) := ψ(d L(p)).

  ∂L ∂

  L (t, x, v) = ( dx i )

  ❊♠ ❝♦♦r❞❡♥❛❞❛s✱ µ ⊗ ❡ ∂v i ∂t

  X ∂L ∂ θ = [Ldt + (dx i i dt)] .

  − v ⊗ ∂v i ∂t

  i

  1

  (π) ❊♥✜♠✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ 1✲❢♦r♠❛ Θ s♦❜r❡ J ♣♦r

  Θ = θ ⊼ ω ✱

  1

  (π) ♦♥❞❡ ω é ✉♠❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ ❡♠ I ❡ ⊼ ❡st❛❜❡❧❡❝❡ ✉♠ ♣r♦❞✉t♦ ❡♥tr❡ h✲❢♦r♠❛s s♦❜r❡ J ✱

  1

  (π) ❝♦♠ ✈❛❧♦r❡s ❡♠ T I✱ ❡ r✲❢♦r♠❛s s♦❜r❡ J ❢♦r♥❡❝❡♥❞♦ ❝♦♠♦ r❡s✉❧t❛❞♦ (h + r − 1)✲❢♦r♠❛s

  1

  (π) s♦❜r❡ J ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ s❡ α = ξ ⊗ ρ✱ ❡♥tã♦ α ⊼ β = ρ ∧ ξyβ✳ Θ s❡ ❝❤❛♠❛ ❢♦r♠❛ ❞❡ P♦✐♥❝❛ré✲❈❛rt❛♥ ✭♦✉ ❛♣❡♥❛s ❞❡ ❈❛rt❛♥✮ q✉❡✱ ❡♠ ❝♦♦r❞❡✲

  ♥❛❞❛s✱ é ❡①♣r❡ss❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ s❡ ω = dt✱

  X ∂L

  Θ = Ldt + (dx i i dt) − v

  ∂v i

  i

  [s]

  L

  ➱ ❢á❝✐❧ ✈❡r q✉❡ ❛s ❢♦r♠❛s Ldt ❡ Θ ❞❡✜♥❡♠ ♦ ♠❡s♠♦ ❢✉♥❝✐♦♥❛❧ A ✿ Z Z

  ∗ ∗ A L [s] = j (s) j (s) Θ.

  1

  1 L = A A

  X Θ = Ldt + α (dx

  i i

  ■ss♦ t❛♠❜é♠ s❡r✐❛ ✈❡r❞❛❞❡ ♣❛r❛ q✉❛❧q✉❡r 1✲❢♦r♠❛ ❞♦ t✐♣♦ e −

  i

  v dt)

  i

  ✳ ▼❛s ❛ ❡s❝♦❧❤❛ ❞❛ ❢♦r♠❛ ❞❡ ♣♦✐♥❝❛ré✲❈❛rt❛♥ ❧❡✈❛ à s❡❣✉✐♥t❡ ❝❛r❛❝t❡r✐③❛çã♦ ✐♥✈❛r✐❛♥t❡

  ✹✻ ❚❡♦r❡♠❛ ✷✳✸✳✷✳ ❯♠❛ s❡çã♦ s ∈ Γ(π) é L✲❝rít✐❝❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

  ∗

  j (s) (XydΘ) = 0,

  1

  1

  (π)) ♣❛r❛ t♦❞♦ X ∈ D(J ✳

  τ (t) = s(t) τ

  Pr♦✈❛✿ ❯♠❛ ✈❛r✐❛çã♦ s ♣♦❞❡ s❡r ❝♦♥str✉í❞❛ ♣♦r ♠❡✐♦ ❞♦ ✢✉①♦ {A } ❞❡ ✉♠ ❝❛♠♣♦ ✈❡rt✐❝❛❧ X q✉❡ ❞❡✐①❛ ✜①❛❞♦s ♦s ❡①tr❡♠❛✐s s(a) ❡ s(b)✳ ❖ ❢❛t♦ ❞❡ X s❡r

  

τ τ (t)

  ✈❡rt✐❝❛❧ ♣❡r♠✐t❡ q✉❡ ❛ ✈❛r✐❛çã♦ A ◦ s s❡❥❛ ❞♦ t✐♣♦ s ✳ ❙❡ τ ✈❛r✐❛ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ (s (t))

  1 τ

  s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥❛ ❞❡ ✵ ❡♥tã♦ ❛s s❡çõ❡s j ❢♦r♥❡❝❡♠ ✉♠❛ ✈❛r✐❛çã♦ ❞❛ s❡çã♦

  (1) (1)

  j (s) τ τ (π)

  1

  1

  ✳ ❆❧é♠ ❞✐ss♦✱ {A } ✐♥❞✉③ ✉♠ ✢✉①♦ ❧♦❝❛❧ {A } ❞❡ ✉♠ ❝❛♠♣♦ X ❡♠ J ✳ ❊st❡

  (1) (1)

  X )(X ) = X

  1,0

  é ♦ ♣r✐♠❡✐r♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ X✳ ❊st❡ ❝❛♠♣♦ é t❛❧ q✉❡ (π ✳ ❆❧é♠

  (1) τ

  (s τ ) = A (s) ❞✐ss♦✱ j

  1 1 ❡✱ ♣♦rt❛♥t♦✱ t❡♠♦s q✉❡

  ◦ j Z Z d d d

  ∗ ∗

  A L [s τ ] τ = j (s τ ) Θ τ = j (s τ ) Θ τ | =0

  1 | =0 1 | =0

  dτ dτ dτ

  I I

  Z Z d d

  (1) ∗ ∗ (1)∗

  = (A (s)) Θ τ = j (s) A Θ τ

  τ

1 =0

1 τ =0

  ◦ j | | dτ dτ

  I I

  Z Z

  (1) (1)

∗ ∗

  y y

  (1)

  = j (s) ( Θ) = j (s) (X dΘ + d(X Θ))

  1 L

  I I

  Z Z

  (1) (1) ∗ ∗

  y y = j (s) (X dΘ) + d(j (s) (X Θ)).

  1

  1 I

  I ∂I = 0

  ❆❣♦r❛✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❙t♦❦❡s ❡ ❧❡♠❜r❛♥❞♦ q✉❡ X| ❡✱ ♣♦rt❛♥t♦✱

  (1)

  X ∂I = 0 | ✱ ♦❜t❡♠✲s❡ q✉❡

  Z Z Z d

  (1) (1) (1) ∗ ∗ ∗

  y y y A L [s τ ] τ = j (s) (X dΘ) + j (s) (X Θ) = j (s) (X dΘ)

  =0

  1

  1

  1

  | ✳ dτ

  I ∂I

  I

  ■ss♦ s✐❣♥✐✜❝❛ q✉❡ s é L✲❝rít✐❝❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

  (1) ∗

  1 (s) (X dΘ) = 0,

  y j

  ∀X ∈ V (π)✳

  1

  (π) 1,0 ❊♥tr❡t❛♥t♦✱ ✉♠ ❝❛♠♣♦ s♦❜r❡ J ♣♦❞❡ s❡r ♣r♦❥❡t❛❞♦✱ ✈✐❛ π ✱ ❡♠ ✉♠ ❝❛♠♣♦

  ♥✉❧♦ ♦✉ q✉❡ t❡♠✱ ♣❡❧♦ ♠❡♥♦s✱ ✉♠❛ ♣❛rt❡ ❤♦r✐③♦♥t❛❧ ♦✉ ✈❡rt✐❝❛❧ ✭❝♦♠ r❡s♣❡✐t♦ ❛ π✮ ♥ã♦

  ∗

  (s) (Y ydΘ) (π))

  1

  1

  ♥✉❧❛✳ ❱❛♠♦s ❛♥❛❧✐s❛r ❛s s✐t✉❛çõ❡s ♣♦ssí✈❡✐s ♣❛r❛ j ✱ q✉❛♥❞♦ Y ∈ D(J ✳

  1

  (π)) (Y ) = 0

  1,0 ∗

  ✭❛✮ ❙❡ Y ∈ D(J é t❛❧ q✉❡ π ✱ ♣♦❞❡♠♦s ♣r♦✈❛r ✉s❛♥❞♦ ❝♦♦r❞❡♥❛❞❛s ♦✉ ✉♠ ❝á❧❝✉❧♦ ✐♥✈❛r✐❛♥t❡ ✭✈✳ ❬✺✵❪✮ q✉❡

  ∗ ∗

  τ (i Y dΘ) = tr[τ ( Y (µ L ) )]Ω

  1 L ◦ ω ✱

  ) ♣❛r❛ t♦❞❛ s❡çã♦ τ ∈ Γ(π

  1 ✳

  (s)

  1

  ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ τ = j ♦❜t❡♠✲s❡ j (s)(i Y dΘ) = 0

  1

  ✹✼

  ∗

  (s) ω = 0

  1

  1

  ♣♦✐s j ✳

  1

  (π)) ∗ (Y ) = X (X) = 0 ✭❜✮ ❙❡ Y ∈ D(J é t❛❧ q✉❡ π 1,0 ❝♦♠ π ✱ ❡♥tã♦ Y é ✉♠❛ ❝♦♠❜✐♥❛çã♦

  ∗ (1) i

  ❞❡ ❝❛♠♣♦s ❞❛ ❢♦r♠❛ X i ✱ ❝♦♠ X ∈ V (π)✳

  1

  (π)) (Y )

  1 ∗

  ✭❝✮ ❙❡ Y ∈ D(J é t❛❧ q✉❡ π 6= 0✱ ❡♥tã♦

  ∗

  j (s) (Y ydΘ) = 0

  1

  ✳ ❯♥✐♥❞♦ ♦ ❢❛t♦ ✐♥✐❝✐❛❧ ❞❛ ❞❡♠♦♥str❛çã♦ às ♦❜s❡r✈❛çõ❡s (a), (b) ❡ (c) ♦❜t❡♠✲s❡ ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✸✳✷✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♦s 1✲❥❛t♦s ❞❡ s❡çõ❡s L✲❝rít✐❝❛s ✭✐st♦ é✱

  ❛s s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✮ sã♦ ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ❞❛ ❞✐str✐❜✉✐çã♦ q✉❡ ❛♥✉❧❛ ♦ s✐st❡♠❛ ❞❡ ❢♦r♠❛s

  1 L = Y ydΘ : Y (π) .

  I ∈ D J ❯s❛♥❞♦ ❛s ❝♦♦r❞❡♥❛❞❛s (t, x, v)✱ ❡st❛ ❞✐str✐❜✉✐çã♦ é✱ ♣♦rt❛♥t♦✱ ❛ ♠❡s♠❛ ❣❡r❛❞❛ ♣❡❧♦ ❝❛♠♣♦

  ∂ ∂ ∂

  X L = + v i + F i (t, x, v) ✭✷✳✶✸✮

  ∂t ∂x i ∂v i ❝✉❥♦ ✢✉①♦ é ❞❛❞♦ ♣❡❧❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡s❝r✐t❛s ♥❛ ❢♦r♠❛

  ( ˙x = v

  i i ˙v i = F i (t, x, v).

  ❙❡ L s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡

  2

  ∂ L det 6= 0,

  ∂v i v j é s❡♠♣r❡ ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ❛s ❡q✉❛çõ❡s ❞♦ ✢✉①♦ ❞♦ ❝❛♠♣♦ ✷✳✶✸ ♥❡st❛ ❢♦r♠❛✳

  ❆s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ sã♦✱ ♣♦rt❛♥t♦✱ ❛ ❢♦r♠❛ ❧♦❝❛❧ ❞❛ ❝♦♥❞✐çã♦

  1 ∗

  j (s) (XydΘ) = 0, (π))

  

1 ✳

  ∀X ∈ D(J

  ✹✽

  ✷✳✹ ❙✐♠❡tr✐❛s ❱❛r✐❛❝✐♦♥❛✐s ❡ ❚❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r

  P♦r ♠❡✐♦ ❞❛ ❝❛r❝t❡r✐③❛çã♦ ❞❛s s❡çõ❡s L✲❝rít✐❝❛s✱ ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✸✳✷✱ s❡rá ♣♦ssí✈❡❧ ❞❡♠♦♥str❛r ♥✉♠❛ ❢♦r♠❛ s✐♠♣❧❡s ♦ t❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r q✉❡ ❡st❛❜❡❧❡❝❡ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ s✐♠❡tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s ❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s✳

  1

  (π) ❉❡✜♥✐çã♦ ✷✳✹✳✶✳ ❯♠ ❝❛♠♣♦ Y s♦❜r❡ DJ é ✉♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ♣❛r❛ ✉♠❛ ❧❛✲ ❣r❛♥❣❡❛♥❛ L s❡

  1 ∞ Y Θ = df, f J (π) .

  L ∈ C

  (1)

  ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ❝❛♠♣♦ Y é ❝❤❛♠❛❞♦ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ❞♦ t✐♣♦ ▲✐❡ s❡ Y = X ❝♦♠ X ∈ D(E)✳ ❈❛s♦ ❝♦♥trár✐♦✱ Y é ❝❤❛♠❛❞♦ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ❈❛rt❛♥✳

  ❯♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ♣r❡s❡r✈❛ ❛ ❞✐str✐❜✉✐çã♦ q✉❡ ❛♥✉❧❛ ♦ s✐st❡♠❛ ❞❡ ❢♦r♠❛s

  L

  I ✳ ❉❡ ❢❛t♦✱ t❡♠♦s ❛ s❡❣✉✐♥t❡

  1

  (π)) Pr♦♣♦s✐çã♦ ✷✳✹✳✷✳ ❙❡ Y ∈ D(J é ✉♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧✱ ❡♥t❛ã♦ Y é ✉♠❛ s✐♠❡✲ tr✐❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✳

  Y Θ = df Y (XydΘ) =

  Pr♦✈❛✿ ❙❡ Y é s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧✱ L ❡✱ ♣♦rt❛♥t♦✱ L [Y, X]ydΘ

  L )

  ✳ ▲♦❣♦✱ Y é s✐♠❡tr✐❛ ❞❛ ❞✐str✐❜✉✐çã♦ Ann (I ❡✱ ♣♦rt❛♥t♦✱ é s✐♠❡tr✐❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✳

  1

  2

  1

  2 Y , Y ] = ,Y [Y ]

  ❊♠ ✈✐rt✉❞❡ ❞❛ ♣r♦♣r✐❡❞❛❞❡ [L L L ✱ ✈❡r✐✜❝❛✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❞❡ ❞✉❛s s✐♠❡tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s é ✉♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ❡ q✉❡✱ ♣♦rt❛♥t♦✱ ❛s s✐♠❡tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s ❢♦r♠❛♠ ✉♠❛ á❧❣❡❜r❛ ❝♦♠ r❡s♣❡✐t♦ ❛ ❡st❡ ♣r♦❞✉t♦ ❞❡ ▲✐❡✳

  ❖ t❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r é ♦ s❡❣✉✐♥t❡

  Y Θ = df, f

  ❚❡♦r❡♠❛ ✷✳✹✳✸✳ ✭❞❡ ◆ö❡t❤❡r✮ ❙❡ Y é ✉♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧✱ t❛❧ q✉❡ L ∈

  1 ∞

  C (J (π)) ✱ ❡♥tã♦

  F = f − Y yΘ

  é ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✳ ❆❧é♠ ❞✐ss♦✱ F é t❛♠❜é♠ ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ♣❛r❛ ♦ ❝❛♠♣♦ Y ✱ ♦✉ s❡❥❛✱ Y (F ) = 0✳

  Pr♦✈❛✿ ❱✐♠♦s✱ ❛♦ ✜♥❛❧ ❞❛ s❡çã♦ ❛♥t❡r✐♦r✱ q✉❡ s❡ s é ✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

  (1)

  1 ∗

  y

  1 (s) (i

  ✈❛r✐❛❝✐♦♥❛❧ ❡♥tã♦ j ♣❛r❛ t♦❞♦ X ∈ D(J ✳ ❉❡ss❛ ❢♦r♠❛✱ s❡ X (π)

  X X dΘ) = 0 (π))

  ∈ DJ é ✉♠❛ s✐♠❡tr✐❛ ❞❡ ◆ö❡t❤❡r✱ ✉s❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❈❛rt❛♥✱ ♦❜t❡♠♦s

  (1) (1) ∗ ∗

  y y

  (1)

  0 = j (s) (X dΘ) = j (s) Θ Θ

  1

  

1

X

  L − d X

  (1) ∗

  y = j

  1 (s) df Θ

  − d X

  (1) ∗

  y = j (s) Θ)

  

1

  ◦ d(f − X

  (1) ∗

  y = d[j (s) (f Θ)]

  1

  ✹✾

  (1)

  y Θ

  ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ F := X − f é ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ♣❛r❛ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦✳ ▼❛✐s

  (1) (1)

  y (F ) = X dF = 0

  ❛✐♥❞❛✱ ♣❡❧❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ❞❛ F ✱ s❡❣✉❡✲s❡ q✉❡ X ✳ ❉❡ ❢❛t♦✱ ✉s❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ❈❛rt❛♥✱ ♦❜t❡♠♦s

  (1) (1) (1)

  y y y X dF = X (d X Θ

  − df)

  (1) (1)

  y y

  (1)

  = X ( Θ dΘ L

  X − X − df)

(1) (1)

  y y = X (df dΘ

  − X − df) = 0.

  ❯♠ ✐♠♣♦rt❛♥t❡ t✐♣♦ ❞❡ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ sã♦ ♦s ❝❛♠♣♦s ❞❡ ❑✐❧❧✐♥❣✱ ✐st♦ é✱ s✐♠❡tr✐❛s ✐♥✜♥✐t❡s✐♠❛✐s ❞❛ ♠étr✐❝❛✳ ❉❡ ❢❛t♦✱ ♦ r❡s✉❧t❛❞♦ ❛❞✐❛♥t❡ ♠♦str❛ q✉❡ s❡ X é ✉♠

  (1)

  ❝❛♠♣♦ ❞❡ ❑✐❧❧✐♥❣ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ▼✱ ❡♥tã♦ X é ✉♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ❞♦ t✐♣♦

  X

  1 g ij v i v j ▲✐❡ ♣❛r❛ ❛ ❧❛❣r❛♥❣❡❛♥❛ L = ✳ ◆♦ ❝❛s♦ ❞❡ss❛ ▲❛❣r❛♥❣❡❛♥❛✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡

  2

  i,j ∂

  ♦ ❝❛♠♣♦ é ✉♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧✳ ❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r✱ ✈❡r✐✜❝❛✲s❡ q✉❡

  ∂t

  L é ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❛ss♦❝✐❛❞❛ ❛ ❡ss❛ s✐♠❡tr✐❛✳

  ❚❡♦r❡♠❛ ✷✳✹✳✹✳ ❙❡❥❛♠ (M, g) ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❡ X ❝❛♠♣♦ ❞❡ ❑✐❧❧✐♥❣ s♦❜r❡ M✳

  X

  1

  (1)

  g ij v i v j ❊♥tã♦ X é ✉♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ ♣❛r❛ L = ✳

  2

  i,j

  ∂

  1

  , ..., x , v , ..., v (π)

  1 n 1 n i

  Pr♦✈❛✿ ❙❡❥❛♠ t, x ❝♦♦r❞❡♥❛❞❛s s♦❜r❡ J ✱ X = ξ ❝❛♠♣♦ ∂x i

  ∂L ∂L (π) v i )dt + dx i s♦❜r❡ J t❛❧ q✉❡ X é ❝❛♠♣♦ ❞❡ ❑✐❧❧✐♥❣ s♦❜r❡ M ❡ Θ = (L − ✳ P❡❧❛

  ∂v i ∂v i ❢ór♠✉❧❛ ✷✳✺✱

  X ∂ξ i ∂

  (1) X = X + v j .

  ∂x j ∂v i

  i,j

  P♦rt❛♥t♦✱

  X

  (1)

  y X Θ = g ij v j ξ i

  

i,j

  ❡

  X X ∂g ij ∂ξ i

  (1)

  y d X Θ = + [ + g ij v j ]dx k [g ik ξ i ]dv k .

  ∂x k ∂x k

  

i,j,k ik

  P♦r ♦✉tr♦ ❧❛❞♦✱

  X X 1 ∂g ij dΘ = ( v i v j )dx k (g ik v i )dv k

  − ∧ dt − ∧ dt 2 ∂x k

  

i,j,k i,k

  X X ∂g ij + ( v + j )dx k i g ik dv k i .

  ∧ dx ∧ dx ∂x k

  • X
  • X
  • [g kj v j
  • ∂g ij

  P♦rt❛♥t♦✱ X

  2

  ❊①❡♠♣❧♦ ✷✳✹✳✺✳ ❈♦♥s✐❞❡r❡ ❛ ♠étr✐❝❛ ❞❡ ❙❝❤✇❛r③s❝❤✐❧❞ ds

  ∂L ∂v i .

  g i,j v j ξ i = ξ i

  i,j

  X

  = 2

  y Θ

  X = X (1)

  H

  é ✉♠❛ s✐♠❡tr✐❛ ❞❡ ▲✐❡✳ P❡❧♦ t❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r✱ ❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❛ss♦❝✐❛❞❛ ❛♦ ❝❛♠♣♦ ❞❡ ❑✐❧❧✐♥❣ X é

  (1)

  X (1) Θ = 0.

  2m x

  L

  ✭✷✳✶✺✮ ❙✉❜st✐t✉✐♥❞♦ ✷✳✶✺ ❡♠ ✷✳✶✹✱ ♦❜t❡♠♦s

  ∂ξ k ∂x j g ik .

  k

  X

  −

  ∂ξ k ∂x i g kj

  k

  X

  ξ k = −

  ∂g ij ∂x k

  k

  X

  = (1 −

  2

  X (g) = 0

  )

  2 ).

  )

  4

  d(x

  3

  x

  2

  2

  )

  3

  (d(x

  2

  2

  )d(x

  − (x

  

2

  )

  2

  d(x

  2

  2m x

  1 −

  1

  −

  2

  )

  1

  ✱ ♣♦✐s X é ❝❛♠♣♦ ❞❡ ❑✐❧❧✐♥❣✱ ♦❜t❡♠♦s

  ), ❡ q✉❡ L

  ✭✷✳✶✻✮

  ∂x k v j ξ k )dx i +

  (1)

  Θ = X

  X (1)

  ▲♦❣♦✱ L

  i,j,k (g ik ξ i )dv k .

  X

  −

  g ik v j ∂ξ k ∂v k dx i

  i,j,jk

  ( ∂g ij ∂x k v j ξ i )dx k

  i,j,k

  X

  ( ∂g ij

  (1)

  i,j,k

  )dt

  (g ik v i v j ξ k x j

  i,j,k

  X

  −

  ( ∂g ij ∂x k v i v j ξ k )dt

  i,j,k

  1

  y dΘ = −

  (1)

  X

  ✺✵ ❡✱ ♣♦rt❛♥t♦✱

  y dΘ + d X

  y Θ

  ∂ ∂x j

  ∂x j )) = (

  X

  , L

  ∂ ∂x i

  ∂x j ) + g(

  , ∂

  ∂ ∂x i

  X

  L

  ∂x j ) + g(

  , ∂

  ∂ ∂x i

  X g)(

  L

  , ∂

  ✭✷✳✶✹✮ −

  ∂ ∂x i

  ❆❣♦r❛✱ ✉s❛♥❞♦ ♦ ❢❛t♦ q✉❡ X(g(

  ∂x k v j ξ k + g ik v j ∂ξ k ∂v k ]dx i .

  ∂ξ k ∂x i

  x j ]dt

  k

  ∂x k v i v j ξ k + g ik v i v j ξ

  

ij

  2 ∂g

  1

  [

  i,j,k

  X

  • sen

  4 .

  3

  ∂x

  = ∂

  5

  4 X

  ∂ ∂x

  4

  cos x

  3

  ∂ ∂x

  ❆s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s✱ ♦❜t✐❞❛s ♣♦r ♠❡✐♦ ❞♦ t❡♦r❡♠❛ ❞❡ ◆ö❡t❤❡r sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❛❞❛s ♣♦r

  4

  = sin x

  4

  4 X

  ∂ ∂x

  4

  sin x

  3

  − cot x

  4 .

  F

  ∂ ∂x

  4

  = v

  5

  4 F

  )v

  4

  cos x

  3

  3

  )v

  = (sin x

  1

  4

  4 F

  3 sin x 4 )v

  − (cot x

  

3

  1 F 3 = (cos x 4 )v

  = v

  2

  = L F

  

3

  4

  ✺✶ ❆ ♠❡♥♦s ❞♦ ❢❛t♦r

  1

  2

  − (x

  2

  )

  

2

  (v

  2

  2m x

  1 −

  −

  2

  2

  )

  1

  )(v

  2

  − 2m x

  ✱ ❛ ▲❛❣r❛♥❣❡❛♥❛ q✉❡ ❞❡s❝r❡✈❡ ❛s ❣❡♦❞és✐❝❛s é ❞❛❞❛ ♣♦r L = (1

  2

  1

  )

  (v

  = cos x

  ✭✷✳✶✼✮ é ♣♦ssí✈❡❧ ♠♦str❛r ✭✈✐❞❡ ❬✺✶❪✮ q✉❡ ❛ á❧❣❡❜r❛ ❞❛s s✐♠❡tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s ❞♦ t✐♣♦ ▲✐❡

  3

  ∂ ∂x

  2 =

  X

  ∂t

  = ∂

  1

  X

  ❞❡ss❛ ▲❛❣r❛♥❣❡❛♥❛ é 5✲❞✐♠❡♥s✐♦♥❛❧ ❡ ❣❡r❛❞❛ ♣❡❧♦s ❝❛♠♣♦s

  2 .

  3

  )

  4

  (v

  3

  x

  2

  − sen

  2

  )

  • cot x
  • (cot x

  ❈❛♣ít✉❧♦ ✸ ■♥t❡❣r❛çã♦ ❞❡ ❋❧✉①♦s ●❡♦❞és✐❝♦s ❝♦♠ ▼ét♦❞♦s ❙✐♠♣❧ét✐❝♦s

  ❆ ❣❡♦♠❡tr✐❛ s✐♠♣❧ét✐❝❛ ❞❡❞✐❝❛✲s❡ ❛ ❡st✉❞❛r ✈❛r✐❡❞❛❞❡s q✉❡ ❛❞♠✐t❡♠ ✉♠❛ ❡str✉t✉r❛ s✐♠♣❧ét✐❝❛✱ ♦✉ s❡❥❛✱ ✈❛r✐❡❞❛❞❡s q✉❡ ❛❞♠✐t❡♠ ✉♠❛ 2✲❢♦r♠❛ ♥ã♦ ❞❡❣❡♥❡r❛❞❛ ❡ ❢❡❝❤❛❞❛✳ P❛r❛ ✈❛r✐❡❞❛❞❡s ❝♦♠ ✉♠❛ t❛❧ ❡str✉t✉r❛✱ ❡①✐st❡ ✉♠❛ ❡①t❡♥s❛ t❡♦r✐❛ ✭♣♦r ❡①❡♠♣❧♦✱ ❬✺❪✱❬✶✸❪✱❬✶✹❪✱❬✹❪✮ ❞❛ q✉❛❧ ❝♦♠❡ç❛♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s ❡ q✉❡ s❡rã♦ út❡✐s ❛♦ ♥♦ss♦ ❡st✉❞♦✳ ❆té ❛❣♦r❛✱ ❛❜♦r❞❛♠♦s ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❝♦♠♦ ✉♠❛ ❞✐♥â♠✐❝❛ ▲❛❣r❛♥❣❡❛♥❛✳ ❆ ❣❡♦♠❡tr✐❛ s✐♠♣❧ét✐❝❛ ♣❡r♠✐t❡ ✉♠❛ ❞❡s❝r✐çã♦ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❞✐♥â♠✐❝❛ ❝❤❛♠❛❞❛ ❍❛♠✐❧t♦♥✐❛♥❛ q✉❡✱ ❛♥❛❧♦❣❛♠❡♥t❡ à ❞✐♥â♠✐❝❛ ▲❛❣r❛♥❣❡❛♥❛✱ é ♣r♦✈❡♥✐❡♥t❡ ❞❡ ✉♠❛ ❢✉♥çã♦ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛✳ ❱❡r❡♠♦s ❛q✉✐ q✉❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡ ❢❛③ ❛ ❧✐❣❛çã♦ ❡♥tr❡ ❛ ❞✐♥â♠✐❝❛ ▲❛❣r❛♥❣❡❛♥❛ ❡ ❛ ❞✐♥â♠✐❝❛ ❍❛♠✐❧t♦♥✐❛♥❛✳ ■ss♦ ♥♦s ♣♦ss✐❜✐❧✐t❛rá tr❛t❛r ❛s ❣❡♦❞és✐❝❛s ❝♦♠ ❛s ❢❡rr❛♠❡♥t❛s ❞❛ ❣❡♦♠❡tr✐❛ s✐♠♣❧ét✐❝❛✳ P♦rt❛♥t♦✱ ❞❡✐①❛r❡♠♦s

  

1

  (I, M ) ❞❡ tr❛t❛r ❛s ❣❡♦❞és✐❝❛s ♥♦ ❡s♣❛ç♦ ❞❡ ❥❛t♦s J ♣❛r❛ tr❛tá✲❧❛s ♥♦ ❛♠❜✐❡♥t❡ s✐♠♣❧ét✐❝♦✳ ◆♦ ❝❛s♦ ❞❛s ❣❡♦❞és✐❝❛s✱ ♦♥❞❡ ❛ ▲❛❣r❛♥❣❡❛♥❛ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ 3.1✱ ✐st♦ é ✉♠❛ ✈✐❛ ❞❡ ♠ã♦ ❞✉♣❧❛✳

  ◆❛ ♣❛rt❡ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✱ ♠♦str❛r❡♠♦s ❝♦♠♦ ♦ ♠ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ à ✐♥t❡❣r❛çã♦ ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❡ ❛♣❧✐❝❛r❡♠♦s ❡st❡ ♠ét♦❞♦ ❛ ❛❧❣✉♠❛s ♠étr✐❝❛s ❞❡ ❊✐♥st❡✐♥✳

  ✸✳✶ ■♥tr♦❞✉çã♦

  ◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ♦❜s❡r✈❛♠♦s q✉❡✱ s♦❜ ❤✐♣ót❡s❡s ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ❛ ❢✉♥çã♦ ❧❛❣r❛♥❣❡❛♥❛ L ✭✐st♦ é✱ s♦❜r❡ ❛ ❝♦♥❞✐çã♦ 3.1✮✱ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ❡q✉❛çõ❡s q✉❡ ❞❡s❝r❡✈❡♠ ♦ ✢✉①♦ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s s♦❜r❡ ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡✳ ❊♥tr❡t❛♥t♦✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ▲❛❣r❛♥❣❡❛♥❛ L✱ ❡st❛s ❡q✉❛çõ❡s ♣♦❞❡♠ ❛ss✉♠✐r ✉♠ ❛s♣❡❝t♦ ♠✉✐t♦ ♣♦✉❝♦ tr❛tá✈❡❧✳ ❊♥tr❡t❛♥t♦✱ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s

  ✺✸ ❛❞❡q✉❛❞❛✱ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ♣♦❞❡♠ ❛ss✉♠✐r ✉♠❛ ❢♦r♠❛ ❜❡♠ ♠❛✐s ❡❧❡❣❛♥t❡✳

  ∂L i :=

  ❉❡ ❢❛t♦✱ s❡ p ✱ ❡♥tã♦✱ ♥❛ ❤✐♣ót❡s❡ q✉❡

  ∂v i

  

2

  ∂ L det 6= 0, ✭✸✳✶✮

  ∂v j ∂v i ❛ tr❛♥s❢♦r♠❛çã♦

  (x, v) ✭✸✳✷✮

  7−→ (x, p) ❞❡✜♥❡ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ✭❞✐❢❡♦♠♦r✜s♠♦✮✳

  ◆❡st❛s ❝♦♦r❞❡♥❛❞❛s✱ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ s❡ t♦r♥❛♠ ♠❛✐s s✐♠♣❧❡s✳ ❉❡

  X p i v i ❢❛t♦✱ s❡ H := − L ❡ ❛ss✉♠✐r♠♦s q✉❡ ✈❛❧❡ ✸✳✶✱ ❡♥t❛ã♦ v = F (x, p)✳ ▲♦❣♦✱

  i

  dH = i dv i + vdp i −dL + p

  ∂L ∂L = dx i dv i + p i dv i + v i dp i . − −

  ∂x i ∂v i

  ∂L i :=

  ❡✱ ❧❡♠❜r❛♥❞♦ q✉❡ p ∂v ✱ ♦❜t❡♠♦s

  i

  ∂L dH = dx i i dv i + p i dv i + v i dp i − − p

  ∂x i ∂L

  = dx i + v i dp i −

  ∂x i ♦♥❞❡ s✉❜í♥❞✐❝❡s r❡♣❡t✐❞♦s ✐♥❞✐❝❛♠ s♦♠❛tór✐♦s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ✐♥t❡r♣r❡t❛♥❞♦ H = H(x, p)✱

  ∂H ∂H

  • ∂x ∂p

  i i

  dx i dp i t❡♠♦s q✉❡ dH = ❡✱ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ❛ ú❧t✐♠❛ ✐❞❡♥t✐❞❛❞❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s

  

  

∂H ∂L

   =

  −

  

∂x ∂x

i i

  ∂H

   = v i .

  ∂p i

  ▲♦❣♦✱ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ s❡ ❡s❝r❡✈❡♠ ♥❛ ❢♦r♠❛✿ 

  ∂H

   ˙x i =

  ∂p i

  ∂H

   ˙p i =

  −

  ∂x i

  ❊st❛s ❡q✉❛çõ❡s sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❊q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ ❡✱ ♥❛ ❤✐♣ót❡s❡ q✉❡ ✸✳✶ é s❛t✐s❢❡✐t❛✱ sã♦ ❡q✉✐✈❛❧❡♥t❡s às ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✳ ❆ ❢♦r♠❛ ❞❡ ❍❛♠✐❧t♦♥ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ é ♠✉✐t♦ s✐♠♣❧❡s ❡ ❛❞♠✐t❡

  ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ q✉❡ s❡rá tr❛t❛❞❛ ♥❛ s❡çã♦ s❡❣✉✐♥t❡✳ ❱❡r❡♠♦s✱ ❛❞✐❛♥t❡✱ q✉❡ ❡ss❛ ✐♥t❡r♣r❡t❛çã♦✱ ❛♣❧✐❝❛❞❛ ❛♦ ✢✉①♦ ❣❡♦❞és✐❝♦✱ ♣♦ss✐❜✐❧✐t❛rá tr❛t❛r ❡ss❡ ✢✉①♦ ❝♦♠♦ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ✉♠❛ ❞✐♥â♠✐❝❛ ❍❛♠✐❧t♦♥✐❛♥❛ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛✳

  ✸✳✷ ❱❛r✐❡❞❛❞❡s ❙✐♠♣❧ét✐❝❛s ❡ ❈❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s

  ◆❡st❛ s❡çã♦ ✈❡r❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s ❞❛ ❣❡♦♠❡tr✐❛ s✐♠♣❧ét✐❝❛ út❡✐s ♣❛r❛ ❛ ❛❜♦r❞❛❣❡♠ ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❝♦♠♦ ✉♠❛ ❞✐♥â♠✐❝❛ ❍❛♠✐❧t♦♥✐❛♥❛✳ ❈♦♠❡ç❛♠♦s

  ✺✹ ❉❡✜♥✐çã♦ ✸✳✷✳✶✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦ ♣❛r✳ ❯♠❛ ❡str✉t✉r❛ s✐♠♣❧ét✐❝❛ s♦❜r❡ M é ✉♠❛ 2✲❢♦r♠❛ ω ❢❡❝❤❛❞❛ ❡ ♥ã♦ ❞❡❣❡♥❡r❛❞❛ ❡♠ M✱ ✐st♦ é✱ t❛❧ q✉❡ dω = 0 ❡ ξyω p M

  6= 0 ♣❛r❛ t♦❞♦ ξ ∈ T ♥ã♦ ♥✉❧♦ ❡ t♦❞♦ p ∈ M ❖ ♣❛r (M, ω) é ❝❤❛♠❛❞♦ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛✳ ❖ ❢❛t♦ ❞❛ ❞✐♠❡♥sã♦ ❞❛ ✈❛r✐❡❞❛❞❡ M s❡r ♣❛r é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ ❞❡❝♦rr❡

  ❞❛ ♥ã♦ ❞❡❣❡♥❡r❛çã♦ ❞❛ 2✲❢♦r♠❛ ω✳ ❉❡ ❢❛t♦✱ ❡♠ t♦❞♦ ♣♦♥t♦ x ∈ M✱ ♣♦❞❡♠♦s ✜①❛r ✉♠❛ ❜❛s❡

  1 n

  v , ..., v x M

  ij )

  ❞❡ T ✳ ◆❡st❛ ❜❛s❡✱ ω ✜❝❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ♠❛tr✐③ Ω := (ω ✱

  ij = ω(v i , v j )

  ♦♥❞❡ ω ✳ ❊st❛ ♠❛tr✐③ é ❛♥t✐✲s✐♠étr✐❝❛ ❡ ♥ã♦ ❞❡❣❡♥❡r❛❞❛✱ ❥á q✉❡ ω é ♥ã♦

  T

  ❞❡❣❡♥❡r❛❞❛✳ ■ss♦ ✐♠♣❧✐❝❛ ✐♠❡❞✐❛t❛♠❡♥t❡ q✉❡ ❛ ❞✐♠❡♥sã♦ ❞❡ M é ♣❛r ❥á q✉❡ Ω = Ω ❡

  T dim (M ) det(Ω) = det(Ω ) = det( det(Ω).

  −Ω) = (−1) ✭✸✳✸✮ ❖✉tr♦ ❢❛t♦ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ q✉❡ ❞❡❝♦rr❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❡str✉t✉r❛ s✐♠♣❧ét✐❝❛ é q✉❡ t♦❞❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ é ♦r✐❡♥tá✈❡❧✳ ❉❡ ❢❛t♦✱ s❡❥❛ (M, ω) ✉♠❛ ✈❛r✐❡❞❛❞❡

  p p M

  s✐♠♣❧ét✐❝❛✳ ❊♠ ❝❛❞❛ ♣♦♥t♦ p ∈ M✱ ω é ✉♠❛ 2✲❢♦r♠❛ ❜✐❧✐♥❡❛r ❛♥t✐s✐♠étr✐❝❛ s♦❜r❡ T ✳

  n

  X

  ∗ i M p = α i i

  P♦rt❛♥t♦✱ s❡ dim(M) = 2n✱ ❡①✐st❡ ✉♠❛ ❜❛s❡ {α } ❞❡ T p t❛❧ q✉❡ ω ∧ α +n ✳ P♦r

  i =1

  ✐♥❞✉çã♦✱ é ❢á❝✐❧ ♣r♦✈❛r q✉❡

  n

  X

  k

  ω = ω = α i i i i

  p

  1 1 +n k k +n

  ∧ ... ∧ ω ∧ α ∧ ... ∧ α ∧ α | {z }

  i ,...,i kvezes 1 k =1 n n

  = n!α k k p

  ❡✱ ♣♦rt❛♥t♦✱ ω ∧ α ∧ ... ∧ α ∧ α ✳ ▲♦❣♦✱ ω 6= 0 ♣❛r❛ t♦❞♦ p ∈ M ❡ ω é ✉♠❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡✳

  p 1 1+n +n

  2n

  =

  1 , ..., x n , p 1 , ..., p n )

  ❯♠ ❡①❡♠♣❧♦ ❞❡ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ é R {(x } ❡q✉✐♣❛❞♦

  n

  X dp i i ❝♦♠ ❛ 2✲❢♦r♠❛ ω = ∧ dx ✳

  i =1

  ❯♠ s❡❣✉♥❞♦ ❡①❡♠♣❧♦ é ♦ ✜❜r❛❞♦ ❝♦t❛♥❣❡♥t❡ ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ N ❡q✉✐♣❛❞♦ ❝♦♠

  ∗

  N ♦ ❞✐❢❡r❡♥❝✐❛❧ ❡①t❡r♥♦ ❞❛ 1✲❢♦r♠❛ ✉♥✐✈❡rs❛❧ ♦✉ t❛✉t♦❧ó❣✐❝❛✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ M = T ❡

  1 ∗

  π : T N (M )

  −→ N ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳ ❆ 1✲❢♦r♠❛ ✉♥✐✈❡rs❛❧ ρ ∈ Λ é ❞❡✜♥✐❞❛ ♣♦r ρ θ = θ(π ξ) θ M.

  

∗ ∀θ ∈ M e ∀ξ ∈ T

i i , p i

  ❙❡ {x } sã♦ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❡♠ N ❡ {x } ❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♦r❞❡♥❛❞❛s

  X X

  θ = p i (θ)dx i θ ρ i dx i

  ❝❛♥ô♥✐❝❛s ❡♠ M✱ t❡♠♦s q✉❡ ρ | ✱ ✐st♦ é✱ ρ = ✳ (M, dρ) é✱ ♣♦rt❛♥t♦✱

  i i

  ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛✳ ❖✉tr♦s ❡①❡♠♣❧♦s s✐♠♣❧❡s ❞❡ ✈❛r✐❡❞❛❞❡s s✐♠♣❧ét✐❝❛s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧❛s s✉✲

  ♣❡r❢í❝✐❡s ♦r✐❡♥tá✈❡✐s✳ ◆❡ss❡s ❝❛s♦s✱ ✉♠❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ ω q✉❛❧q✉❡r é s❡♠♣r❡ ❢❡❝❤❛❞❛ ❡✱ ♣♦rt❛♥t♦✱ ❞❡✜♥❡ ✉♠❛ ❡str✉t✉r❛ s✐♠♣❧ét✐❝❛✳

  X dp i ◆♦s ❞♦✐s ♣r✐♠❡✐r♦s ❡①❡♠♣❧♦s✱ ❛ ❢♦r♠❛ s✐♠♣❧ét✐❝❛ t✐♥❤❛ ❛ ❢♦r♠❛ ❝♦♦r❞❡♥❛❞❛ ∧

  i

  dx i ✳ ❖ t❡♦r❡♠❛ ❞❡ ❉❛r❜♦✉① ❛✜r♠❛ q✉❡✱ ❧♦❝❛❧♠❡♥t❡✱ ❡♠ t♦❞❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ ❡①✐st❡♠

  X

  i , p i )

  dp i i ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s (x ♥❛s q✉❛✐s ❛ ❢♦r♠❛ s✐♠♣❧ét✐❝❛ é s✐♠♣❧❡s♠❡♥t❡ ∧ dx ✳

  ✺✺ ❉❡✜♥✐çã♦ ✸✳✷✳✷✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ∈ D(M, ω) é ❞✐t♦ ❍❛♠✐❧t♦♥✐❛♥♦ s❡ i

  X ω =

  −dH, ✭✸✳✹✮

  ∞

  (M ) ♦♥❞❡ H ∈ C ✳

  ❈♦♠♦ ω é ♥ã♦ ❞❡❣❡♥❡r❛❞❛✱ ❛ ❛♣❧✐❝❛çã♦

  1

  Γ : (M ) D(M) −→ Λ

  X 7−→ Xyω

  é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❖s ❝❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s sã♦ ♦s ❝❛♠♣♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✱ ♣♦r ♠❡✐♦ ❞❡ Γ✱ às 1✲❢♦r♠❛s ❡①❛t❛s✳

  ∞

  (M ) , p )

  i i

  ❙❡ H ∈ C ✱ ✉s❛♥❞♦ ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s (x ✱ é ❢á❝✐❧ ✈❡r q✉❡ ♦ ❝❛♠♣♦

  −1 H = Γ (

  ❍❛♠✐❧t♦♥✐❛♥♦ X −dH) t❡♠ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

  n

  X ∂H ∂ ∂H ∂ X H = .

  − ✭✸✳✺✮ ∂p i ∂x i ∂x i ∂p i

  i =1 ✸✳✸ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡

  ◆♦ ✐♥í❝✐♦ ❞❡st❡ ❝❛♣ít✉❧♦✱ ✈✐♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s q✉❡ ❣❡r❛ ✉♠❛ ❝♦rr❡✲ s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✳ ❊♥tr❡t❛♥t♦✱ ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ ❡r❛ ❧♦❝❛❧ ❡✱ ♣♦rt❛♥t♦✱ ❞❡♣❡♥❞❡♥t❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❱❡r❡♠♦s ❛q✉✐ ✉♠❛ ✈❡rsã♦ ✐♥✈❛r✐❛♥t❡ ❞❡str❛ tr❛♥s❢♦r♠❛çã♦ q✉❡ é ❝♦♠✉♠❡♥t❡ ❝❤❛♠❛❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✳

  ∗

  M, ω) ❈♦♠❡ç❛♠♦s ❧❡♠❜r❛♥❞♦ q✉❡ (T ✱ ❝♦♠ ω = dρ✱ é ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛✳

  ∗

  M ❆❧é♠ ❞✐ss♦✱ ♥❛s ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s (x, p) ❡♠ T ✱ ρ s❡ ❡s❝r❡✈❡ ❝♦♠♦

  X ρ = p i dx i .

  ✭✸✳✻✮

  

i

x M x M

  ❈♦♠♦ T é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡♥tã♦✱ ♣❛r❛ t♦❞♦ ξ ∈ T ✱ t❡♠♦s ✉♠ ✐s♦♠♦r✲

  x M ξ (T x M ) ξ

  ✜s♠♦ T ≃ T q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r α ✳

  i , v i ) ξ

  ❯s❛♥❞♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s (x ❡♠ T M✱ ♦ ✐s♦♠♦r✜s♠♦ α é t❛❧ q✉❡ α ξ : T x M ξ (T x M )

  −→ T ∂ ∂ x ξ .

  | 7−→ | ∂x i ∂v i

  X ∂

  ξ = ξ i x

  ✐st♦ é✱ α | ⊗ dx | ✳ ∂v

  i i

  ❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡ é ❛ ❛♣❧✐❝❛çã♦

  ∗

  M L(L) : T M −→ T ✭✸✳✼✮

  ξ

  ✺✻ t❛❧ q✉❡

  V

  (L) ξ ξ , ✭✸✳✽✮

  L(L)(ξ) := d | ◦ α

  V

  ♦♥❞❡ d é ♦ ❞✐❢❡r❡♥❝✐❛❧ ❡①t❡r♥♦ ✈❡rt✐❝❛❧

  V

  1 ∞

  d : C (T M ) (T M ) −→ Λ

  V

  f (f ), 7−→ d

  ❞❡✜♥✐❞♦ ❝♦♠♦

  V

  d (f ) ξ = (df ) T M ,

  π

  | | (ξ) ❝♦♠ π : T M −→ M ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳

  ❊♠ ❝♦♦r❞❡♥❛❞❛s (x, v)✱ ∂f

  V d (f ) = dv i .

  ∂v i P♦rt❛♥t♦✱

  V

  (L) ξ ξ L(L)(ξ) = d | ◦ α

  ∂L ∂ = (ξ)dv i ξ ξ i π )

  (ξ)

  | ◦ ( | ⊗ dx | ∂v i ∂v i

  ∂L = (ξ)dx i π .

  (ξ)

  | ∂v i

  ▲♦❣♦✱

  X ∂L (ξ)dx i π .

  (ξ)

  L(L)(ξ) = | ✭✸✳✾✮ ∂v i

  i ∗ i , p i ) M

  ❙❡✱ ♣♦rt❛♥t♦✱ ✉s❛♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s (x ❞❡ T ✱ t❡♠♦s q✉❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❝♦♦r❞❡♥❛❞❛ ❞❡ L(L) é ( x i = x i

  ∂L p i = . ∂v i

  ∂L

  ❱✐♠♦s ♥♦ ✐♥í❝✐♦ ❞❡st❛ s❡çã♦ q✉❡✱ s❡ L é r❡❣✉❧❛r✱ ✐st♦ é✱ s❡ det ∂v ∂v 6= 0✱ ❡♥tã♦

  j i

  ❡st❛ tr❛♥s❢♦r♠❛çã♦ tr❛♥s❢♦r♠❛ ❛s ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ♥❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥

  X p i v i ❝♦♠ ❍❛♠✐❧t♦♥✐❛♥❛ H = − L✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ❝❛♠♣♦ q✉❡ ❞❡s❝r❡✈❡ ❛s ❡q✉❛çõ❡s

  i

  ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❝♦♠ ▲❛❣r❛♥❣❡❛♥❛ L é tr❛♥s❢♦r♠❛❞♦ ✈✐❛ L(L) ∗ ♥♦ ❝❛♠♣♦ ❍❛♠✐❧t♦♥✐❛♥♦

  X p i v i ❛ss♦❝✐❛❞♦ à ❍❛♠✐❧t♦♥✐❛♥❛ H = − L✳

  i

  1

  (I, M ) P❛r❛ ♦ ❡st✉❞♦ ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❡♠ J ≈ I × T M ♣♦r ♠❡✐♦ ❞♦ ♣r♦❜❧❡♠❛

  1

  g π (ξ, ξ) ✈❛r✐❛❝✐♦♥❛❧✱ ❢♦✐ ♥❡❝❡ssár✐♦ ✐♥tr♦❞✉③✐r ❛ ❧❛❣r❛♥❣❡❛♥❛ L(t, ξ) = (ξ) ✳ ❱❡r❡♠♦s ❛❣♦r❛

  2

  ❝♦♠♦ ❡st❛ ▲❛❣r❛♥❣❡❛♥❛ s❡ tr❛♥s❢♦r♠❛ ♣♦r ♠❡✐♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✳ ▼❛✐s

  ✺✼ ❣❡♦❞és✐❝♦ ♣♦r ♠❡✐♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ♣r✐♠❡✐r♦ ♠♦str❛r q✉❡ ❛ ❧❛❣r❛♥❣❡❛♥❛ L q✉❡ ❞❡s❝r❡✈❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛✳ ❉❡ ❢❛t♦✱ s❡

  X ∂

  ξ = v i ∂x i

  i

  X ∂L

  = g kj v j ✭✸✳✶✵✮

  ∂v k

  

j

  ❡

  2

  ∂ L = g kl .

  ✭✸✳✶✶✮ ∂v l v k

  2

  ∂ L = det (g kl )

  ▲♦❣♦✱ det 6= 0✳ ∂v l v k

  ❚❡♠♦s q✉❡

  X ∂L

  jk

  v j = g (x), ✭✸✳✶✷✮

  ∂v k

  k jk

  ♦♥❞❡ g sã♦ ❝♦❡✜❝✐❡♥t❡s ❞❛ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❛ ♠étr✐❝❛✳

  X

  ∂L

  p i v i ❆❣♦r❛✱ ❧❡♠❜r❛♥❞♦ q✉❡ p = ✱ ❛ ❍❛♠✐❧t♦♥✐❛♥❛ H = − L s❡ ❡s❝r❡✈❡ ❝♦♠♦

  ∂v i

  X

  ik

  H(x, p) = p i p k g − L(x, v(x, p))

  ik

  X X

  X

  1

  ik ik jl

  = p i p k g p k g p l g g ij −

  2

  

ik ij kl

  X

  1

  

kl

= p k p l g .

  2

  kl ✸✳✹ ❈❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s ❡ ❡str✉t✉r❛ ❞❡ P♦✐ss♦♥

  ◆❡st❛ s❡çã♦ ❞✐s❝✉t✐r❡♠♦s ♦ ✐♠♣♦rt❛♥t❡ ❢❛t♦ ❞❡ q✉❡ ❝❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s ❝♦♠✲ ♣♦❡♠ ✉♠ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ❆❧é♠ ❞✐ss♦✱ ✐♥tr♦❞✉③✐♠♦s ❛ ♥♦çã♦ ❞❡ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥ ❡♥tr❡ ❢✉♥çõ❡s ❞❡✜♥✐❞❛s s♦❜r❡ ✈❛r✐❡❞❛❞❡s s✐♠♣❧ét✐❝❛s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❈♦♠❡ç❛♠♦s ❝♦♠ ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✸✳✹✳✶✳ ❙❡❥❛♠ f ❡ g ❞✉❛s ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ y y (M, ω)

  f

  X g ω = ✳ ❖ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥ {f, g} ❡♥tr❡ f ❡ g é ❞❡✜♥✐❞♦ ♣♦r {f, g} := −X

  (g)

  f f g

  −X ✱ ♦♥❞❡ X ❡ X ❞❡♥♦t❛♠ ♦s ❝❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s ❛ss♦❝✐❛❞♦s ❛ f ❡ g r❡s♣❡❝t✐✈❛✲ ♠❡♥t❡✳

  ∞

  (M ) ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞♦ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ f ∈ C é ✉♠❛

  H H (f ) = 0

  ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❞❡ X ✱ ✐st♦ é✱ X ✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ {H, f} = 0✳ y y ω = ω =

  1

  2

  1

  1

  2

  2

  ❙❡❥❛♠ X ❡ X t❛✐s q✉❡ X −dH ❡ X −dH ✱ t❡♠♦s q✉❡ y [X

  1 , X 2 ]yω =

  L

  1

  X X 2 ω

  y y = (X ω) ω

  X

  2

  2 X

  L

  1 − X L

  1

  y y y y = X d(X ω) + d(X X ω) = , H

  1

  2

  1

  2

  1

  2

  ✺✽ ■ss♦ ♣r♦✈❛ ❛ s❡❣✉✐♥t❡ Pr♦♣♦s✐çã♦ ✸✳✹✳✷✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s Ham(ω) é ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡

  ∞ H , X H ] = X ,H 1 , H 2 (M )

  ❡ [X

  1 2 {H

  1 2 } ✱ ♣❛r❛ t♦❞❛s H ∈ C ✳

  ❚❡♠♦s t❛♠❜é♠ ❛ s❡❣✉✐♥t❡ Pr♦♣♦s✐çã♦ ✸✳✹✳✸✳ ❖ ❝♦❧❝❤❡t❡ ❞❡ ♣♦✐ss♦♥ é R✲❜✐❧✐♥❡❛r✱ ❛♥t✐ss✐♠étr✐❝❛ ❡ s❛t✐s❢❛③ ❛ ✐❞❡♥t✐✲ ❞❛❞❡ ❞❡ ❏❛❝♦❜✐✿

  {g, {f, h}} + {h, {g, f}} + {f, {h, g}} = 0;

  ∞

  (M ), ✳ P♦rt❛♥t♦✱ (C { , }) é ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳

  Pr♦✈❛✿ ❆ ❜✐❧✐♥❡❛r✐❞❛❞❡ s♦❜r❡ R ❡ ❛ ❛♥t✐s✐♠❡tr✐❛ ❞♦ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥ sã♦ , X

  ❡✈✐❞❡♥t❡s✳ Pr♦✈❜❛r❡♠♦s✱ ♣♦rt❛♥t♦✱ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✳ P❛r❛ ✐ss♦✱ s❡❥❛♠ X

  1 2 ❡ X

  3

  ❝❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s t❛✐s q✉❡ y X ω =

  1

  1

  −dH y X ω =

  2

  2

  −dH y X ω = .

  3

  3

  −dH ❆♣❧✐❝❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦❧❝❤❡t❡ ❞❡ P♦✐ss♦♥ ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡✱ ♦❜t❡♠✲ s❡ q✉❡

  , H ,H (H )

  1

  2

  3

  3

  {{H }, H } = X {H

  1 2 }

  = [X , X ]ydH

  1

  2

  

3

  y =

  1 , X 2 ]yX 3 ω

  −[X y =

  2

  

3

  −(L

  1

  X X )yX ω

  y y y y =

  X (X

  X ω) + X

  X (X ω)

  2

  

3

  2

  3

  −L

  1

  y y y = X ( , H [X , X ]yω + X

X X ω

  1

  2

  3

  2

  1

  3

  2

  3

  1

  {H }) + X L =

  1 , 2 , H

  

3

2 , 1 , H

  3

  {H {H }} − {H {H }}, ω = 0

  X

  ❖♥❞❡ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡ L

  1 ✳ ▲♦❣♦✱

  , H , H , H

  1

  2

  3

  3

  2

  1

  2

  3

  1 {{H }, H } + {{H }, H } + {{H }, H } = 0.

  ✺✾

  ✸✳✺ ▼ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐

  P❛r❛ ❛ ♣❛rt❡ s❡❣✉✐♥t❡ ❞❡st❛ s❡çã♦ s❡rá ❢✉♥❞❛♠❡♥t❛❧ ❛ s❡❣✉✐♥t❡ , ω ) , ω )

  1

  1

  2

  2

  1

  2

  ❉❡✜♥✐çã♦ ✸✳✺✳✶✳ ❙❡❥❛♠ (M ❡ (M ✈❛r✐❡❞❛❞❡s s✐♠♣❧ét✐❝❛s ❡ T : M −→ M ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✳ T é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ s❡ s❛t✐s❢❛③

  ∗ T ω = ω .

  2

  1 ∗

  ω = ω

  1

  1

  1

  1

  ❊♠ ♣❛rt✐❝✉❧❛r✱ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ F : M −→ M é ❞✐t♦ s✐♠♣❧ét✐❝♦ s❡ F ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ✐♥✜♥✐t❡s✐♠❛❧✱ t❡♠♦s ❛ s❡❣✉✐♥t❡

  ❉❡✜♥✐çã♦ ✸✳✺✳✷✳ ❯♠ ❝❛♠♣♦ X s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ (M, ω) é ✉♠❛ tr❛♥s❢♦r✲ ♠❛çã♦ s✐♠♣❧ét✐❝❛ ✐♥✜♥✐t❡s✐♠❛❧ s❡ X ω = 0.

  L ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ X é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ ✐♥✜♥✐t❡s✐♠❛❧ s❡✱ ❡ s♦♠❡♥t❡

  ∗

  ω = ω s❡✱ s❡✉ ✢✉①♦ é ❝♦♠♣♦st♦ ♣♦r tr❛♥s❢♦r♠❛çõ❡s ✐♥✜♥✐t❡s✐♠❛✐s✱ ✐st♦ é✱ A t ✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝❛♠♣♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❡ ❛ ❢ór♠✉❧❛ ❞❡ ❈❛rt❛♥ é ♣♦ssí✈❡❧

  ♣r♦✈❛r ♦ s❡❣✉✐♥t❡ ❚❡♦r❡♠❛ ✸✳✺✳✸✳ ✭❞❡ ▲✐♦✉✈✐❧❧❡✮ ❖s ❝❛♠♣♦s ❍❛♠✐❧t♦♥✐❛♥♦s sã♦ tr❛♥s❢♦r♠❛çõ❡s s✐♠♣❧ét✐❝❛s ✐♥✜♥✐t❡s✐♠❛✐s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦s ✢✉①♦s ❍❛♠✐❧t♦♥✐❛♥♦s ♣r❡s❡r✈❛♠ ❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡

  n n

  Ω = ω ✱ ♦♥❞❡ ω ❡♥t❡♥❞❡✲s❡ ♣♦r ω ∧ ... ∧ ω ✳

  | {z }

  nvezes

  ❆s tr❛♥s❢♦r♠❛çõ❡s s✐♠♣❧ét✐❝❛s ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ ❢♦r♠❛♠ ✉♠ ❣r✉♣♦✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ❧❡♠❜r❛♥❞♦ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡

  = [

  X , Y ]

  L [X,Y ] L L s❡ ✈❡r✐✜❝❛ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s tr❛♥s❢♦r♠❛çõ❡s s✐♠♣❧ét✐❝❛s ✐♥✜♥✐t❡s✐♠❛✐s ❛❞♠✐t❡ ✉♠❛ ❡s✲ tr✉t✉r❛ ❞❡ á❧❣❡❜r❛ ❞❡ ▲✐❡✳

  ✸✳✺✳✶ ❋✉♥çõ❡s ❣❡r❛❞♦r❛s ❞❡ tr❛♥s❢♦r♠❛çõ❡s s✐♠♣❧ét✐❝❛s

  ❱❡r❡♠♦s ❛❣♦r❛ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❣❡r❛r ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝❧❛ss❡ ❞❡ tr❛♥s❢♦r♠❛çõ❡s s✐♠♣❧ét✐❝❛s✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❡ T : (M, ω) −→ (M, ω) ✉♠❛ tr❛♥s❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ ❡ (q, p) ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s s♦❜r❡ M ✭t❛✐s q✉❡ ❛ ❢♦r♠❛ s✐♠♣❧ét✐❝❛ ω s❡ ❡s❝r❡✈❡ ❝♦♠♦ ω = dp

  ∧ dq✮✳ ▲♦❣♦✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ T é ❡①♣r❡ss❛ ❡♠ ❝♦♦r❞❡♥❛❞❛s ♣♦r T (q, p) = (¯ q(q, p), ¯ p(q, p))

  ✱ t❡♠♦s q✉❡

  

  d¯ p(q, p) (d¯ p

  ✻✵ ♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ d (¯ p(q, p)d¯ q(q, p)

  − pdq) = 0. P♦rt❛♥t♦✱ ♣❡❧♦ ♠❡♥♦s ❧♦❝❛❧♠❡♥t❡✱ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❝♦❜❡rt❛ ♣❡❧❛s ❝♦♦r❞❡♥❛❞❛s (q, p)✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r p(q, p)d¯ ¯ q(q, p)

  − pdq = −df(q, p), ✭✸✳✶✸✮

  ∂ q ¯ ∞

  (U ) ❝♦♠ f ∈ C ✳ ▼❛s s❡ ❝♦♥s✐❞❡r❛r♠♦s q✉❡ det 6= 0✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r p ❡♠ ❢✉♥çã♦

  ∂p

  ❞❛s ❝♦♦r❞❡♥❛❞❛s q ❡ ¯q ❡ s✉❜st✐t✉✐r ❡♠ ✸✳✶✸✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ❞❡✜♥✐♥❞♦ W (q, ¯ q) := f (q, p(q, ¯ q)),

  ♦❜t❡♠♦s q✉❡ ∂W ∂W p(q, ¯ ¯ q)d¯ q dq d¯ q

  − p(q, ¯q)dq = − − ∂q ∂ ¯ q

  ❡✱ ♣♦rt❛♥t♦✱ (

  ∂W

  p =

  ∂q

  ✭✸✳✶✹✮

  ∂W p = ¯ .

  −

  ∂ q ¯

  ❖ s✐st❡♠❛ ✸✳✶✹✱ ♣♦r ❝♦♥str✉çã♦✱ ❞❡✜♥❡ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ (q, p)

  7−→ (¯q, ¯p) s❡ W (q, ¯q) s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦

  2

  ∂ W det 6= 0. ✭✸✳✶✺✮

  ∂ ¯ q∂q ❯♠❛ ❢✉♥çã♦ ❞❡ss❡ t✐♣♦ é ❝❤❛♠❛❞❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ ❞♦ t✐♣♦ ❧✐✈r❡✳

  

✸✳✺✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ❙✐♠♣❧ét✐❝❛s ❡ ♦ ▼ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥ ❏❛❝♦❜✐

  ❖ ♠ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ ❧✐✈r❡ (q, p) 7−→ (¯q, ¯p) t❛❧ q✉❡ H(q(¯ q, ¯ p), p(¯ q, ¯ p)) = ¯ H(¯ q).

  ✭✸✳✶✻✮ ❆ ✈❛♥t❛❣❡♠ ❞❡ss❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛s (¯q, ¯p) é ❞❡✈✐❞❛ ❛♦ ❢❛t♦ ❞❡ q✉❡ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ ❛❞q✉✐r❡♠ ✉♠❛ ❢♦r♠❛ ❢❛❝✐❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳ ❆ s❛❜❡r✱ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥

  H ♣❛r❛ ❛ ♥♦✈❛ ❍❛♠✐❧t♦♥✐❛♥❛ ¯ sã♦ ❞❛❞❛s ♣♦r

  (

  ∂ ¯ H

  ˙¯q = = 0

  ∂ p ¯

  ✭✸✳✶✼✮

  

∂ ¯ H

p = ˙¯ = ν(¯ q).

  − ∂ q

  

¯

  ✻✶ )

  ❈♦♠♦ ˙¯q = 0✱ ♦❜t❡♠♦s q✉❡ ¯q(t) = ¯q é ❝♦♥st❛♥t❡ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ˙¯p = ν(¯q) = ν(¯q é ❝♦♥st❛♥t❡✳ ▲♦❣♦✱ ♦ s✐st❡♠❛ ✸✳✶✼ é ❢❛❝✐❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱ ♦♥❞❡ ❛s s♦❧✉çõ❡s sã♦ ❞❛❞❛s ♣♦r

  ( ¯ q(t) = ¯ q

  ✭✸✳✶✽✮ p(t) = ν(¯ ¯ q )t + ¯ p . P❛r❛ ♦❜t❡r ♦ ✢✉①♦ ♥❛s ❝♦♦r❞❡♥❛❞❛s (q, p) é s✉✜❝✐❡♥t❡ ✐♥✈❡rt❡r ❛ tr❛♥s❢♦r♠❛çã♦✳

  ❯s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞❛ s❡çã♦ ❛♥t❡r✐♦r✱ s❛❜❡♠♦s q✉❡✱ ♣❛r❛ ❝♦♥str✉✐r ✉♠❛ tr❛♥s✲ ❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ ❧✐✈r❡ q✉❡ s❛t✐s❢❛ç❛ ✸✳✶✻ é ♣r❡❝✐s♦ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ W = W (q, ¯q) ❞❛ ❡q✉❛çã♦

  ∂W

H(q, ) = H(¯ q)

  ✭✸✳✶✾✮ ∂q q✉❡ s❛t✐s❢❛ç❛ t❛♠❜é♠ ❛ ❝♦♥❞✐çã♦ ✸✳✶✺✳ ❆ ❡q✉❛çã♦ ✸✳✶✾ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❡q✉❛çã♦ ❞❡

  ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ❯♠ ❞♦s ♠ét♦❞♦s ♠❛✐s ❡✜❝✐❡♥t❡s ♣❛r❛ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ✸✳✶✾ é ♦

  ♠ét♦❞♦ ❞❛ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✳ ❖s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r ♠♦str❛♠ ❝♦♠♦ ❡st❡ ♠ét♦❞♦ ❢✉♥❝✐♦♥❛ ♥❛ ♣rát✐❝❛✳ ❊♠ ❣❡r❛❧✱ ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❞❡st❡ ♠ét♦❞♦ ♥❡♠ s❡♠♣r❡ é ❣❛r❛♥t✐❞❛ ♣♦✐s ♣♦❞❡ ❞❡♣❡♥❞❡r ❞❡ ✉♠❛ ♣❛rt✐❝✉❧❛r ❡s❝♦❧❤❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ▼❛✐♦r❡s ❞❡t❛❧❤❡s s♦❜r❡ ❡st❡ ❛s♣❡❝t♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ♥♦s s❡❣✉✐♥t❡s tr❛❜❛❧❤♦s ❬✶✼✱ ✷✺✱ ✸✷❪✳ ❊①❡♠♣❧♦ ✸✳✺✳✹✳ ✭❋❧✉①♦ ❣❡♦❞és✐❝♦ ♥✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ r❡✈♦❧✉çã♦✮ ❯♠❛ s✉♣❡r❢í❝✐❡ ❞❡ r❡✈✲ ♦❧✉çã♦ S ♣♦❞❡ s❡r ♣❛r❛♠❡tr✐③❛❞❛ ♣♦r

    x = f (r)cosϕ

  1

    x = f (r)senϕ

  2

     x

  3 = g(r).

  ❆ ♠étr✐❝❛ s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ r❡✈♦❧✉çã♦✱ ❞❛❞❛ ♣❡❧❛ ♣❛r❛♠étr✐③❛çã♦ ❛❝✐♠❛✱ ✐♥❞✉③✐❞❛

  3

  ❞❛ ♠étr✐❝❛ ❝❛♥ô♥✐❝❛ ❞♦ R é ❞❛❞❛ ♣♦r

  2

  

2

  2

  2

  2 ′ ′

  g S = f (r) + g (r) dr + f (r) dϕ . ❆ ▲❛❣r❛♥❣❡❛♥❛ L s♦❜r❡ S q✉❡ ❞❡s❝r❡✈❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ❞❛❞❛ ♣♦r

  1

  2

  2

  2

  2

  2

′ ′

  L(r, ϕ, v , v ) = f (r) + g (r) v + f (r) v

  1

  2

  1

  2

  2 ❡ ❛ ❍❛♠✐❧t♦♥✐❛♥❛ H ♦❜t✐❞❛ ♣❡❧❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡ é ❞❛❞❛ ♣♦r

  2

  2

  1 p p

  1

  2 H(r, ϕ, p , p ) = + .

  1

  2

  

2

  2

  2 ′ ′

  2 f (r) + g (r) f (r) ❆❣♦r❛✱ ♣❛r❛ ❛♣❧✐❝❛r♠♦s ♦ ♠ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✱ ♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ W (r, ϕ, ¯ r, ¯ ϕ)

  ❞❛ ❡q✉❛çã♦  

  2 ∂W 2 ∂W

  ∂ϕ

  1

  ∂r

    ϕ +

    = ¯

  2

  2

  2 ′ ′

  2 f (r) + g (r) f (r)

  ✻✷ ❙✉♣♦♥❞♦ q✉❡ ❛ s♦❧✉çã♦ W é ❞❛ ❢♦r♠❛

  W (r, ϕ, ¯ r, ¯ ϕ) = W (r, ¯ r, ¯ ϕ) + W (ϕ, ¯ r, ¯ ϕ),

  1

  2

  ❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ é r❡❡s❝r✐t❛ ❞❛ ❢♦r♠❛  

  2 ∂W

  2

  2 ∂W

  1 ∂ϕ

  1

  ∂r

   

  • ϕ   = ¯

  2

  2

  2 ′ ′

  2 f (r) + g (r) f (r) ❘❡♦r❣❛♥✐③❛♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s

   

  2 ∂W

  2

  2 ∂W

  1 ∂ϕ ∂r

     − 2ϕ = −  . ✭✸✳✷✵✮

  2

  2

  2 ′ ′

  f (r) + g (r) f (r) ❖❜s❡r✈❡ q✉❡ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ❡q✉❛çã♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ ϕ✳ ▲♦❣♦✱

  2 ∂W

  2 ∂ϕ = α(¯ r, ¯ ϕ).

  2

  f (r)

  

∂W

  1

  ❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ✸✳✷✵ ❝♦♠ r❡s♣❡✐t♦ ❛ ✱ ♦❜t❡♠♦s

  

∂r

  s

  2

  ∂W α

  1

  2

  2 ′ ′

  = 2 ¯ ϕ (f (r) + g (r) ) −

  

2

  ∂r f (r) ❡ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ✐♥t❡❣r❛❧ ❝♦♠♣❧❡t❛ ❞❡ ✸✳✷✵ ❞❛❞❛ ♣♦r

  Z s

  2

  α

  2

  2 ′ ′

  W (r, ϕ, ¯ r, ¯ ϕ) = 2 ¯ ϕ (f (r) + g (r) )dr + α(¯ r, ¯ ϕ)ϕ.

  −

  2

  f (r) ❊①❡♠♣❧♦ ✸✳✺✳✺✳ ✭●❡♦❞és✐❝❛s ❞♦ ❊❧✐♣só✐❞❡✮ ❈♦♥s✐❞❡r❡ ♦ ❡❧✐♣só✐❞❡ ❞❛❞♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ♣♦r

  2

  2

  2

  x x x

  1

  2

  3 + + = 1.

  2

  2

  2

  a b c ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ♣❛r❛♠❡tr✐③❛çã♦ ❞♦ ❡❧✐♣só✐❞❡

   p √

  2

   x = acos(θ) ǫ + (1 (ϕ)

  1

   − ǫ)cos 

  √ x = bsen(θ)cos(ϕ)

  2

   p √

  

  2

   x = csen(ϕ) 1 (ϕ)

  3

  − ǫcos ♦♥❞❡ θ ∈ (0, 2π] ❡ ϕ ∈ (0, 2π]✳ ❖❜s❡r✈❡ q✉❡✱ ❝♦♠ ❡st❛ ♣❛r❛♠❡tr✐③❛çã♦✱ ♦ ❡❧✐♣só✐❞❡ é ❝♦❜❡rt♦ ❞✉❛s ✈❡③❡s✳

  )

  ij

  ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ s❡ (g sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ♠étr✐❝❛ ❞♦ ❡❧✐♣só✐❞❡ ✐♥❞✉③✐❞❛

  3

  ❞❛ ♠étr✐❝❛ ❝❛♥ô♥✐❝❛ ❞♦ R ❡♥tã♦

  2

  1

  2

  

2

  = [v A(θ) + v B(ϕ)][C(θ) + D(ϕ)],

  1

  

2

  ✻✸ ♦♥❞❡

  2

  (c θ − a) + (b − a)cos

  A(θ) =

  2

  a + (b θ − a)cos

  2

  (b ϕ − a) + (c − b)cos

  B(ϕ) =

  2

  2

  bsen ϕ + ccos ϕ

  2 C(θ) = (b θ

  − a)sen

  2 D(ϕ) = (c ϕ.

  − b)cos P♦r ♠❡✐♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✱ ♦❜t❡♠♦s ❛ ❍❛♠✐❧t♦♥✐❛♥❛

  2

  2

  1 p p

  1

  1

  2 H = . +

  ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ∂W ∂W

  H θ, ϕ, , = ¯ ϕ ✭✸✳✷✶✮

  ∂θ ∂ϕ ❡ s✉♣♦♥❤❛ q✉❡ ❛ ❢✉♥çã♦ W (θ, ϕ, ¯θ, ¯ϕ) t❡♥❤❛ ❛ ❢♦r♠❛ ♣❛rt✐❝✉❧❛r W = W (θ, ¯ θ, ¯ ϕ) + W (ϕ, ¯ θ, ¯ ϕ).

  1

  2

  ❆❣♦r❛✱ ❞❡ ✸✳✷✶✱ ♦❜t❡♠♦s q✉❡ " #

  2

  2

  1 1 ∂W

  1

  1 ∂W

  2

  2

  • = ¯ ϕ 2(C(θ) + D(ϕ)) A(θ) θ B(ϕ) ϕ

  ❘❡♦❣❛r♥✐③❛♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s !

  2

  2

  1 ∂W

  1

  1 ∂W

  2 ϕ = ϕ .

  − 2C(θ) ¯ − − 2D(ϕ) ¯ A(θ) θ B(ϕ) ϕ

  ❖❜s❡r✈❡ q✉❡ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ ❞❡♣❡♥❞❡ ❞❡ θ ❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ ϕ ❡ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❡♣❡♥❞❡ ❞❡ ϕ ❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ θ✳ ▲♦❣♦✱ t❡♠♦s q✉❡

  

  2 ∂W

  1

  1

   ϕ = α(¯ θ, ¯ ϕ)

  

  A θ − 2C(θ) ¯ (θ) 2 ∂W

  1

  2

   + 2D(ϕ) ¯ ϕ = ϕ) −α(¯θ, ¯

   B (ϕ) ϕ ❡ ❛ ✐♥t❡❣r❛çã♦ ❞❡st❛s ❞✉❛s ❡q✉❛çõ❡s ♥♦s ❢♦r♥❡❝❡ ❛ ❢✉♥çã♦ W

  2 W = α ¯ θ, ¯ ϕ + 2C (θ) ¯ ϕ A (θ) dθ

  1 Z

  • 2 2D (ϕ) ¯ ϕ ϕ B (ϕ) dϕ + β ¯ θ, ¯ ϕ .

  − α ¯θ, ¯ = ¯ θ

  1 Z

  ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ α ¯θ, ¯ϕ ❡ q✉❡ β = 0✳ ▲♦❣♦✱

  1

  2

  2

  ¯ W = θ + 2C (θ) ¯ ϕ A (θ) dθ +

  2D (ϕ) ¯ ϕ B (ϕ) dϕ − ¯θ

  1 Z Z

  ❡ ❝♦♥❝❧✉✐♠♦s q✉❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ♥♦ ❡❧✐♣só✐❞❡ é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳

  ✻✹

  

✸✳✻ ❚❡♦r✐❛ ●❡♦♠étr✐❝❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐

  M ❈❧❛ss✐❝❛♠❡♥t❡✱ ❞❛❞❛ ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ H s♦❜r❡ ♦ ❡s♣❛ç♦ ❝♦t❛♥❣❡♥t❡ T

  i )

  ❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s (q ✱ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲

  ∞

  (M ) ❏❛❝♦❜✐ é ✉♠❛ ❝♦♥❞✐çã♦ s♦❜r❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ S ∈ C ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

  ∂S H q , = c

  i

  ∂q i ♦♥❞❡ c é ✉♠❛ ❝♦♥st❛♥t❡ ✜①❛❞❛✳

  ❯♠❛ ✈❡③ q✉❡✱ ❡♠ ✈✐st❛ ❞♦ t❡♦r❡♠❛ ❞❡ ❉❛r❜♦✉①✱ q✉❛❧q✉❡r ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛

  ∗

  (M, ω)

M

  ♣♦❞❡ s❡r ❧♦❝❛❧♠❡♥t❡ ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠ ❛❧❣✉♠ ❛❜❡rt♦ ❞♦ ✜❜r❛❞♦ ❝♦t❛♥❣❡♥t❡ T ✱ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ♣♦❞❡♠ s❡r ❡st✉❞❛❞❛s ❞❡ ❢♦r♠❛ ❣❡r❛❧ ❡♠ ✈❛r✐❡❞❛❞❡s s✐♠✲ ♣❧ét✐❝❛s✳

  ❆q✉✐✱ ❞❛r❡♠♦s ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡st❛ ❝❧❛ss❡ ❞❡ ❡q✉❛çõ❡s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ✉♠ ♠ét♦❞♦ ❣❡♦♠étr✐❝♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ s♦❧✉çõ❡s✳

  P❛r❛ ❡st❡ ✜♠✱ ♣r✐♠❡✐r♦ r❡❧❡♠❜r❛♠♦s ❛ ❞❡✜♥✐çã♦ ❣❡♦♠étr✐❝❛ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r✲ ❡♥❝✐❛❧ ♣❛r❝✐❛❧ ✭❊❉P✮ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❉❡✜♥✐çã♦ ✸✳✻✳✶✳ ❯♠❛ ❊❉P ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❡s❝❛❧❛r é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ E ❞♦ ♣r✐♠❡✐r♦

  1

  (n, N ) ❡s♣❛ç♦ ❞❡ ❥❛t♦s J ❞❛s s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ n + 1✲ ❞✐♠❡♥s✐♦♥❛❧ N✳

  1 J (n, N )

  é ✉♠❛ ✈❛r✐❡❞❛❞❡ 2n+1✲❞✐♠❡♥s✐♦♥❛❧ ❡q✉✐♣❛❞❛ ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛r✲

  1

  t❛♥ C ✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s✱ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ N ❡q✉✐♣❛❞❛ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s

  1

  (q , ..., q n ) (n, N ) , ..., q n , u, v , ..., v n )

  1 ✱ J é ♥❛t✉r❛❧♠❡♥t❡ ❡q✉✐♣❛❞❛ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s (q

  1 1 t❛✐s

  X

  1 , ..., q n )

  v i dq i q✉❡ u = u(q ❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❈❛rt❛♥ é ❞❡s❝r✐t❛ ♣♦r C = Ann{du − }✳

  i n 1 i dq i

  (n, N ) ❆ ✶✲❢♦r♠❛ θ := du − u ❞❡✜♥❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❝♦♥t❛t♦ ❡♠ J ✱ ✐st♦ é✱ θ ∧ (dθ)

  n

  = dθ é ✉♠❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ ✭❛q✉✐✱ (dθ) ∧ ... ∧ dθ ♥✲✈❡③❡s✮✳

  ◆❡ss❛s ❝♦♦r❞❡♥❛❞❛s✱ t❡♠♦s q✉❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❧♦❝❛❧ ❞❡ ✉♠❛ ❊❉P ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é F (q i , u, v i ) = 0.

  

1

  (N, R) sã♦ n✲❞✐♠❡♥s✐♦♥❛✐s ❡ sã♦ ❝❤❛♠❛❞❛s ❱❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ♠❛①✐♠❛✐s ❞❡ C s♦❜r❡ J ▲❡❣❡♥❞r✐❛♥❛s✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✱ ✉♠❛ ✈❛r✐❡❞❛❞❡ ▲❡❣❡♥❞r✐❛♥❛ t❡♠ ❛ ❢♦r♠❛

  (q i , u, v i ) : u = f (q i ), v j = ∂ q j f (q i ) ✳

  ❚❡♠♦s ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✸✳✻✳✷✳ ❙♦❧✉çõ❡s ❞❡ ✉♠❛ ❊❉P ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ E sã♦ s✉❜✈❛r✐❡❞❛❞❡s ▲❡❣❡♥✲ ❞r✐❛♥❛s ❞❡ E✳

  ✻✺

  1

  (n, N ) ▲♦❝❛❧♠❡♥t❡✱ J ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣r♦❞✉t♦

  ∗

  T N × R✳ ◆❡st❛ ✐❞❡♥t✐✜❝❛çã♦ ❧♦❝❛❧✱ ❛ ❝♦♦r❞❡♥❛❞❛ u é ✉♠❛ ❝♦♦r❞❡♥❛❞❛ s♦❜r❡ ✉♠❛ ✈✐③✐♥✲

  ∗ i , v i )

  N ❤❛♥ç❛ ❞❡ R ❡ (q sã♦ ❝♦♦r❞❡♥❛❞❛s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ T ✳ ◆❡st❛s ❝♦♦r❞❡♥❛❞❛s

  i i

  ❧♦❝❛✐s✱ dθ = dq ∧ dv é ❛ ❢♦r♠❛ s✐♠♣❧ét✐❝❛ ❝❛♥ô♥✐❝❛✳

  u

  ◗✉❛♥❞♦ ✭✸✳✷✷✮ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ u✱ ✐st♦ é✱ ∂ é ✉♠❛ s✐♠❡tr✐❛ ❞❛ ❊❉P ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❡♥tã♦✱ s❡ u = u(q) é ✉♠❛ s♦❧✉çã♦✱ u(q) + constante é t❛♠❜é♠ ✉♠❛ s♦❧✉çã♦✳ ◆❡st❡

  ∗

  N ❝❛s♦✱ ❛ ❡q✉❛çã♦ é s✐♠♣❧❡s♠❡♥t❡ ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ T ❥á q✉❡ ♦ ❢❛t♦ ❞❡ F

  , v )

  i i

  ♥ã♦ ❞❡♣❡♥❞❡r ❞❡ u ✐♠♣❧✐❝❛ q✉❡ F = F (q ✳ ❆ss✐♠✱ ❞❡✈✐❞♦ à ❡str✉t✉r❛ s✐♠♣❧ét✐❝❛ ❞❡st❛ ✈✐③✐♥❤❛♥ç❛✱ ♥❡st❡s ❝❛s♦s ❛ ❡q✉❛çã♦ é ❝❤❛♠❛❞❛ s✐♠♣❧ét✐❝❛✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❡st❛ t❡r♠✐♥♦❧♦❣✐❛✱ ❡ ❡♠ ✈✐st❛ ❞❛ ❞❡s❝r✐çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❞❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❢❡✐t❛ ❛♥t❡r✐♦r♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❥✉st❛♠❡♥t❡ ❝♦♠♦

  u

  ✉♠❛ ❡q✉❛çã♦ s✐♠♣❧ét✐❝❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❆ ♣r❡s❡♥ç❛ ❞❛ s✐♠❡tr✐❛ ∂ é✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♠✉❧t✐✲✈❛❧✉❛❞❛s ♣❛r❛ ❡st❛ ❝❧❛ss❡ ❞❡ ❡q✉❛çõ❡s✳ ❉❡ss❛ ❢♦r♠❛✱ r❡❧❡♠❜r❛♥❞♦ q✉❡✱ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ 2n✲❞✐♠❡♥s✐♦♥❛❧

  (M, Ω)

  L = 0

  ✱ ✈❛r✐❡❞❛❞❡s ▲❛❣r❛♥❣✐❛♥❛s sã♦ ❛s s✉❜✈❛r✐❡❞❛❞❡s n✲❞✐♠❡♥s✐♦♥❛✐s L ❞❡ M t❛✐s ω| ✭✈❛r✐❡❞❛❞❡s ▲❛❣r❛♥❣❡❛♥❛s sã♦ ✈❛r✐❡❞❛❞❡s ❞❡ ❞✐♠❡♥sã♦ ♠❛①✐♠❛❧ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ω L = 0

  | ✮✱ ♣♦❞❡♠♦s ✉s❛r ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✸✳✻✳✸✳ ❉❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ (M, ω)✱ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲ ❏❛❝♦❜✐ Γ é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❞❡ M ❡ s✉❛s s♦❧✉çõ❡s sã♦ s✉❜✈❛r✐❡❞❛❞❡s ▲❛❣r❛♥❣✐❛♥❛s ❛s q✉❛✐s ❡stã♦ ❝♦♥t✐❞❛s ❡♠ E✳

  ❊♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✱ ✉♠❛ ✈❛r✐❡❞❛❞❡ ▲❛❣r❛♥❣✐❛♥❛ t❡♠ ❛ ❢♦r♠❛

  i

  (q i , p i ) : p j = ∂ q j W (x )

  1 , ..., q n )

  ✱ ♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ W = W (q ✳ ❖ ❡♥tã♦ ❝❤❛♠❛❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ é ♦ ❞❡ ❡♥❝♦♥tr❛r s♦❧✉çõ❡s ♣❛r❛

  ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ❡st❡ é ✉♠ ♣r♦❜❧❡♠❛ ✐♥✈❛r✐❛♥t❡✳ ❆ ❡♥tã♦ ❝❤❛♠❛❞❛ t❡♦r✐❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ✐♥✈❡st✐❣❛ ❛ ❝♦♥❡①ã♦ ❡♥tr❡ ❛s s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❡ ❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ✐♥✈❛r✐❛♥t❡ ❡st❛ ❝♦♥❡①ã♦ ❡ ❞❛❞❛ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛ ❛❜s♦rçã♦✱ ♦ q✉❛❧ ♣♦❞❡ s❡r ✐❧✉str❛❞♦ ❝♦♠♦ s❡❣✉❡✳

  P❛r❛ q✉❛❧q✉❡r ❤✐♣❡rs✉♣❡r❢í❝✐❡ Γ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ (M, ω)✱ ❡♠ t♦❞♦

  q (Γ)

  ♣♦♥t♦ q é ❞❡✜♥✐❞♦ ✉♠ s✉❜❡s♣❛ç♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ✭♣❡❧❛ ♥ã♦✲❞❡❣❡♥❡r❛çã♦ ❞❡ ω✮ l ⊂

  T q Γ q M

  q Γ q (Γ) =

  ⊂ T ♦ q✉❛❧ é✱ ♣♦r ❞❡✜♥✐çã♦✱ ♦ ❝♦♠♣❧❡♠❡♥t♦ ω✲♦rt♦❣♦♥❛❧ ❞❡ T ✱ ✐st♦ é✱ l

  q M ; ω(ξ, η) = 0, q Γ q (Γ)

  {ξ ∈ T ∀η ∈ T }✳ ❖ s✉❜❡s♣❛ç♦ l é ❝❤❛♠❛❞♦ ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ Γ ♥♦ S l q (Γ)

  ♣♦♥t♦ q ❡ l(Γ) = é ❛ ❞✐str✐❜✉✐çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ Γ✳

  q (Γ) q Γ

  ❖ ❡s♣❛ç♦ ❝❛r❛❝t❡ríst✐❝♦ l é ❝♦♥t✐❞♦ ❡♠ T ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ω s❡r✐❛ ❞❡❣❡♥✲

  q (Γ) q Γ q M

  ❡r❛❞❛✳ ❉❡ ❢❛t♦✱ s❡ l ❡st✐✈❡ss❡ ❝♦♥t✐❞♦ ♥♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ T ❡♠ T t❡rí❛♠♦s q✉❡ M Γ

  q q

  ❡①✐st❡ ✉♠ ξ ∈ T t❛❧ q✉❡ ω(ξ, ·) = 0 ❡♠ T ❡✱♣♦rt❛♥t♦✱ ξ s❡r✐❛ ✉♠ ✈❡t♦r ♥ã♦✲♥✉❧♦ t❛❧

  ξ ω = 0

  ✻✻

  q (Γ) q L

  s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ s❡ L ⊂ Γ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ▲❛❣r❛♥❣✐❛♥❛✱ ❡♥tã♦ l ⊂ T ✱ ∀q ∈ L✳

  q (Γ) q (Γ)

  ❆ r❛sã♦ ♣❛r❛ l s❡r ❛❜s♦r✈✐❞♦ ♣♦r L é ❛ s❡❣✉✐♥t❡✿ s❡ l ❢♦ss❡ tr❛♥s✈❡rs❛❧ ❛ L✱ ❡♥tã♦ l q (Γ) q L

  q

  ⊕ T s❡r✐❛ ✉♠ (n + 1)✲❞✐♠❡♥s✐♦♥❛❧ s✉❜❡s♣❛ç♦ ♥♦ q✉❛❧ ω s❡ ❛♥✉❧❛✳ ▼❛s ❛ ❞✐♠❡♥sã♦

  q M V = 0

  ♠❛①✐♠❛❧ ❞♦ s✉❜❡s♣❛ç♦s V ⊂ T t❛✐s q✉❡ ω| é n ❡ ❝♦♥✐❝✐❞❡ ❝♦♠ ❛ ❞✐♠❡♥sã♦ ❞♦s s✉❜❡s♣❛ç♦s ▲❛❣r❛♥❣❡❛♥♦s✳

  H

  ❆❣♦r❛✱ ❥á q✉❡ X é t❛♥❣❡♥t❡ ❛ H = c ❡ é t❛♠❜é♠ ✉♠❛ s✐♠❡tr✐❛ ✐♥✜♥✐t❡s✐♠❛❧ ❞❛ ❡str✉t✉r❛ s✐♠♣❧ét✐❝❛ ω✱ s❡✉ ✢✉①♦ tr❛♥s❢♦r♠❛ ✈❛r✐❡❞❛❞❡s ❧❛❣r❛♥❣❡❛♥❛s ❡♠ ✈❛r✐❡❞❛❞❡s y

  ω = dH

  H

  ❧❛❣r❛♥❣❡❛♥❛s ❡ s♦❧✉çõ❡s ❞❡ Γ = {H = c} ❡♠ s♦❧✉çõ❡s✳ ▼❛s✱ ❛ ❝♦♥❞✐çã♦ X

  H

  ✐♠♣❧✐❝❛ q✉❡ X é ω✲♦rt♦❣♦♥❛❧ ❛ t♦❞♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡ ❛ H = c✱ ❡ ♣♦rt❛♥t♦

  X H H é ω✲♦rt♦❣♦♥❛❧ ❛ Γ✳ P♦rt❛♥t♦✱ X ❣❡r❛ ❛ ❞✐str✐❜✉✐çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡ ♦ ♣r✐♥❝í♣✐♦ ❞❛

  H H

  ❛❜s♦rçã♦ ✐♠♣❧✐❝❛ q✉❡ X é t❛♥❣❡♥t❡ ❛ t♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ Γ✳ P♦rt❛♥t♦✱ X tr❛♥s❢♦r♠❛ t♦❞❛ s♦❧✉çã♦ ❞❡ Γ ♥❡❧❛ ♠❡s♠❛✳ ■ss♦ ♥♦s ❞á ✉♠❛ ❢♦r♠❛ ❞❡ ❝♦♥str✉✐r ❡①♣❧✐❝✐t❛♠❡♥t❡ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ❊ss❛ ❝♦♥str✉çã♦ s❡❣✉❡ ❛ ♠❡s♠❛ ✐❞❡✐❛ ❞♦ ♠ét♦❞♦ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ♣❛r❛ ❊❉P✬s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❉❡ ❢❛t♦✱ ❡❧❛ ♣♦❞❡ s❡r ✐♥t❡r♣r❡t❛❞❛ ❝♦♠♦ ✉♠❛ ❡s♣❡❝✐❛❧✐③❛çã♦ ❞♦ ♠ét♦❞♦ ♣❛r❛ ♦ ❝❛s♦ s✐♠♣❧ét✐❝♦✳

  ′ t H

  ❙❡❥❛ {A } ♦ ✢✉①♦ ❞❡ X ❡ L ⊂ M ✉♠❛ ✈❛r✐❡❞❛❞❡ (n − 1)✲❞✐♠❡♥s✐♦♥❛❧ ♣ré✲

  ′ L = 0

  H

  ▲❛❣r❛♥❣✐❛♥❛ ✭✐st♦ é✱ t❛❧ q✉❡ ω| ✮ ❝♦♥t✐❞❛ ❡♠ Γ✳ ❙❡ ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X é S

  ′ ′

  A t (L ) tr❛♥s✈❡rs❛❧ ❛ L ✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ✈❛r✐❡❞❛❞❡ L = t ✳ ❊st❛ é ✉♠❛ s♦❧✉çã♦

  q L =

  ❞❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ q ∈ L✱ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ T

  ′ ′

  T q (A t (L )) H > = 0 t H ✱ ♦♥❞❡ ω| ✱ ❥á q✉❡ A é ✉♠❛ s✐♠❡tr✐❛ ❞❡ ω✳ ▲♦❣♦✱ X

  ⊕ < X | x T q (A t (L ))

  ′ q (A t (L ))

  é ω✲♦rt♦❣♦♥❛❧ ❛ T ❡ L é ▲❛❣r❛♥❣❡❛♥❛✳ P♦rt❛♥t♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ s❡ r❡❞✉③ ❛♦s s❡❣✉✐♥t❡s ❞♦✐s ♣r♦❜❧❡♠❛s✿

  ′

  (1) (n

  ❞❡s❝r❡✈❡r t♦❞❛ s✉❜✈❛r✐❡❞❛❞❡ ♣ré✲▲❛❣r❛♥❣❡❛♥❛ L − 1)✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ Γ❀

  ′

  (2) ❝♦♥str✉✐r L ♣❛rt✐♥❞♦ ❞❡ L ✳

  ❚❡❝♥✐❝❛♠❡♥t❡✱ (2) ❝♦♥s✐st❡ ♥❛ ✐♥t❡❣r❛çã♦ ❞♦ ✢✉①♦ ❞♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✱ ♦ q✉❛❧ é ✉♠ ♣r♦❜❧❡♠❛ s♦❧ú✈❡❧ ❛ ♣r✐♥❝í♣✐♦✳

  P♦r ♦✉tr♦ ❧❛❞♦✱ ❛s ❝♦♥s✐❞❡r❛çõ❡s ❛❝✐♠❛ ♣r♦✈❛♠ q✉❡ ✈❛r✐❡❞❛❞❡s ♣ré✲▲❛❣r❛♥❣✐❛♥❛s s♦❜r❡ Γ sã♦ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ ✈❛r✐❡❞❛❞❡s ▲❛❣r❛♥❣❡❛♥❛s✳ ▲♦❣♦✱ ❝♦♠♦ ✉♠❛ ✈❛r✐❡❞❛❞❡ (q i , p i ) : p j = ∂ q W

  ▲❛❣r❛♥❣❡❛♥❛ t❡♠ ❛ ❢♦r♠❛ ❧♦❝❛❧ j ✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❝♦♦r❞❡♥❛❞❛s ♥❛s q✉❛✐s ❛s ✈❛r✐❡❞❛❞❡s ♣ré✲▲❛❣r❛♥❣❡❛♥❛s q✉❡ ❡stã♦ ❝♦♥t✐❞❛s ❡♠ Γ ♣♦❞❡♠ s❡r ❧♦❝❛❧✲

  ′

  = (q i , p i ) : q n = 0, p j = ∂ q φ, p n = φ n

  j

  ♠❡♥t❡ r❡♣r❡s❡♥t❛❞❛s ❝♦♠♦ L ✱ ♣❛r❛ ❛❧❣✉♠❛s

  , ..., q n ) n = φ n (q , ..., q n )

  1

  1

  ❢✉♥çõ❡s φ = φ(q −1 ❡ φ −1 ✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ❡♠ ❝❛❞❛ ❝❛rt❛ ❝♦♦r✲

  n

  ❞❡♥❛❞❛✱ ✈❛r✐❛♥❞♦ ❛s ❢✉♥çõ❡s φ ❡ φ ✱ ♣♦❞❡♠♦s ❧♦❝❛❧♠❡♥t❡ ❞❡s❝r❡✈❡r t♦❞❛s ❛s ✈❛r✐❡❞❛❞❡s

  ′

  ♣ré✲▲❛❣r❛♥❣❡❛♥❛s L ✳ ■ss♦ ❞á ✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ (1)✳ ❆❣♦r❛✱ é út✐❧ ✐♥tr♦❞✉③✐r ❛ s❡❣✉✐♥t❡ t❡r♠✐♥♦❧♦❣✐❛✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ♣ré✲▲❛❣r❛♥❣❡❛♥❛

  ′

  (n − 1)✲❞✐♠❡♥s✐♦♥❛❧ L ⊂ Γ s❡rá ❝❤❛♠❛❞♦ ✉♠ ✈❛❧♦r ✐♥✐❝✐❛❧ ❞❡ ❈❛✉❝❤② ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡

  ′ H

  ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ Γ✳ ❙❡ L é tr❛♥s✈❡rs❛❧ à ❝❛r❛❝t❡ríst✐❝❛ X ✱ ♦ ✈❛❧♦r ✐♥✐❝✐❛❧ ❞❡ ❈❛✉❝❤② s❡rá

  ✻✼ ❝❤❛♠❛❞♦ ♥ã♦✲❝❛r❛❝t❡ríst✐❝♦✳

  ❯s❛♥❞♦ ❡st❛ t❡r♠✐♥♦❧♦❣✐❛✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ♠ét♦❞♦ ❞❛ s♦❧✉çã♦ ❡①♣❧í❝✐t❛ ♠❡♥✲ ❝✐♦♥❛❞♦ ❛❝✐♠❛ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ✈❛✐ ❛♦ ❧♦♥❣♦ ❞♦s s❡❣✉✐♥t❡s três ♣❛ss♦s✿

  (a) ❝♦♥str✉✐r ✉♠ ✈❛❧♦r ✐♥✐❝✐❛❧ ❞❡ ❈❛✉❝❤② ♥ã♦✲❝❛r❛❝t❡ríst✐❝♦❀

  (b) H ✐♥t❡❣r❛r X ❀

  S

  ′

  (c) A t (L ) ❝♦♥str✉✐r L = t ✳

  ◆♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ à ❝♦♥❡①ã♦ ❡♥tr❡ ❛s s♦❧✉çõ❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲ ❏❛❝♦❜✐ Γ = {H = c} ❡ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ s✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦✱ é ♣r❡❝✐s♦ ❧❡♠❜r❛r ❛ ❞❡✲ s❝r✐çã♦ q✉❡ ❞❡♠♦s ❞❡ tr❛♥s❢♦r♠❛çõ❡s s✐♠♣❧ét✐❝❛s ❡♠ t❡r♠♦s ❞❡ ❢✉♥çõ❡s ❣❡r❛tr✐③❡s✳

  ❉❡ ❢❛t♦✱ ✉♠❛ s♦❧✉çã♦ ❞❡ Γ t❡♠ ❛ ❢♦r♠❛ L = i , p i = ∂ q i W

  {q } ❡✱ ♣♦r ❞❡✜♥✐çã♦✱ é t❛❧ q✉❡

  ∂W H q, = c.

  ∂q ❆❣♦r❛✱ ❝♦♥s✐❞❡r❡ ✉♠❛ ❢❛♠í❧✐❛ Γ(¯q) ❞❡ ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐

  ∂W H q, = K(¯ q

  1 , ..., ¯ q n )

  ∂q , ..., ¯ q n )

  ❛ q✉❛❧ é ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ K ❞❡ n ♣❛râ♠❡tr♦s (¯q

  1 ✳ ❚❛❧ ❢✉♥çã♦

  K ♣♦❞❡r✐❛ ❛té s❡r ❝♦♥st❛♥t❡✳

  , ..., q n , ¯ q , ..., ¯ q n )

  1

  1

  ❯♠❛ ✐♥t❡❣r❛❧ ❝♦♠♣❧❡t❛ ❞❡ Γ(¯q) é ✉♠❛ s♦❧✉çã♦ W = W (q t❛❧ q✉❡

  2 ∂ W

  det( ) 6= 0✳

  ∂ q∂q ¯

  ❉❛❞❛ ✉♠❛ ✐♥t❡❣r❛❧ ❝♦♠♣❧❡t❛✱ ♦ s✐st❡♠❛ (

  ∂W

  p =

  ∂q ∂W

  p = ¯ − ∂ q

  ¯

  ❞❡✜♥❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ T : (¯q, ¯p) 7→ (q, p)✳ ❊♠ t❡r♠♦s ❞❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛s (¯q, ¯p)✱ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ sã♦ ❞❡s❝r✐t❛s

  ♣♦r ♠❡✐♦ ❞❛ ♥♦✈❛ ❍❛♠✐❧t♦♥✐❛♥❛ K (¯q) ♣❡❧❛s ❡q✉❛çõ❡s (

  ∂K

  ˙¯q = = 0,

  ∂ p ¯

∂K

  p = ˙¯ = ν(¯ q) −

  

∂ q

¯

  ❡ ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡ ✐♥t❡❣r❛❞❛s ♥❛ ❢♦r♠❛   q = ¯ ¯ q

  ∂W  p = ν(¯ ¯ q t + ¯ p ) = (q, ¯ q),

  − ∂ ¯ q

  n

  , ¯ p

  ✻✽ ❊♥tã♦✱ t♦♠❛♥❞♦ ❛ ✐♠❛❣❡♠ ♣♦r ♠❡✐♦ ❞❡ T ❞❡st❛ s♦❧✉çã♦✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ ♦r✐❣✐♥❛❧✳ ❊st❡ ♠ét♦❞♦ ❞❡ ✐♥t❡❣r❛çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡

  ❍❛♠✐❧t♦♥ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ♠ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ❯♠❛ s✐♠♣❧❡s ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ✐♥t❡❣r❛❧ ❝♦♠♣❧❡t❛ é q✉❡ ❛s ♥♦✈❛s

  i H

  ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s {¯q } ❢♦r♠❛♠ ✉♠ s✐st❡♠❛ ❞❡ n ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❡ X ✐♥✈♦❧✉t✐✈♦✳ ◆❛ ♣rát✐❝❛✱ ♦ ♠❛✐s ❡✜❝✐❡♥t❡ ♠ét♦❞♦ ♣❛r❛ ❝❛❧❝✉❧❛r ✐♥t❡❣r❛✐s ❝♦♠♣❧❡t❛s ❞❡ ✭✸✳✷✷✮ é

  ♣♦r s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✳ ❊♥tr❡t❛♥t♦✱ ❛ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ❞❡♣❡♥❞❡ ❞❡ ✉♠❛ ❡s❝♦❧❤❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡ ♥❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ❛♣❧✐❝á✲❧❛✳ ❯♠❛ ❞✐s❝✉ssã♦ ❞❡st❡ ❛s♣❡❝t♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✼✱ ✷✺✱ ✸✷❪✳

  ✸✳✼ ❆♣❧✐❝❛çã♦ ❛ ❛❧❣✉♠❛s ▼étr✐❝❛s ❞❡ ❊✐♥st❡✐♥

  ◆❡ss❛ s❡çã♦ ❛♣❧✐❝❛r❡♠♦s ♦ ♠ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ à ✐♥t❡❣r❛çã♦ ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❞❛s s❡❣✉✐♥t❡s ❝❧❛ss❡s ❞❡ ♠étr✐❝❛s ❞❡ ❊✐♥st❡✐♥✿

  2 q ζ (q

  1

  2

  2

  2

  2

  1 2 −A)

  dq dq + ǫ q (dq + F (q ) dq )

  1

  2

  3

  ✭❛✮ g = ǫ

  1 −

  2

  1

  3 4 ✱ q q 1 −A

  1

  2

  2

  2

  2

  , ǫ = = ) = sinh (q ), (q ), sin (q ) ♦♥❞❡ ǫ

  1 2 ±1 ❡ ζ ±1 ❡ F (q

  3 3 −cosh

  3 3 ✳

  1

  2

  2

  2

  2

  2 √ 1 (dq dq ) + ǫ 2 q 1 (dq + ǫdq )

  ✭❜✮ g = ǫ − ζ ✱

  q

  1

  2

  3

  4

  1

  2

  , ǫ , ζ =

  1

  2 ♦♥❞❡ ǫ, ǫ ±1.

  ❊st❛s ♠étr✐❝❛s sã♦ s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥ ♥♦ ✈á❝✉♦ Ric(g) = 0 s♦❜ ❛s s❡❣✉✐♥t❡s ❤✐♣ót❡s❡s✿ ✭■✮ ♦ ❡s♣❛ç♦ ❛❞♠✐t❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ ❑✐❧❧✐♥❣ 3✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ór❜✐t❛s 2✲❞✐♠❡♥s✐♦♥❛✐s✳

  ✭■■✮ ❛ ♠étr✐❝❛ g é ♥ã♦ ❞❡❣❡♥❡r❛❞❛ s♦❜r❡ ❡ss❛s ór❜✐t❛s 2✲❞✐♠❡♥s✐♦♥❛✐s ❡ ❛ ❞✐str✐❜✉✐çã♦ ♦rt♦❣♦♥❛❧ às ór❜✐t❛s é ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳

  ❊st❛s ♠étr✐❝❛s ❛❞♠✐t❡♠✱ ❝♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✱ ❛❧❣✉♠❛s ♠étr✐❝❛s ♠✉✐t♦ s✐❣✲

  2

  2

  ) = sen (q ), ζ = = ǫ = 1

  3

  3

  1

  2

  ♥✐✜❝❛t✐✈❛s✳ ❉❡ ❢❛t♦✱ s❡ ❝♦♥s✐❞❡r❛♠♦s ❡♠ (a) F (q −1, ǫ ❡

  GM

  A = 2

  2 c ✭G é ❛ ❝♦♥st❛♥t❡ ❞❡ ◆❡✇t♦♥✱ c ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ❡ M ✉♠❛ ♠❛ss❛✮ ❡♥tã♦ ♦❜t❡✲

  ♠♦s ❛ s♦❧✉çã♦ ❞❡ ❙❝❤✇❛r③❝❤✐❧❞ ♣❛r❛ ♦ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❝♦♠ s✐♠❡tr✐❛ ❡s❢ér✐❝❛ ♣r♦❞✉③✐❞❛ ♣♦r ✉♠❛ ♠❛ss❛ M ♥ã♦ r♦t❛♥t❡✳ ❆s á❧❣❡❜r❛s ❞❡ ❑✐❧❧✐♥❣ ❞❡st❛s ♠étr✐❝❛s sã♦ ❝♦♠♣❧❡t❛♠❡♥t❡

  2

  2

  ) = senh q q

  3

  3

  3

  ❞❡t❡r♠✐♥❛❞❛s ❝♦♠♦ s❡❣✉❡✳ ◆♦ ❝❛s♦ (a)✱ Kill(g) = SO(2, 1) s❡ F (q ✱ −cosh ✱

  2

  ) = sen q

  3

  3

  ❛♦ ❝♦♥trár✐♦✱ s❡ F (q ✱ Kill(g) = SO(3)✳ ◆♦ ❝❛s♦ (b)✱ ❛ á❧❣❡❜r❛ ❞❡ ❑✐❧❧✐♥❣ é

  2

  ± dη ✳ ▼❛✐♦r❡s ❞❡t❛❧❤❡s s♦❜r❡ ❡st❛s ♠étr✐❝❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ♥♦s ❛rt✐❣♦s ❬✹✸❪✱ ❬✹✹❪✱ ❬✹✺❪✳ ◆❡st❡s tr❛❜❛❧❤♦s ❢♦r❛♠ t❛♠❜é♠ ❝❧❛ss✐✜❝❛❞❛s ❛s ♠étr✐❝❛s ❝♦♠ ór❜✐t❛s 2✲❞✐♠❡♥s✐♦♥❛✐s ❝♦rr❡s♣♦♥❞❡♥t❡s às ♠❡s♠❛s ❤✐♣ót❡s❡s ✭r❡❧❛t✐✈❛♠❡♥t❡ à ♥ã♦ ❞❡❣❡♥✲

  ✻✾ ❖ ❝❛s♦ ❞❡ ♠étr✐❝❛s ❞❡❣❡♥❡r❛❞❛s s♦❜r❡ ❛s ór❜✐t❛s ❞❡ ❑✐❧❧✐♥❣✱ ❛♦ ❝♦♥trár✐♦✱ ❢♦✐ ❝♦♠♣❧❡t❛✲ ♠❡♥t❡ ❝❧❛ss✐✜❝❛❞♦ ❡♠ ❬✷✸❪✱ ❬✷✹❪✳ ❊♠ ❬✺✸❪ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ r❡s✉♠♦ ❝♦♠♣❧❡t♦ ❞❡ss❡s r❡s✉❧t❛❞♦s ❥✉♥t♦ às ♣♦ssí✈❡✐s ❛♣❧✐❝❛çõ❡s ❢ís✐❝❛s ❞❡ss❛s ♠étr✐❝❛s ❡♠ r❡❧❛t✐✈✐❞❛❞❡ ❣❡r❛❧✳

  ✸✳✼✳✶ ❈❛s♦ (a)

  ◆❡ss❡ ❝❛s♦✱ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✱ ♠♦str❛r❡♠♦s ❛ ✐♥t❡❣r❛çã♦ ♣♦r q✉❛❞r❛t✉r❛s ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❞❡ ♠étr✐❝❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛

  2

  q

  1 ζ (q

  1

  − A)

  2

  2

  2

  2

  2

  g = ǫ dq dq + ǫ q dq + F (q )dq ,

  1 −

  2

  3

  1

  2

  1

  3

  4

  q q

  1

  1

  − A

  2 1 , ǫ 2 = =

  ♦♥❞❡ ǫ ±1 ❡ ζ ±1✳ P♦r ♠❡✐♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✱ ❛ ❢✉♥çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ s❡ ❡s❝r❡✈❡ q q

  1

  1

  1

  1

  − A

  2

  2

  2

  2 H = + + p p p p . 1 −

  2

  3

  4

  2

  2

  2

  ǫ q ǫ ζ (q ǫ q ǫ q F (q )

  1

  1

  1

  1

  2

  2

  3

  − A)

  1

  1

  ◆♦ss♦ ♣r✐♠❡✐r♦ ♦❜❥❡t✐✈♦ s❡rá ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ tr❛♥s❢♦r♠❛çã♦ s✐♠♣❧ét✐❝❛ W = W (q, ¯q) t❛❧ q✉❡✱ ♥❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛s (¯q, ¯p)✱ ❛ ❢✉♥çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ H = ¯ q

  1

  t❡♠ ❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s ¯ ✳ ◆❡st❛s ❝♦♦r❞❡♥❛❞❛s✱ ❛s ❡q✉❛çõ❡s ❞❡ ❍❛♠✐❧t♦♥ s❡ ❡s❝r❡✈❡♠

  (

  i = 0

  ˙¯q p ˙¯ i = i

  1i

  −δ ∈ {1, 2, 3, 4} ❡✱ ♣♦rt❛♥t♦✱ sã♦ ❢❛❝✐❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ♣♦r q✉❛❞r❛t✉r❛s✿

  ( q ¯ i = ¯ q i p ¯ i = t + ¯ p i i

  1i

  −δ ∈ {1, 2, 3, 4} , ¯ p i

  ❝♦♠ ¯q i ∈ R✳

  ❆ ❢✉♥çã♦ W ❞❡✈❡ s❡r ✉♠❛ ✐♥t❡❣r❛❧ ❝♦♠♣❧❡t❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✿

  2

  2

  2

  2

  q ∂W q ∂W 1 ∂W 1 ∂W

  1

  1

  − A

  • = ¯ q .

  1

  −

  2

  2

  2

  ǫ q ∂q ǫ ζ (q ∂q ǫ q ∂q ǫ q F (q ) ∂q

  1

  1

  1

  1

  1

  2

  

2

  3

  2

  3

  4

  − A)

  1

  1

  ❊st❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ✐♥t❡❣r❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞❡ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✳ ❉❡ ❢❛t♦✱ s❡ s✉♣♦r♠♦s q✉❡

W = W (q , ¯ q) + W (q , ¯ q) + W (q , ¯ q) + W (q , ¯ q)

  1

  1

  2

  2

  3

  3

  4

  4

  ✭✸✳✷✷✮ , ¯ q , ¯ q , ¯ q )

  1

  2

  3

  4

  ♦♥❞❡ ¯q = (¯q ✱ ❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❛ss✉♠❡ ❛ ❢♦r♠❛

  2

  2

  2

  2

  q ∂W q ∂W 1 ∂W 1 ∂W

  1

  1

  1

  2

  3

  4

  − A + = ¯ + q .

  1

  −

  2

  2

  2

  ǫ q ∂q ǫ ζ (q ∂q ǫ q ∂q ǫ q F (q ) ∂q

  1

  1

  1

  1

  1

  2

  2

  3

  2

  3

  4

  − A)

  1

  1

  ✼✵

  2

  2

  ❆❣♦r❛✱ ❞❡r✐✈❛♥❞♦ ❡ss❛ ❡q✉❛çã♦ ❝♦♠ r❡s♣❡✐t♦ ❛ q ✱ ♦❜t❡♠♦s q✉❡ W ❞❡♣❡♥❞❡ ❧✐♥✲ ❡❛r♠❡♥t❡ ❞❡ q

  2 ❡✱ ♣♦rt❛♥t♦✱ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛

  W

  2 (q 2 , ¯ q) = α

21 (¯ q)q

2 + α 22 (¯ q).

  ✭✸✳✷✸✮

  4

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡r✐✈❛♥❞♦ ❝♦♠ r❡s♣❡✐t♦ ❛ q ✱ ♦❜t❡♠♦s W (q , ¯ q) = α (¯ q)q + α (¯ q).

  4

  4

  41

  4

  42

  ✭✸✳✷✹✮ ❈♦♠ ✐ss♦✱ ❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❛ss✉♠❡ ❛ ❢♦r♠❛

  !

  2

  2

  3

  2

  2

  q (q ∂W q α (¯ q) 1 ∂W α (¯ q)

  1

  1

  1

  21

  3

  41

  − A)

  1

  2

  q = , +

  

1

  − − ¯q

  1 −

  2

  ǫ ∂q ǫ ζ (q ǫ ∂q F (q )

  1

  1

  1

  1

  2

  3

  3

  − A)

  3

  1

  ♦♥❞❡ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ q ❡ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ q ✳ ■ss♦ ♠♦str❛ q✉❡ ❛♠❜♦s ♦s ❧❛❞♦s sã♦ ✐❣✉❛✐s ❛ ✉♠❛ ❢✉♥çã♦ B(¯q)✳ ▲♦❣♦✱ s❡ ♦❜t❡♠ ❢❛❝✐❧♠❡♥t❡ q✉❡

  Z s

  2

  2

  1 q α (¯ q) ǫ B(¯ q)

  21

  1

  1

  • W = ǫ q ¯ dq + R (¯ q), + q

  1

  1

  1

  1

  1

  1

  ± ✭✸✳✷✺✮

  2

  q

  1 ζ (q 1 q

  1

  − A − A) Z s

  2

  α (¯ q)

  41 W 3 =

  2 B(¯ q) dq 3 + R 3 (¯ q),

  ± −ǫ − ✭✸✳✷✻✮ F (q

  3 )

  1

  3

  ♦♥❞❡ R ❡ R sã♦ ❢✉♥çõ❡s ❛r❜✐trár✐❛s✳ P♦rt❛♥t♦✱ s✉❜st✐t✉✐♥❞♦ ✸✳✷✺✱✸✳✷✸✱✸✳✷✻ ❡ ✸✳✷✹ ❡♠ ✸✳✷✷✱ ♦❜t❡♠♦s ✉♠❛ ✐♥t❡❣r❛❧ ❝♦♠✲

  = α = 0 = ¯ q

  22

  44

  21

  2

  ♣❧❡t❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡s❝♦❧❤❡♥❞♦ α ✱ α ✱ α = ¯ q = R = 0

  41

  4

  3

  1

  3

  ✱ B = −¯q ❡ R ✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❛ s❡❣✉✐♥t❡ ✐♥t❡❣r❛❧✿ Z s

  Z s

  2

  2

  2

  1 q q ¯ ǫ

  

1 q ¯

3 q ¯

  1

  2

  4

  ǫ q q ¯ dq + ¯ + W = q q ǫ q ¯ + ¯ + dq q q .

  1

  1

  1

  1

  2

  2

  2

  3

  3

  4

  4

  − −

  2

  q ζ (q q F (q )

  1

  1

  

1

  3

  − A − A) P♦rt❛♥t♦✱ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❞❡s❝r✐t♦ ♣❡❧❛s ❡q✉❛çõ❡s

   r

  2

  ¯ ǫ q ∂ ¯

  

  1

  1

  2

  1

  3

   ǫ q q ¯ dq = t +

  1

  1

  1

  2

  1

  1

   − − ¯p

  ∂ q q ζ q ¯ 1 1 (q

  1

  1 −A −A)

     r 

  

2

  2

   R

  

q q ¯

∂ ǫ ¯ q

  

  1

  

1

  2

  1

  3

   q

  2 = 2 ǫ 1 q 1 q ¯ +

  1 2 dq

  1

  −¯p − −

  ∂ q q ζ q ¯ 2 1 (q

  

1

  1 −A −A)

  r

  2

  2

  2

   R q R

  q q q ∂ ¯ ∂ ¯ ǫ q

  

  1 1 ¯

  3

  1

  2

  

  ǫ q ¯ dq = ǫ q q ¯

  • 4

  2 dq

  

  2

  3

  3

  3

  1

  1

  1

  1 ∂ q − F −¯p − ∂ q q ζ − q ¯ 3 (q 3 ) ¯

  3 1 (q

  1

  1

  

  −A −A)

    

  ¯ q

   ∂

  4

   q = ǫ q ¯ dq

  4 −¯p 4 −

  2 3 −

  3 ∂ q F ¯ 4 (q 3 ) i

  ♦♥❞❡ ❛s ¯q ✬s sã♦ ❝♦♥st❛♥t❡s✳ ➱ ❡✈✐❞❡♥t❡ q✉❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❡s❝r❡✈❡r ❡①♣❧✐❝✐t❛♠❡♥t❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ❞❡✈✐❞❛ às ✐♥t❡❣r❛✐s ♣r❡s❡♥t❡s ♥❛ ♣r✐♠❡✐r❛ ❡ t❡r❝❡✐r❛ ❡q✉❛çõ❡s✳

  ✼✶

  ✸✳✼✳✷ ❈❛s♦ (b)

  ◆❡st❡ ❝❛s♦✱ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✱ ♠♦str❛r❡♠♦s ❛ ✐♥t❡❣r❛çã♦ ♣♦r q✉❛❞r❛t✉r❛s ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❞❡ ♠étr✐❝❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ❢♦r♠❛

  1

  2

  2

  2

  2

  2

  g = ǫ (dq dq ) + ǫ q (dq + ǫdq ),

  1

  2

  1 1 − ζ

  2

  3

  4

  √ q

  1

  2

  , ǫ , ζ =

  1

  2 ♦♥❞❡ ǫ, ǫ ±1.

  P♦r ♠❡✐♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✱ ❛ ❢✉♥çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ s❡ ❡s❝r❡✈❡ √ q

  1

  1

  1

  1

  2

  2

  2

  2 H = (p p ) + (p p + ).

  −

  1

  2

  3

  4

  2

  ǫ ζ ǫ q ǫ

  1

  2

  1

  ■♥✐❝✐❛❧♠❡♥t❡✱ ♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ tr❛♥s❢♦r♠❛çã♦ s✐♠✲ ♣❧ét✐❝❛ W = W (q, ¯q) t❛❧ q✉❡✱ ♥❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛s (¯q, ¯p)✱ ❛ ❢✉♥çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ t❡♠ ❛

  H = ¯ q

  1

  ❢♦r♠❛ ¯ ✳ ❆ ❢✉♥çã♦ W ❞❡✈❡ s❡r ✉♠❛ ✐♥t❡❣r❛❧ ❝♦♠♣❧❡t❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✿

  ! !

  2

  2

  2

  2

  √ q ∂W 1 ∂W 1 ∂W 1 ∂W

  1

  • = ¯ q . +

  1

  −

  2

  ǫ

  1 ∂q 1 ζ ∂q 2 q

1 ǫ

2 ∂q 3 ǫ ∂q

  4

  ❊st❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ✐♥t❡❣r❛❞❛ ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞❛ s❡♣❛r❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✳ ❉❡ ❢❛t♦✱ s❡ s✉♣♦r♠♦s q✉❡

  W = W

  1 (q 1 , ¯ q) + W 2 (q

2 , ¯ q) + W

3 (q 3 , ¯ q) + W 4 (q 4 , ¯ q),

  ✭✸✳✷✼✮

  1 , ¯ q 2 , ¯ q 3 , ¯ q 4 )

  ♦♥❞❡ ¯q = (¯q ✱ ❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❛ss✉♠❡ ❛ ❢♦r♠❛ ! !

  √

  2

  2

  2

  2

  q

  1 ∂W

  1

  1 ∂W

  2

  1 ∂W

  3

  1 ∂W

  4

  • = ¯ q .

  1

  −

  2

  ǫ ∂q ζ ∂q q ǫ ∂q ǫ ∂q

  1

  1

  2

  1

  

2

  3

  4

  2

  2

  ❆❣♦r❛✱ ❞❡r✐✈❛♥❞♦ ❡ss❛ ❡q✉❛çã♦ ❝♦♠ r❡s♣❡✐t♦ ❛ q ✱ ♦❜t❡♠♦s q✉❡ W ❞❡♣❡♥❞❡ ❧✐♥✲

  2

  ❡❛r♠❡♥t❡ ❞❡ q ❡✱ ♣♦rt❛♥t♦✱ ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ W (q , ¯ q) = α (¯ q)q + α (¯ q).

  2

  2

  21

  2

  22

  ✭✸✳✷✽✮

  3

  ❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❡r✐✈❛♥❞♦ ❝♦♠ r❡s♣❡✐t♦ ❛ q ✱ ♦❜t❡♠♦s

W (q , ¯ q) = α (¯ q)q + α (¯ q)

  3

  3

  31

  3

  32

  ✭✸✳✷✾✮

  4

  ❡✱ ❞❡r✐✈❛♥❞♦ ❝♦♠ r❡s♣❡✐t♦ ❛ q ✱ ♦❜t❡♠♦s W (q , ¯ q) = α (¯ q)q + α (¯ q).

  4

  4

  41

  4

  42

  ✭✸✳✸✵✮ ❈♦♠ ✐ss♦✱ ❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❛ss✉♠❡ ❛ ❢♦r♠❛

  2 √

  2

  2

  q ∂W

  1 α (¯ q)

  1 α (¯ q)

  1

  21

  41

  2

  = ¯ q α (¯ q) + .

  1

  31

  − −

  2

  ✼✷ ❡ é ❢á❝✐❧ ✈❡r q✉❡

  Z s √

  2

  2

  q ∂W

  1 α (¯ q)

  1 α (¯ q)

  1

  21

  41

  2

  = q + ¯ α (¯ q) dq + R(¯ q),

  1

  31 1 ✭✸✳✸✶✮

  ± − −

  2

  ∂q ǫ ζ q ǫ ǫ

  1

  1

  

1

  2

  ♦♥❞❡ R é ✉♠❛ ❢✉♥çã♦ ❛r❜✐trár✐❛✳ P♦rt❛♥t♦✱ s✉❜st✐t✉✐♥❞♦ ✸✳✸✶✱ ✸✳✷✽✱ ✸✳✷✾ ❡ ✸✳✸✵ ❡♠ ✸✳✷✼✱ ♦❜t❡♠♦s ✉♠❛ ✐♥t❡❣r❛❧ ❝♦♠♣❧❡t❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡s❝♦❧✲

  = α = α = 0

  22

  33

  44

  ❤❡♥❞♦ α ❡ R = 0✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ ✐♥t❡❣r❛❧✿ Z s

  √

  2

  2

  ∂W q

  1 q ¯

  1 ¯ q

  1

  2

  4

  2 W = = + q ¯ q ¯ dq + ¯ q q + ¯ q q + ¯ q q .

  1

  1

  2

  2

  3

  3

  4

  4

  −

  3 −

  2

  ∂q ǫ ζ q ǫ ǫ

  1

  1

  1

  

2

P♦rt❛♥t♦✱ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❞❡s❝r✐t♦ ♣❡❧❛s ❡q✉❛çõ❡s

   r

  √

  2

  

1

  

  2

  2

  4

  2 R q q q ∂ 1 ¯ ¯

   q ¯

  • 2 q ¯ dq = t

  1

  1

  1

   ∂ q ǫ ζ − q ǫ

  3 − ǫ − p ¯

  1

  1

  

1

  

2

     r 

  

  2

  2

   R q q q

  1 ¯ ¯

   ∂

  1

  2

  2

  4

   q = q ¯

  • dq

  2 q ¯

  

  2

  2

  1

  1

  −p − ∂ q ǫ ζ − q ǫ

  3 − ǫ ¯

  2

  

1

  1

  2

  r

  

  2

   ∂

  1

  2

  2

   q = q ¯ ¯ q dq 

  2 R q q q 1 ¯ ¯

  • 4

  3

  3

  1

  2

  1

  −p − −

  3 − ∂ q ¯ ǫ ζ q ǫ ǫ

  3

  

1

  1

  2

     r 

  

  

  2

   ∂

  1

  2

  1

  2

  

  2 R q q q ¯ ¯

  • 4 q = q ¯ ¯ q dq .

  

  4

  4

  1

  2

  1

  −p − −

  3 − ∂ q ¯ ǫ ζ q ǫ ǫ

  4

  

1

  1

  2 i

  ♦♥❞❡ ❛s q ✬s sã♦ ❝♦♥st❛♥t❡s✳ ❖❜s❡r✈❡ q✉❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❡s❝r❡✈❡r ❡①♣❧✐❝✐t❛♠❡♥t❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ❞❡✈✐❞❛ à ✐♥t❡❣r❛❧ ♣r❡s❡♥t❡ ♥❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦✳

  ❈❛♣ít✉❧♦ ✹ ❯s♦ ❞❡ ❙✐♠❡tr✐❛s ♥❛ ■♥t❡❣r❛çã♦ ❞❡ ❋❧✉①♦s ●❡♦❞és✐❝♦s ✹✳✶ ■♥tr♦❞✉çã♦

  ◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r ♠♦str❛♠♦s ❝♦♠♦ ❛ ❣❡♦♠❡tr✐❛ s✐♠♣❧ét✐❝❛ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ tr❛t❛r ❛ ✐♥t❡❣r❛çã♦ ♣♦r q✉❛❞r❛t✉r❛s ❞❛s ❊❉❖✬s ❞♦ t✐♣♦ ✈❛r✐❛❝✐♦♥❛❧✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❝❧áss✐❝♦ ❞❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐✱ ✈✐♠♦s ❝♦♠♦ ❛ ❛❜♦r❞❛❣❡♠ s✐♠♣❧ét✐❝❛ s❡ ❛♣❧✐❝❛ ❛♦ ❝❛s♦ ❞❛s ❡q✉❛çõ❡s q✉❡ ❞❡s❝r❡✈❡♠ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❞❡ ❛❧❣✉♠❛s ♠étr✐❝❛s ❞❡ ❊✐♥st❡✐♥✳

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ❞✐s❝✉t✐r❡♠♦s ♦ ✉s♦ ❞❡ s✐♠❡tr✐❛s ♥❛ ✐♥t❡❣r❛çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♣r♦✈❛r❡♠♦s ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ♣♦r q✉❛❞r❛t✉r❛s ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧✳ ❉❡ss❛ ❢♦r♠❛✱ ❡♥❝♦♥tr❛r❡♠♦s✱ ❝♦♠♦ ❝❛s♦s ♣❛rt✐❝✲ ✉❧❛r❡s✱ ♦s t❡♦r❡♠❛s ❞❡ ▲✐♦✉✈✐❧❧❡ s♦❜r❡ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❝♦♠✉t❛t✐✈❛ ❡ ♥ã♦ ❝♦♠✉t❛t✐✈❛ ❞❡ s✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s ✭✈✳ ❬✶✸❪✱ ❬✶✹❪✱ ❬✷✽❪✱ ❬✸✽❪✮✳ ❊st❡s r❡s✉❧t❛❞♦s s❡rã♦ ❛♣❧✐❝❛❞♦s ❛♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❞✐str✐❜✉✐çã♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ q✉❡✱ s♦❜r❡ ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ é ❛ss♦❝✐❛❞❛ ❛♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✢✉①♦ ❣❡♦❞és✐❝♦✳

  

✹✳✷ ➪❧❣❡❜r❛s ❙♦❧ú✈❡✐s ❞❡ ❙✐♠❡tr✐❛s ❡ ❊str✉t✉r❛s ❙♦❧ú✈❡✐s

  ❆s ❡str✉t✉r❛s s♦❧ú✈❡✐s ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ❣❡♥❡r❛❧✐③❛r ✉♠ r❡✲ s✉❧t❛❞♦ ❝❧áss✐❝♦ q✉❡ ❛✜r♠❛ q✉❡✱ ❝♦♥❤❡❝❡♥❞♦✲s❡ ✉♠❛ á❧❣❡❜r❛ k✲❞✐♠❡♥s✐♦♥❛❧ G ❞❡ s✐♠❡tr✐❛s✱ s♦❧ú✈❡❧ ❡ tr❛♥s✈❡rs❛❧✱ ♣❛r❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ (n − k)✲❞✐♠❡♥s✐♦♥❛❧ D ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡✲ ❣rá✈❡❧✱ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ n✲❞✐♠❡♥s✐♦♥❛❧✱ ❡♥tã♦ D ♣♦❞❡ s❡r ✐♥t❡❣r❛❞❛ ♣♦r q✉❛❞r❛t✉r❛s✳ ❉❡ ❢❛t♦✱ ❛s ❡str✉t✉r❛s s♦❧ú✈❡✐s ❣❡♥❡r❛❧✐③❛♠ ❛ s❡❣✉✐♥t❡ ♥♦çã♦ ❞❡ á❧❣❡❜r❛ s♦❧ú✈❡❧✿ ❉❡✜♥✐çã♦ ✹✳✷✳✶✳ ❯♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ (G, [, ]) s❡ ❝❤❛♠❛ s♦❧ú✈❡❧ s❡ ❛ ❜❛♥❞❡✐r❛ ❞❛s á❧❣❡❜r❛s

  ✼✹ ❞❡r✐✈❛❞❛s é t❛❧ q✉❡

  (0) (1) (l)

  = 0 G = G ⊃ G ⊃ ... ⊃ G

  ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ l ≥ 0✳ ▲❡♠❜r❛♠♦s q✉❡✱ ❞❛❞❛ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ (G, [, ])✱ ❛ s✉❛ ♣r✐♠❡✐r❛ s✉❜á❧❣❡❜r❛

  (1)

  ❞❡r✐✈❛❞❛ G é

  

(1) (0)

= [X, Y ] : X, Y .

  G ∈ G ■♥❞✉t✐✈❛♠❡♥t❡✱ sã♦ ❞❡✜♥✐❞❛s ❛s ❞❡r✐✈❛❞❛s s✉♣❡r✐♦r❡s

  

(h+1) (h)

= [X, Y ] : X, Y .

  G ∈ G

  (l) 1 , ..., Z r

  ❆ ♣❛rt✐r ❞❡ G é s❡♠♣r❡ ♣♦ssí✈❡❧ ❝♦♥str✉✐r ✉♠❛ ❜❛s❡ {Z } ❞❡ G t❛❧ q✉❡ Z h , ..., Z h >

  • 1

  1

  é s✐♠❡tr✐❛ ❞❡ < Z ✱ ♣❛r❛ t♦❞♦ h ∈ {1, ..., r − 1}✳ ❊st❛ ♦❜s❡r✈❛çã♦ ❧❡✈❛ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ❛ ❝♦♥s✐❞❡r❛r ❛ s❡❣✉✐♥t❡

  , ..., X h > ❉❡✜♥✐çã♦ ✹✳✷✳✷✳ ❉❛❞❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ D =< X

  1 ✱ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r✲ 1 , ..., Y n

  ❡♥❝✐á✈❡❧ M✱ ✉♠ s✐st❡♠❛ ❞❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s {Y −h } ❢♦r♠❛ ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ = s =< X , ..., X h , Y , ..., Y s >

  1

  1

  ♣❛r❛ D s❡✱ ❞❡♥♦t❛♥❞♦ D D ❡ D ✱ ❛s s❡❣✉✐♥t❡s ❞✉❛s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿

  n = T M ;

  ✭✐✮ D −h

  Y s s s ✭✐✐✮ L D −1 ⊂ D −1 ✱ ∀s ∈ {1, ..., n − h}.

  , ..., Y n

  1

  ❊♠ ♣❛rt✐❝✉❧❛r✱ ❞❛❞❛ ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ {Y −h } ♣❛r❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ ✐♥t❡❣rá✈❡❧ D✱ t❡♠♦s ❛ ❜❛♥❞❡✐r❛ ❞❡ ❞✐str✐❜✉✐çõ❡s ✐♥t❡❣rá✈❡✐s

  n = T M

  1 D = D ⊂ D ⊂ ... ⊂ D −h

  ❖❜s❡r✈❡ q✉❡ ❛ ❝♦♥❞✐çã♦ (i) ✐♠♣❧✐❝❛ q✉❡ ❡str✉t✉r❛s s♦❧ú✈❡✐s ❣❧♦❜❛✐s só ♣♦❞❡♠ ❡①✐st✐r ♣❛r❛ ✈❛r✐❡❞❛❞❡s ♦r✐❡♥tá✈❡✐s✳ P♦rt❛♥t♦✱ s♦❜r❡ ✈❛r✐❡❞❛❞❡s ♥ã♦ ♦r✐❡♥tá✈❡✐s✱ só ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❡str✉t✉r❛s s♦❧ú✈❡✐s ❞❡✜♥✐❞❛s ❧♦❝❛❧♠❡♥t❡✳ ❏á ❛ ❝♦♥❞✐çã♦ (ii)✱ ✐♠♣õ❡ s♦♠❡♥t❡ q✉❡ Y

  , ..., Y n

  1

  2

  s❡❥❛ s✐♠❡tr✐❛ ❞❡ D✳ ❉❡ ❢❛t♦✱ ♦s ❞❡♠❛✐s ❝❛♠♣♦s Y −h ♣♦❞❡♠ ♥ã♦ s❡r s✐♠❡tr✐❛s ❞❛

  s s

  ❞✐str✐❜✉✐çã♦ D ♠❛s ❝❛❞❛ ❝❛♠♣♦ Y ❞❡✈❡ s❡r s✐♠❡tr✐❛ ❞❛ ❞✐str✐❜✉✐çã♦ D ✳

  −1

  ◆❛ s❡çã♦ ❛ s❡❣✉✐r✱ ❝♦♠ ♦ t❡♦r❡♠❛ ✹✳✸✳✶✱ ♠♦str❛r❡♠♦s q✉❡ ❛s ❡str✉t✉r❛s s♦❧ú✈❡✐s ❞❡s❡♠♣❡♥❤❛♠ ♦ ♠❡s♠♦ ♣❛♣❡❧ ❞❛s á❧❣❡❜r❛s s♦❧ú✈❡✐s ♠❛①✐♠❛✐s ♥❛ ✐♥t❡❣r❛çã♦ ❞❡ ✉♠❛ ❞✐s✲ tr✐❜✉✐çã♦ ❞❡ ❋r♦❜❡♥✐✉s✳ P♦rt❛♥t♦✱ ♣❡❧♦ ❢❛t♦ ❞❡ s❡r❡♠ ♦❜❥❡t♦s ♠❛✐s ❣❡r❛✐s✱ ❛s ❡str✉t✉r❛s s♦❧ú✈❡✐s t♦r♥❛♠✲s❡ ♣❛rt✐❝✉❧❛r♠❡♥t❡ út❡✐s q✉❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❇✐❛♥❝❤✐✲▲✐❡ ♥ã♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦✳

  ✼✺

  

✹✳✸ ■♥t❡❣r❛çã♦ ♣♦r ◗✉❛❞r❛t✉r❛s ♥❛ Pr❡s❡♥ç❛ ❞❡ ❊str✉✲

t✉r❛s ❙♦❧ú✈❡✐s

  ◆❡st❛ s❡çã♦ ✈❡r❡♠♦s ❝♦♠♦ ❡str✉t✉r❛s s♦❧ú✈❡✐s ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ♥❛ ✐♥t❡❣r❛çã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞♦ s❡❣✉✐♥t❡

  , ..., Y n ❚❡♦r❡♠❛ ✹✳✸✳✶✳ ❙❡ {Y

  1 −h } ❢♦r♠❛ ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ ♣❛r❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ 1 , ..., X h >

  D =< X ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M n✲❞✐♠❡♥s✐♦♥❛❧ ❡ ❝♦♠ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ Ω✱ ❡♥tã♦✿ y y y y

  ...yY n X ...yX h Ω

  1

  1

  ✭✶✮ ∆ := Y −h 6= 0❀ , ..., Ω n

  1

  ✭✷✮ D = Ann{Ω −h }✱ ♦♥❞❡

  1 y y y y y Ω i := Y ...y b Y i ...yY n X ...yX h Ω,

  1

  1 −h ✭✹✳✶✮

  ∆ ♣❛r❛ t♦❞♦ i ∈ {1, ..., n − h}

  n = 0

  ✭✸✮ dΩ −h ❀

  i i +1 , ..., Ωn

  ✭✹✮ dΩ ≡ 0 mod{Ω − h}✱ ♣❛r❛ t♦❞♦ i ∈ {1, ..., n − h − 1}✳ Pr♦✈❛✿ (1) ❡ (2) sã♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞♦ ❢❛t♦ q✉❡ Ω é ✉♠❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ ❡ q✉❡

  , ..., Y n , X , ..., X h

  1

  1

  {Y −h } é ✉♠ r❡❢❡r❡♥❝✐❛❧ s♦❜r❡ M✳ (3)

  ❉❡✜♥✐♥❞♦ y y y y y α i = Y

  1 ...y b Y i ...yY n

  X

  1 ...yX h Ω −h k

  • 1

  y Y i α i

  ❡ ♦❜s❡r✈❛♥❞♦ q✉❡ ∆ = (−1) ✱ ♣❛r❛ t♦❞♦ i ∈ {1, ..., n − h}✱ ♦❜t❡♠♦s

  1 dΩ n = d α n

  −h −h

  ∆

  1

  1 = d∆ dα +

  n n

  ∧ α −h −h

  2

  ∆ −∆

  n + ∆dα n

  −d∆ ∧ α −h −h =

  2

  ∆

  n −h+1

  ( −1) y y

  = Y α n n dα n ) n + Y n (α n )dα n ,

  n−h

  −(L −h − Y −h −h ∧ α −h −h −h −h

  2

  ∆ ♦♥❞❡ ✉s❛♠♦s ❛ ❢ór♠✉❧❛ ❞❡ ❈❛rt❛♥ ♣❛r❛ ❛ ❞❡r✐✈❛❞❛ ❞❡ ▲✐❡✳

  n = Ann n

  ❈♦♠♦✱ ♣♦r ❝♦♥str✉çã♦✱ D −h−1 {α −h } é ✐♥t❡❣rá✈❡❧ dα n = ρ n .

  

−h ∧ α −h

n n

  ❡✱ ❝♦♠♦ Y −h é s✐♠❡tr✐❛ ❞❡ D −h−1 ✱ t❡♠♦s q✉❡

  Y α n = f α n , n−h

  ✼✻

  n = 0

  ♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ f✳ ❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s q✉❡ dΩ −h ✳ (4)

  n = dI n

  ❯t✐❧✐③❛♥❞♦ ♦ ▲❡♠❛ ❞❡ P♦✐♥❝❛ré✱ ♦❜t❡♠♦s q✉❡✱ ♣❡❧♦ ♠❡♥♦s ❧♦❝❛❧♠❡♥t❡✱ dΩ ✱

  −h −h n n = Ann n n

  ♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ I −h ✳ ❊♥tr❡t❛♥t♦✱ D −h−1 {Ω −h } ❡✱ ♣♦rt❛♥t♦✱ I −h é ✉♠❛

  n

  ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❞❡ D −h−1 ✳ ❋❛❧t❛ ❛♣❡♥❛s ♦❜s❡r✈❛r q✉❡✱ s♦❜r❡ ❛s ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ ❞❡ I n , Y , ..., Y n

  1

  2

−h ✱ {Y −h−1 } ❢♦r♠❛ ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ ♣❛r❛ D✳ ▲♦❣♦✱ ♣♦❞❡♠♦s ✐t❡r❛r ♦

n

  ♣r♦❝❡❞✐♠❡♥t♦ ❛♥t❡r✐♦r ❡ ❞❡♠♦♥str❛r q✉❡ ❛ r❡str✐çã♦ ❛ ❡ss❛ ✈❛r✐❡❞❛❞❡ ❞❡ ♥í✈❡❧ ❞❡ Ω −h−1 é

  n

  ❢❡❝❤❛❞❛ ❡ ❡♥❝♦♥tr❛r ♠❛✐s ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ I −h−1 ✳ ❈♦♥t✐♥✉❛♥❞♦ ❞❡ss❛ ❢♦r♠❛✱ ♣r♦✈❛✲ , ..., I

  1 n

  ♠♦s ♦ ❞❡s❡❥❛❞♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦❜t❡♠♦s ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ ✐♥t❡❣r❛✐s {I −h }✳ ◆❛ ♣rát✐❝❛✱ ✉t✐❧✐③❛♥❞♦ ❡st❡ t❡♦r❡♠❛✱ ♣♦❞❡♠♦s ✐t❡r❛t✐✈❛♠❡♥t❡ ❝❛❧❝✉❧❛r ✉♠ ♥ú♠❡r♦

  , ..., I n

  1

  ♠❛①✐♠❛❧ ❞❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s {I −h } q✉❡✱ ❡♠ ✈✐rt✉❞❡ ❞❛ ♣r♦♣♦s✐çã♦ 2.1.17✱ ♥♦s ♣❡r♠✐t❡ ❞❡s❝r❡✈❡r ❛s ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ❞❡ D ♥❛ ❢♦r♠❛ ✐♠♣❧í❝✐t❛

   

  I

  1 = c

  

1

   

  , ✳✳✳

    

  I n = c n

  −h −h

  c i ∈ R✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❡ t❡♦r❡♠❛✱ ♦❜t❡♠♦s t❛♠❜é♠ ♦ s❡❣✉✐♥t❡

  ❚❡♦r❡♠❛ ✹✳✸✳✷✳ ✭❞❡ ❇✐❛♥❝❤✐✲▲✐❡✮ ❙❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥t❡❣rá✈❡❧ D ❛❞✲ ♠✐t❡ ✉♠❛ á❧❣❡❜r❛ ♠❛①✐♠❛❧ ❡ s♦❧ú✈❡❧ ❞❡ s✐♠❡tr✐❛s tr❛♥s✈❡rs❛✐s ❛ D✱ ❡♥tã♦ D é ✐♥t❡❣rá✈❡❧ ♣♦r q✉❛❞r❛t✉r❛s✳

  Pr♦✈❛✿ ❯♠❛ á❧❣❡❜r❛ s♦❧ú✈❡❧ ❞❡ s✐♠❡tr✐❛s tr❛♥s✈❡rs❛✐s ♠❛①✐♠❛❧ ❢♦r♠❛ ✉♠❛ ❡s✲ tr✉t✉r❛ s♦❧ú✈❡❧ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ D✳ P♦rt❛♥t♦✱ ❛♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r ❛ ❡st❛ ❡str✉t✉r❛✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡✳

  ✹✳✹ ❆♣❧✐❝❛çõ❡s ❛ ❊❉❖✬s ❞♦ ❚✐♣♦ ❱❛r✐❛❝✐♦♥❛❧

  ❈♦♠♦ ❞✐s❝✉t✐♠♦s ♥♦ ❝❛♣ít✉❧♦ ✷✱ ❛s ❡q✉❛çõ❡s ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✱ q✉❛♥❞♦ ❛ ❧❛✲

  1 L (π)

  ❣r❛♥❣❡❛♥❛ é r❡❣✉❧❛r✱ sã♦ ❛s ❡q✉❛çõ❡s ❞♦ ✢✉①♦ ❞❡ ✉♠ ❝❛♠♣♦ X s♦❜r❡ J ✳ ❆q✉✐✱ ❝♦♠♦ ♥♦ ❝❛♣ít✉❧♦ ✷✱ π ❞❡♥♦t❛ ♦ ✜❜r❛❞♦ tr✐✈✐❛❧ π : R × M −→ R✳ P♦rt❛♥t♦✱

  1

  (π) ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r J ❝♦♠ R × M✳ ❉❡ ❢❛t♦✱ ✉♠❛ s❡çã♦ σ q✉❛❧q✉❡r ❞❡ π ♣♦❞❡ s❡r

  

1

  ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠ ✉♠❛ ❝✉r✈❛ γ ❡♠ M ❡ [σ] ✱ a ∈ R✱ ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠ ♦ ✈❡t♦r

  

a

  t❛♥❣❡♥t❡ à ❝✉r✈❛ γ ♥♦ ♣♦♥t♦ γ(a)✳

  1

  g v v

  ij i j

  ◆♦ ❝❛s♦ ❞❡ ✉♠❛ ❧❛❣r❛♥❣❡❛♥❛ L = ✱ q✉❡ ❞❡s❝r❡✈❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❞❡ ✉♠❛

  2 L

  ✼✼

  

1

  : J (π)

  1

  ♣♦❞❡ ♣r♦❥❡t❛r✱ ❝♦♠ r❡s♣❡✐t♦ à ♣r♦❥❡çã♦ π −→ T M✱ ❡♠ ✉♠ ❝❛♠♣♦ s♦❜r❡ T M q✉❡

  E

  ❞❡♥♦t❛r❡♠♦s ❝♦♠ X ✳ ❆♣❧✐❝❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❞❛ s❡çã♦ ❛♥t❡r✐♦r à ❞✐str✐❜✉✐çã♦ 1✲❞✐♠❡♥s✐♦♥❛❧ s♦❜r❡

  T M E > ❞❡✜♥✐❞❛ ❝♦♠♦ < X ✳ ❆s ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s ❞❡ss❛ ❞✐str✐❜✉✐çã♦ sã♦ ❛s ♣ré✲

  ❣❡♦❞és✐❝❛s ❞❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ M✱ ✐st♦ é✱ ❛ ♠❡♥♦s ❞❡ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ♥❛t✲ ✉r❛❧✱ sã♦ ❝✉r✈❛s ❣❡♦❞és✐❝❛s✳

  E

  ❖ ❝❛♠♣♦ X é ✉♠ ❝❛♠♣♦ s♦❜r❡ T M q✉❡✱ ✈✐❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡✱ s❡ tr❛♥s✲

  ij

  1

  M g p p

  H i j

  ❢♦r♠❛ ♥♦ ❝❛♠♣♦ ❍❛♠✐❧t♦♥✐❛♥♦ X s♦❜r❡ T ❝♦♠ ❍❛♠✐❧t♦♥✐❛♥❛ H = ✳ ❆ss✐♠

  2 E > H >

  ❝♦♠♦ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ < X ✱ t❛♠❜é♠ ♣❛r❛ < X ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦s r❡s✉❧t❛❞♦s ❞❛ s❡çã♦ ❛♥t❡r✐♦r ♣❛r❛ ✐♥t❡❣r❛r ♦ ✢✉①♦ ❣❡♦❞és✐❝♦✳ ◆❡ss❡ ❝❛s♦✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ✹✳✸✳✶ s♦❜r❡ ✐♥t❡❣r❛çã♦ ❝♦♠ ❡str✉t✉r❛s s♦❧ú✈❡✐s✱ ♣♦❞❡♠♦s ❢❛❝✐❧♠❡♥t❡ ❞❡❞✉③✐r ❞♦✐s ✐♠♣♦rt❛♥t❡s r❡s✉❧t❛❞♦s s♦❜r❡ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s ❝♦♠ s✐♠❡tr✐❛s✱ ❛ s❛❜❡r✱ ♦ t❡♦r❡♠❛ ❞❡ ▲✐♦✉✈✐❧❧❡ ❝♦♠✉t❛t✐✈♦ ❡ ♦ t❡♦r❡♠❛ ❞❡ ▲✐♦✉✈✐❧❧❡ ♥ã♦✲❝♦♠✉t❛t✐✈♦✳ ❖ t❡♦r❡♠❛ ❞❡

  H >

  ▲✐♦✉✈✐❧❧❡ ❝♦♠✉t❛t✐✈♦ ❛✜r♠❛ q✉❡✱ s❡ < X é ✉♠ ❝❛♠♣♦ ❍❛♠✐❧t♦♥✐❛♥♦ s♦❜r❡ ✉♠❛ ✈❛r✲ = H, f , ..., f n

  1

  ✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ 2n✲❞✐♠❡♥s✐♦♥❛❧ M q✉❡ ❛❞♠✐t❡ n ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s f −1

  i , f j

  ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡ ❡♠ ✐♥✈♦❧✉çã♦ ✭✐st♦ é✱ t❛❧ q✉❡ {f } = 0✮✱ ❡♥tã♦ ♦ ✢✉①♦

  H

  ❍❛♠✐❧t♦♥✐❛♥♦ ❞❡ X é ✐♥t❡❣rá✈❡❧ ♣♦r q✉❛❞r❛t✉r❛s✳ ❉♦ ♥♦ss♦ ♣♦♥t♦ ❞❡ ✈✐st❛✱ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ♣♦r q✉❛❞r❛t✉r❛s é ✉♠❛ s✐♠♣❧❡s ❝♦♥✲

  H

  s❡q✉ê♥❝✐❛ ❞♦ ❢❛t♦ q✉❡✱ ♥❛s ❤✐♣ót❡s❡s ❞♦ t❡♦r❡♠❛✱ X é t❛♥❣❡♥t❡ às ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ n c = , f = c , ..., f n = c n ✲❞✐♠❡♥s✐♦♥❛✐s Γ

  1

  1

  {H = c −1 −1 } ❡ q✉❡✱ ❛❧é♠ ❞✐ss♦✱ ♦s ❝❛♠♣♦s

  H , X f , ..., X f n− c H >

  {X

  1 1 } sã♦ t♦❞♦s t❛♥❣❡♥t❡s ❛ Γ ❡ ❝♦♠✉t❛♠✳ P♦rt❛♥t♦✱ < X ✐♥❞✉③✱ ❡♠ c

  ❝❛❞❛ ✈❛r✐❡❞❛❞❡ ❞❡ ♥í✈❡❧ Γ ✱ ✉♠❛ ❞✐str✐❜✉✐çã♦ 1✲❞✐♠❡♥s✐♦♥❛❧ q✉❡ ❛❞♠✐t❡ ✉♠❛ ❡str✉t✉r❛

  f , ..., X f n−

  s♦❧ú✈❡❧ ❞❛❞❛ ♣❡❧♦s ❝❛♠♣♦s X

  1 1 ✳ ◆❡st❡ ❝❛s♦✱ ❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ é ❞❛❞❛ ♣♦r ✉♠❛ H >

  á❧❣❡❜r❛ ❛❜❡❧✐❛♥❛ ❞❡ s✐♠❡tr✐❛s ❞❡ < X ✳ ❖ t❡♦r❡♠❛ ❞❡ ▲✐♦✉✈✐❧❧❡ ♥ã♦ ❝♦♠✉t❛t✐✈♦✱ ❞❡♠♦♥str❛❞♦ ♣♦r ❋♦♠❡♥❦♦ ❡ ▼✐s❤❝❤❡♥❦♦

  H

  ❡♠ ❬✸✽❪✱ ❣❡♥❡r❛❧✐③❛ ♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ s❡ X é ✉♠ ❝❛♠♣♦ ❍❛♠✐❧t♦♥✐✲ ❛♥♦ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ 2n✲❞✐♠❡♥s✐♦♥❛❧ M q✉❡ ❛❞♠✐t❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ P♦✐ss♦♥

  (2n ❞❡ s✐♠❡tr✐❛s A ♠✲❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠ m ≥ n ❡ ❝♦♠ ✉♠ ❝❡♥tr♦ A − m)✲❞✐♠❡♥s✐♦♥❛❧

  H

  t❛❧ q✉❡ H ∈ A ✱ ❡♥tã♦ ♦ ✢✉①♦ ❞❡ X s❡ ✐♥t❡❣r❛ ♣♦r q✉❛❞r❛t✉r❛s✳ ❚❛♠❜é♠ ♥❡st❡ ❝❛s♦✱ ♣♦❞❡♠♦s ❞❡❞✉③✐r ❡st❡ r❡s✉❧t❛❞♦ ♣❡❧♦ t❡♦r❡♠❛ ✹✳✸✳✶ ❞❛ s❡çã♦ ❛♥t❡r✐♦r✳ ❉❡ ❢❛t♦✱ s❡ A =<

  = H, f , ..., f m c = , f = c , ..., f m = c m

  1

  1

  1

  {f −1 } >✱ ❡♥tã♦ ❛s ✈❛r✐❡❞❛❞❡s Γ {H = c −1 −1 } sã♦ (2n = = H, g , ..., g H , X g , ..., X g

  1

  

2

  −m)✲❞✐♠❡♥s✐♦♥❛✐s ❡✱ s❡ A {g 2n−m }✱ ♦s ❝❛♠♣♦s {X

  2 2n−m } g , X g ] = 0 c i j

  sã♦ t❛♥❣❡♥t❡s ❛ ❡st❛s ✈❛r✐❡❞❛❞❡s✳ ▲♦❣♦✱ s❡♥❞♦ q✉❡ [X ✱ s♦❜r❡ ❝❛❞❛ Γ ❛✐♥❞❛

  H >

  t❡♠♦s ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ ♣❛r❛ ❛ ❞✐str✐❜✉✐çã♦ ✐♥❞✉③✐❞❛ ♣♦r < X ✳ ❚❛♠❜é♠ ♥❡st❡ ❝❛s♦✱ ❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ é ❢♦r♠❛❞❛ ♣♦r ✉♠❛ á❧❣❡❜r❛ ❛❜❡❧✐❛♥❛ ❞❡ s✐♠❡tr✐❛s✳

  ❈♦♠ ❜❛s❡ ♥❡st❛s ✐♥t❡r♣r❡t❛çõ❡s ❞♦s t❡♦r❡♠❛s ❞❡ ▲✐♦✉✈✐❧❧❡ ❝♦♠✉t❛t✐✈♦ ❡ ♥ã♦ ❝♦✲ ♠✉t❛t✐✈♦✱ ♣♦❞❡♠♦s ♣❡♥s❛r ❡♠ ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ✹✳✸✳✶ ❛ ❝❛s♦s ♠❛✐s ❣❡r❛✐s✱ ❝♦♠♦ ♣r♦♣♦st♦

  ✼✽

  H

  ❡♠ ❬✶✾❪✳ ❉❡ ❢❛t♦✱ s❡ X é ✉♠ ❝❛♠♣♦ ❍❛♠✐❧t♦♥✐❛♥♦ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧ét✐❝❛ 2n✲ ❞✐♠❡♥s✐♦♥❛❧ M q✉❡ ❛❞♠✐t❡ ✉♠❛ á❧❣❡❜r❛ ❞❡ P♦✐ss♦♥ ❞❡ s✐♠❡tr✐❛s A✱ h✲❞✐♠❡♥s✐♦♥❛❧✱ t❛❧ q✉❡

  = H, f

  1 , ..., f h c = , f 1 = c 1 , ..., f h = c h

  A = {f −1 }✳ ❊♥tã♦✱ ❞❡♥♦t❛♥❞♦ ❝♦♠ Γ {H = c −1 −1 } ❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ✹✳✸✳✶ ❛♦s ❝❛s♦s ❡♠ q✉❡✿

  = X H , Z , ..., Z

  c

  1

  2

  ✭✐✮ ❡①✐st❡♠ 2n − h ❝❛♠♣♦s Z 2n−h t❛♥❣❡♥t❡s às ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ Γ ✭❡✱

  i (f j ) = 0 , ..., Z

  1

  ♣♦rt❛♥t♦✱ t❛✐s q✉❡ Z ✮❀ ✭✐✐✮ {Z 2n−h } ❞❡t❡r♠✐♥❛ ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ ♣❛r❛ < X H > c

  ❡♠ ❝❛❞❛ Γ ✳ ❆ss✐♠ ❝♦♠♦ ♦s t❡♦r❡♠❛s ❞❡ ▲✐♦✉✈✐❧❧❡✱ ❝♦♠✉t❛t✐✈♦ ❡ ♥ã♦ ❝♦♠✉t❛t✐✈♦✱ t❡♠ ✉♠

  ❝♦rr❡s♣♦♥❞❡♥t❡ ❧❛❣r❛♥❣❡❛♥♦ ❡♠ T M✱ t❛♠❜é♠ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❛❝✐♠❛ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛

  ∗

  M E H ❡♠ T M ❜❡♠ ❝♦♠♦ ❡♠ T ✳ ❉❡ ❢❛t♦✱ ❡♠ T M✱ s❡ X ✭♦ ❝❛♠♣♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ X ✮ ❛❞♠✐t❡ ✉♠ s✐st❡♠❛ ❞❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞♦ t✐♣♦ ◆ö❡t❤❡r✱ ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ y

  = L = X E Θ, f

  1 , ..., f h

  {f −1 }✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ 4.3.1 ❛♦s ❝❛s♦s ❡♠ q✉❡✿ ✭✐✮ = X E , Z , ..., Z c =

  1

  2

  ❡①✐st❡♠ 2n −h ❝❛♠♣♦s Z 2n−h t❛♥❣❡♥t❡s às ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ Γ {L = c , f = c , ..., f h = c h , ..., Z

  1

  1

  1 −1 −1 }❀ ✭✐✐✮ {Z 2n−h } ❞❡t❡r♠✐♥❛ ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ ♣❛r❛

  < X E > c ❡♠ ❝❛❞❛ Γ ✳

  ◆❛ ♣ró①✐♠❛ s❡çã♦✱ tr❛❜❛❧❤❛♥❞♦ ❡♠ T M✱ s❡rã♦ ❞❛❞♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡st❡ ♠ét♦❞♦✳ ❖✉tr♦s ❡①❡♠♣❧♦s ♣♦❞❡rã♦ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✶✾❪✳

  ✹✳✺ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s

  ◆❡st❛ s❡çã♦ ✐❧✉str❛r❡♠♦s✱ ♣♦r ♠❡✐♦ ❞❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s✱ ♦ ♠ét♦❞♦ ❞❡s❝r✐t♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✳ ❯s❛r❡♠♦s✱ ♣❛r❛ ❡st❡ ✜♠✱ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❛❞❛ ♣♦r ❇✐❛♥❝❤✐ ❡♠ ❬✾❪ ❞❛s ♠étr✐❝❛s tr✐❞✐♠❡♥s✐♦♥❛✐s q✉❡ ❛❞♠✐t❡♠ ✉♠ ❣r✉♣♦ ❞❡ ✐s♦♠❡tr✐❛s✳ ◆❡st❡ tr❛❜❛❧❤♦ ❞❡ ❇✐❛♥❝❤✐✱ ❢♦r❛♠ ✐❞❡♥t✐✜❝❛❞♦s q✉✐♥③❡ t✐♣♦s ❞✐st✐♥t♦s ❞❡ ♠étr✐❝❛s✳ ❊♠ q✉❛s❡ t♦❞♦s ❡st❡s ❝❛s♦s✱ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ❢❛❝✐❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ♣♦r q✉❛❞r❛t✉r❛ ♣♦✐s ❡①✐st❡♠ á❧❣❡❜r❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❛❜❡❧✐❛♥❛s ❞❡ s✐♠❡tr✐❛s✳ ◆♦s s❡❣✉✐♥t❡s ❝✐♥❝♦ ❝❛s♦s✱ ❛♦ ❝♦♥trár✐♦✱ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡✱ s❡ ❢♦r ♣♦ssí✈❡❧✱ é ♠❡♥♦s ó❜✈✐❛✳

  2

  2

  2

  2

  • αdx + 2 (β ) dx dx + (αx + γ) dx

  2

  2

  3

  2

  ✭✐✮ g = dx

  1 2 − αx 2 − 2βx 3 ✱

  ❝♦♠ α, β ❡ γ ❢✉♥çõ❡s ❞❡ x

  1 ❀

  2

  2

  2

  2

  2

  • ϕ (x

  1 ) (dx + sin (dx ))

  ✭✐✐✮ g = dx

  1

  2 3 ❀

  2

  2 2 2x

  2

  2

  • ϕ (x ) (dx + e dx )

  1

  ✭✐✐✐✮ g = dx

  1

  2 3 ✱

  ♦♥❞❡ ϕ é ✉♠❛ ❢✉♥çã♦ ❛r❜✐trár✐❛✳

  ✼✾ ✭✐✈✮

  (4) ′ ′′

Q (x ) Q (x ) Q (x ) h

  1

  1

  1

  2

  2

  2

  2

  • g = dx + Q(x )dx + Q(x )x x dx

  1

  1

  2

  1

  2 2 − −

  3

  24

  2

  2

  2

  ′′ ′′′ ′′

  Q (x ) Q (x ) Q (x )

  1

  1

  1

  • 2 + h dx dx + 2 + h x dx dx

  1

  2

  2

  1

  3

  −

  12

  24

  12

  ′

  Q (x

  1 )

  • 2 ) x dx dx ,

  − Q (x

  1

  2

  2

  3

  4 )

  1

  1

  ❝♦♠ Q(x ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ ❣r❛✉ q✉❛tr♦ ❡♠ x ❝♦♠ ♣r✐♠❡✐r♦ ❝♦❡✜❝✐❡♥t❡ ♥ã♦ ♥❡❣❛t✐✈♦ ❡ h ✉♠❛ ❝♦♥st❛♥t❡❀

  X a ik dx i dx k ✭✈✮ g = ✱ ❝♦♠

  i,k

  2

  2

  (a + d ) a = 2e cos(2x ) + 2f sen(2x ) + ;

  11

  3

  3

  2

  2

  2

  2

  a

  22 = 2sen(x 1 )cos(x 1 )(b sen(x 3 ) 3 )) 11 sen (x 1 ) + a + d sen (x 1 );

  − c cos(x − a

  2

  a = a ;

  33

  a = cos(x )(b cos(x ) + c sen(x )) + 2sen(x )(e sen(2x ) ))

  12

  1

  3

  3

  1

  3

  3

  − f cos(2x a = b cos(x ) + c sen(x )

  13

  3

  3

  2

  a = a cos(x ) + sen(x )(b sen(x ) ))

  23

  1

  1 3 − c cos(x

  3

  ❉❡ ❢❛t♦✱ ♥❡st❡s ❝❛s♦s✱ é ♥❡❝❡ssár✐❛ ✉♠❛ ❛♥á❧✐s❡ ♠❛✐s ❞❡t❛❧❤❛❞❛ ♣❛r❛ ❡st❛❜❡❧❡❝❡r s❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ é ✐♥t❡❣rá✈❡❧ ♣♦r q✉❛❞r❛t✉r❛s✳ ❖ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ ❛❜❛✐①♦ é r❡❧❛t✐✈♦ ❛ ✉♠❛ ❞❛s ♠étr✐❝❛s ❝❧❛ss✐✜❝❛❞❛s ❡♠ ❬✾❪ ♣❡❧❛s q✉❛✐s✱ ❛♦ ❝♦♥trár✐♦ ❞❛s ♠étr✐❝❛s (i)−(v)✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ♣♦r q✉❛❞r❛t✉r❛s

  é ♠❛✐s ❡✈✐❞❡♥t❡✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ ❡①❡♠♣❧♦ é✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♦ ❞❡ ✐❧✉str❛r ♦ ♠❡❝❛♥✐s♠♦ ❞❡ ✐♥t❡❣r❛çã♦ ♣♦r ♠❡✐♦ ❞❡ ❡str✉t✉r❛s s♦❧ú✈❡✐s ♥♦ ❝❛s♦ ♠❛✐s s✐♠♣❧❡s ❥á ❝♦❜❡rt♦ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ▲✐❡✲❇✐❛♥❝❤✐✳ ◆♦ s❡❣✉♥❞♦ ❡①❡♠♣❧♦✱ ❛♦ ❝♦♥trár✐♦✱ s❡rá ♠♦str❛❞❛ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ♣♦r q✉❛❞r❛t✉r❛s ❞❛s ♠étr✐❝❛s ❞♦ t✐♣♦ (iii)✳ ◆❡st❡ ❝❛s♦✱ ❥á q✉❡ ♦ t❡♦r❡♠❛ ❞❡ ▲✐❡✲❇✐❛♥❝❤✐ ♥ã♦ s❡ ❛♣❧✐❝❛✱ ♦ ♠ét♦❞♦ ❞❡ ✐♥t❡❣r❛çã♦ ❞✐s❝✉t✐❞♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r s❡rá ✐❧✉str❛❞♦ ❝♦♠ ✉♠ ♠❛✐♦r ❣r❛✉ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✳ ❊♠ ♣❛rt✐❝✉❧❛r ♥❡st❡ ❡①❡♠♣❧♦✱ ✈❡r❡♠♦s ❛ ✈❛♥t❛❣❡♠ ❞❡ ❝♦♥s✐❞❡r❛r s✐♠❡tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s ♠❛✐s ❣❡r❛✐s ❞❛q✉❡❧❛s ❞❡ ❑✐❧❧✐♥❣ ❥✉st✐✜❝❛♥❞♦✱ ❛ss✐♠✱ ❛ ❞❡s❝r✐çã♦ ❞❛q✉❡❧❡ t✐♣♦ ❞❡ s✐♠❡tr✐❛ ♥♦ ❝❛♣ít✉❧♦ ✷✳

  2

  2 1 dx 2 dx

  1

  2 ✹✳✺✳✶ ❆♣❧✐❝❛çã♦ às ▼étr✐❝❛s ❞♦ ❚✐♣♦ g = dx

  3 + + dx + 2x

  2

  2 1 + x 1 dx

  3

  ◆♦s ❞❡♠❛✐s ❝❛s♦s ❡st✉❞❛❞♦s ❡♠ ❬✾❪✱ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ é ❣❛r❛♥t✐❞❛ ♣♦✐s ❛s ♠étr✐❝❛s

  1 , X 2 > = E, f 1 , f

  2

  ❛❞♠✐t❡♠ ✉♠❛ á❧❣❡❜r❛ ❛❜❡❧✐❛♥❛ ❞❡ s✐♠❡tr✐❛s G =< X ✳ ▲♦❣♦✱ s❡ {f }

  ✽✵

  1

  sã♦ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❛❞❛s ♣❡❧♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ ❡ ♣❡❧❛s s✐♠❡tr✐❛s ❞♦ t✐♣♦ ❑✐❧❧✐♥❣ X ❡ X

  2 s♦❜r❡ ❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❛r✐❡❞❛❞❡s ✐♥t❡❣r❛✐s tr✐❞✐♠❡♥s✐♦♥❛✐s✱ t❡♠♦s ✉♠❛ ❡str✉t✉r❛ E , X , X 1 >

  s♦❧ú✈❡❧ ❛❜❡❧✐❛♥❛ ✐♥❞✉③✐❞❛ ♣❡❧❛ á❧❣❡❜r❛ ❛❜❡❧✐❛♥❛ < X ✳ ❈♦♠♦ ✐❧✉str❛çã♦ ❞❡st❡

  2

  2

  • dx +2x dx dx
  • 1

  1

  2

  3

  ❝❛s♦✱ ✐♥t❡❣r❛♠♦s ♣♦r q✉❛❞r❛t✉r❛s ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❞❛ ♠étr✐❝❛ g = dx

  2

  2

  2

  (1 + x ) dx

  1 3 ✳ ❊st❛ ♠étr✐❝❛ ❛❞♠✐t❡ ✉♠❛ á❧❣❡❜r❛ ❛❜❡❧✐❛♥❛ ❞❡ s✐♠❡tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s ❞♦ t✐♣♦

  , X2 >

  1

  ❑✐❧❧✐♥❣ Kill(g) =< X ❣❡r❛❞❛ ♣❡❧♦s ❝❛♠♣♦s ∂ ∂

  X = , X =

  1

  2

  2

  3

  ∂x ∂x q✉❡ s❛t✐s❢❛③❡♠ [X , X ] = 0.

  1

  2

  ❆ ❧❛❣r❛♥❣❡❛♥❛ ❘✐❡♠❛♥♥✐❛♥❛ L✱ ♥❡st❡ ❝❛s♦✱ é ❞❛❞❛ ♣♦r

  2

  2

  1

  1

  2

  2 L = v + v + 2x v v + (1 + (x ) )v ,

  

2

  3

  1

  2 3 ✭✹✳✷✮ i

  1

  , v i ) (π) ♦♥❞❡ (t, x sã♦ ❝♦♦r❞❡♥❛❞❛s s♦❜r❡ J ✳

  X ∂L

  (dx i i dt) ❯s❛♥❞♦ ❡ss❛ L ❡ ❧❡♠❜r❛♥❞♦ q✉❡ Θ = − v − Ldt ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r

  ∂v i

  i (1) (1)

  y y = X Θ = X Θ

  ❛s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s f

  1 ❡ f 2 ✉t✐❧✐③❛♥❞♦ ♦s ♣r✐♠❡✐r♦s ♣r♦❧♦♥❣❛♠❡♥t♦s

  1

  

2

(1)

  X = X

  1

  1 (1)

  X = X

  2

  2

  2

  = v + x v = x v + v + v x

  1

  2

  1

  3

  2

  1

  2

  

3

  3

  ♦❜t❡♠♦s q✉❡ f ❡ f

  1 ✳

  ❖✉tr❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ é ❞❛❞❛ ♣❡❧♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ E = L✳ ❊st❛ é ❛ ✐♥t❡❣r❛❧

  

E

  ♣r✐♠❡✐r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ♠❡s♠♦ ❝❛♠♣♦ X q✉❡ ❞❡s❝r❡✈❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ∂ ∂ ∂ ∂

  1 X E = v + v + v + (v v + x v )

  1

  2

  3

  2

  3

  3

  1

  2

  

3

  ∂x ∂x ∂x ∂v

  1

  ∂ ∂

  1

  

1

  2

  1 + ( v + x v v + (x ) v v ) v + x v v ) .

  3

  1

  1

  2

  1

  3

  1

  2

  1

  3

  −v − (v ∂v ∂v

  2

  3 c = , f = c , f = c

  ❆s ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ Γ {E = c

  1

  1

  2 2 }✱ ♥❡st❡ ❝❛s♦✱ sã♦ tr✐❞✐♠❡♥✲

  1

  2

  s✐♦♥❛✐s ❥á q✉❡✱ ❞❡ ❢❛t♦✱ ❛s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s E, f ❡ f sã♦ ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❈♦♠♦ ❛ á❧❣❡❜r❛ ❞❡ s✐♠❡tr✐❛s Kill(g) é ❛❜❡❧✐❛♥❛ ❡ t❛♥❣❡♥t❡ às ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧✱

  ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦ ♠ét♦❞♦ ❞❡ ✐♥t❡❣r❛çã♦ ❝♦♠ ❡str✉t✉r❛s s♦❧ú✈❡✐s ❝♦♠ ❡st❛ á❧❣❡❜r❛ ❡ ❝♦♠ ❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ Ω := dx

  1

  2 3 ✳ ❆ss✐♠✱ t❡♠♦s q✉❡

  ∧ dx ∧ dx y y y ∆ := X E

  X

  1

  1 X 2 Ω = v

  2

  ❡ ❛ r❡str✐çã♦ ¯Ω ❞❛ 1✲❢♦r♠❛ 1 dx

  1

  y y Ω =

  X E X Ω = + dx

  2

  1

  3

  3

  −v

  ✽✶

  c

  à ✈❛r✐❡❞❛❞❡ Γ é ❢❡❝❤❛❞❛✳ h i

  v v

  1

  1

  = d x := x ➱ ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r q✉❡ ¯Ω

  2 3 ✳ ▲♦❣♦✱ I

  2 3 é ✉♠❛

  − v v − v v

  • x +x

  2

  1

  3

  2

  1

  3 E , X 1 , X 2 > c

  1

  ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❞❡ < X s♦❜r❡ Γ ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❛ r❡str✐çã♦ ¯Ω ❞❛ 1✲❢♦r♠❛ 1 dx

  1

  y y Ω =

  X E X Ω = v

  1

  2

  2

  2

  − dx ∆ v

  1 c = c Ω = dI

  2

  3

  1

  1

  à ✈❛r✐❡❞❛❞❡ Γ ∩ {I } é ❢❡❝❤❛❞❛ ❡ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ ¯ ❝♦♠

  2

  √

  (x1v2+(1+x 1)v3)

  2 (v +x v ) (x )

  2

  1

  3 1 − v2+x1v3

  2

  2

  2

  2

  2

  (2v + 4x v v + 2x v + v + v )arctg( )

  1

  2

  3

  2

  1

  3

  3 1 v

  1 I = 1 p

  2

  2

  2 (v + x v ) (v + x v )

  2

  1

  

3

  2

  1

  3

  2

  2

  2

  2

  2x v + 4x x v v + 2x x v + 2x v v + v v + 2x v v

  2

  2

  1

  2

  3

  2

  1

  1

  2

  3

  1

  1

  3

  2

  1

  3

  1 .

  p −

  2

  2

  2 (v + x v ) (v + x v )

  2

  1

  3

  2

  1

  3

  ▲♦❣♦✱ ❛s ❣❡♦❞és✐❝❛s sã♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❞❡s❝r✐t❛s ♣❡❧♦ s✐st❡♠❛ , f = c , f = c , I = c , I = c

  1

  1

  2

  

2

  1

  3

  2

  4

  {E = c }

  2 2 2x

  2

  2

  1 1 + φ (x ) dx 2 + e dx

  3 ✹✳✺✳✷ ❆♣❧✐❝❛çã♦ às ▼étr✐❝❛s ❞♦ t✐♣♦ g = dx

  ❈♦♠♦ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ❞❛s ❡str✉t✉r❛s s♦❧ú✈❡✐s ♥♦ ❝❛s♦ ❡♠ q✉❡ ♥ã♦ t❡♠♦s ✉♠❛ á❧❣❡❜r❛ ❞❡ s✐♠❡tr✐❛s ❛❜❡❧✐❛♥❛✱ ♠♦str❛r❡♠♦s ❝♦♠♦ é ♣♦ssí✈❡❧ ✐♥t❡❣r❛r ♣♦r

  2 2 2x

  2

  2

  • φ (x ) (dx + e dx )

  1

  q✉❛❞r❛t✉r❛s ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❞❛ ♠étr✐❝❛ g = dx

  1

  2 3 ✳ P❛r❛ ❡st❛

  , X , X >

  1

  2

  3

  ♠étr✐❝❛ é ❝♦♥❤❡❝✐❞❛ ❛ á❧❣❡❜r❛ ❞❡ s✐♠❡tr✐❛s ❞♦ t✐♣♦ ❑✐❧❧✐♥❣ Kill(g) =< X ❣❡r❛❞❛ ♣❡❧♦s ❝❛♠♣♦s

  ∂ ∂ ∂ X = , X =

  1

  2

  3

  − x

  3

  3

  ∂x ∂x ∂x

  2

  ∂ 1 ∂

  2 −2x

  X = x e

  • 2

  3

  3

  − x

  3

  2

  ∂x 2 ∂x

  3

  q✉❡ s❛t✐s❢❛③❡♠ [X , X ] =

  1

  2

  1

  −X [X , X ] =

  1

  3

  2

  −X [X , X ] = X

  2

  3

  1

  ❆ ❧❛❣r❛♥❣❡❛♥❛ ❘✐❡♠❛♥♥✐❛♥❛ L✱ ♥❡st❡ ❝❛s♦✱ é ❞❛❞❛ ♣♦r

  1

  1

  i

  2 2 2x

  2

  , v i ) = v φ(x ) v + e v ,

  1

  2 L(t, x +

  1

  2 3 ✭✹✳✸✮

  2

  2

  i

  1

  , v i ) (π) ♦♥❞❡ (t, x sã♦ ❝♦♦r❞❡♥❛❞❛s s♦❜r❡ J ✳

  ✽✷

  X ∂L

  (dx dt)

  i i

  ❯s❛♥❞♦ ❡ss❛ L ❡ ❧❡♠❜r❛♥❞♦ q✉❡ Θ = −v −Ldt ♣♦❞❡♠♦s ❢❛❝✐❧♠❡♥t❡ ∂v i

  i

  ❝❛❧❝✉❧❛r ❛s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s

  

(1) (1) (1)

  1

  y y y f = X Θ, f = X Θ, f = X Θ

  2

  3

  1

  

2

  3

  ✉t✐❧✐③❛♥❞♦ ♦s ♣r✐♠❡✐r♦s ♣r♦❧♦♥❣❛♠❡♥t♦s

  (1)

  X = X

  1 ;

  1

  ∂

  (1)

  X = X ;

  2

  3 2 − v

  ∂v

  3

  ∂ ∂

  (1) −2x

  3

  3

  2 X = X + v v + x v )

  2

  3

  3 3 − (e

  ∂v ∂v

  2

  3

  ❡ ♦❜t❡♠♦s q✉❡

  2 2x

  2

  f

  1 = φ e v 3 ; 2 2x

  2

  f = (x e v )

  2

  3

  3

  2

  −phi − v

  1

  

2 2x

  2

  3

  

3

  2 f = φ ( + e x v ).

  3

  2

  − −v

  3 − 2x

  2 ❖✉tr❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ é ❞❛❞❛ ♣❡❧♦ ❢✉♥❝✐♦♥❛❧ ❡♥❡r❣✐❛ E = L✳ ❊st❛ é ❛ ✐♥t❡❣r❛❧

  

E

  ♣r✐♠❡✐r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ♠❡s♠♦ ❝❛♠♣♦ X q✉❡ ❞❡s❝r❡✈❡ ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ∂ ∂ ∂ ∂

  2 2x

  2 ′

  1

  2

  

3

  2 X E = v + v + v + φφ (v + e v )

  2

  3

  ∂x ∂x ∂x ∂v

  1

  2

  3

  1 ′

  ∂ 2v (v φ + φv ) ∂

  3

  1

  2 2x

  2 2 ′ + (φ v v φ ) .

  1

  2 3 − 2v −

  ∂v φ ∂v

  2

  3 E

  ❖ ♥♦ss♦ ♣r♦❜❧❡♠❛ ❝♦♥s✐st❡ ❡♠ ✐♥t❡❣r❛r ♦ ❝❛♠♣♦ X ✳ n o

  (1) (1) (1)

  X E , X , X , X ❙❛❜❡♠♦s q✉❡ ♦s ❝❛♠♣♦s

  1

  2 3 ❣❡r❛♠ ✉♠❛ á❧❣❡❜r❛ s❛t✐s❢❛③❡♥❞♦ (1)

  [X E , X ] = 0

  i

(1) (1) (1) (1)

  [X , X ] = [X , X ] =

  1

  2

  1 2 −X

  1

(1) (1) (1) (1)

  [X , X ] = [X , X ] = X

  1

  3

  1

  3

  2

(1) (1) (1)

(1)

  [X , X ] = [X

  2 , X 3 ] = .

  −X

  2

  3

  3

  ▲♦❣♦✱ ❡st❛ á❧❣❡❜r❛ ♥ã♦ é s♦❧ú✈❡❧✳ ❆❧é♠ ❞✐ss♦✱ ♥❡♠ t♦❞♦s ❡st❡s ❝❛♠♣♦s sã♦ t❛♥✲ , f = c , f = c , f = c

  1

  1

  2

  2

  3

  3

  ❣❡♥t❡s às ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ {E = c } q✉❡ sã♦ ❜✐❞✐♠❡♥s✐♦♥❛✐s✳ , f

  E

  ❉❡ ❢❛t♦✱ ❛s ❢✉♥çõ❡s E, f

  1 2 ❡ f 3 sã♦ ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❖ ❝❛♠♣♦ X é t❛♥✲ i

  ❣❡♥t❡ à ❡ss❛ ✈❛r✐❡❞❛❞❡ ❞❡ ♥í✈❡❧ ❡✱ ❛♦ ❝♦♥trár✐♦ ❞❡st❡✱ ♥❡♥❤✉♠ ❞♦s ♦✉tr♦s ❝❛♠♣♦s X é t❛♥❣❡♥t❡✳ ▼♦str❛r❡♠♦s ❞♦✐s ♣♦ssí✈❡✐s ♠ét♦❞♦s ❞❡ ✐♥t❡❣r❛çã♦ ❞♦ ✢✉①♦ ❣❡♦❞és✐❝♦ ❛♣r♦✈❡✐✲ t❛♥t♦✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❛ ♣r❡s❡♥ç❛ ❞❡ s✐♠❡tr✐❛s ❞❡ ❈❛rt❛♥✳ P♦❞❡✲s❡ ✈❡r✐✜❝❛r✱ ♥♦ ❝❛s♦

  ❞❡st❛ ♠étr✐❝❛✱ q✉❡ ❛ á❧❣❡❜r❛ ❞❛s s✐♠❡tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s ❞♦ t✐♣♦ ▲✐❡ ❝♦✐♥❝✐❞❡ ❝♦♠ Kill(g)✳ ❆♦ ❝♦♥trár✐♦✱ ❛ á❧❣❡❜r❛ ❞❡ s✐♠❡tr✐❛s ✈❛r✐❛❝✐♦♥❛✐s ❞♦ t✐♣♦ ❈❛rt❛♥ é ♠❛✐s ❛♠♣❧❛ ❡ ✉s❛r❡♠♦s

  ✽✸ ❆❜♦r❞❛❣❡♠ ■ ✿ ■♥t❡❣r❛çã♦ s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ tr✐❞✐♠❡♥s✐♦♥❛✐s

  ◆❡st❡ ❝❛s♦✱ ❝♦♥str✉✐r❡♠♦s ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ t❛♥❣❡♥t❡ às ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ Γ c = , f = c , f = c

  1

  1

  2

  2 {E = c } .

  1 P❛r❛ ❡ss❡ ✜♠✱ ♣r♦❝✉r❛♠♦s ✉♠❛ s✐♠❡tr✐❛ ✈❛r✐❛❝✐♦♥❛❧ Z ❞♦ t✐♣♦ ❈❛rt❛♥✱ t❛♥❣❡♥t❡ c E

  ❛ Γ ❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ X ✳ ❯♠❛ s✐♠❡tr✐❛ ❞❡ss❡ t✐♣♦ é ❞❛❞❛ ♣❡❧♦ ❝❛♠♣♦ ∂ ∂ ∂ ∂

  2 2x

  2

  1

  2

  3

  2 Z = φ ( v + 2v v ).

  2

  3

  −v − v − e

  3

  ∂x ∂x ∂v ∂v

  2

  3

  2

  4 E

  ❊st❡ ❝❛♠♣♦ ❝♦♠✉t❛ ❝♦♠ X ✳ P♦rt❛♥t♦✱ ♣❛r❛ ❝♦♥str✉✐r ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ ❞♦

  c

  2

  t✐♣♦ q✉❡ ♣r❡❝✐s❛♠♦s ♣♦❞❡♠♦s ♣r♦❝✉r❛r ✉♠ ❝❛♠♣♦ Z t❛♥❣❡♥t❡ ❛ Γ ❡ t❛❧ q✉❡ [Z

  2 , Z

1 ] E , Z

1 >

  ∈ < X [Z , X E ] E , Z > .

  2

  1

  ∈ < X ❯♠❛ ❡s❝♦❧❤❛ ♣♦❞❡ s❡r

  3

  φ ∂ ∂ ∂ ∂

  2x 2 2x

  2 2 2x

  2

  2

  2

  3

  3

  2

  2

  3

  2 Z = v + φ e v v + v v φ e .

  −e − φ

  ′

  φ ∂x ∂x ∂x ∂v

  1

  

2

  3

  2 E , Z , Z > c

  1

  2

  ❉❡ss❛ ❢♦r♠❛✱ < X é ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ s♦❜r❡ Γ ✳ P♦rt❛♥t♦✱ ♣♦❞❡✲ ♠♦s ❛♣❧✐❝❛r ♦ ♠ét♦❞♦ ❞❡ ✐♥t❡❣r❛çã♦ ❝♦♠ ❡str✉t✉r❛s s♦❧ú✈❡✐s ✉s❛♥❞♦ ❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡

  Ω = dx

  1

  2

  3

  ∧ dx ∧ dx

  

c , f = c , f = c , v

  s♦❜r❡ Γ ✳ ❉❡ ❢❛t♦✱ {E = c

  1

  1

  2

  2

  1 2 ❡ v 3 ❡✱

  } ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛ ❝♦♠ r❡s♣❡✐t♦ ❛ v

  1 , x 2 , x

  

3

  ♣♦rt❛♥t♦✱ t❡♠ ❝♦♦r❞❡♥❛❞❛s ✐♥t❡r♥❛s x ✳ ❉❡ ❢❛t♦✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ♣❛r❛♠❡tr✐③❛çã♦

  c

  ❞❡ Γ ✿ c

  1

  v

  3 = 2 2x

  2

  φ e c x + c

  1

  3

  2

  v =

  2

  2

  φ

  1

  ǫ

  2 2 2x 2 2x 2 2x 2x

  2

  2

  2

  2

  2

  2

  v = ( e x c x e e + c e φ ) ,

  1

  

1

  2

  3

  −c − c

  1 3 − 2c − c

  2 2x

  2

  e ♦♥❞❡ ǫ = ±1✳ ❆ss✐♠✱ t❡♠♦s q✉❡

  4 2 2x

  2

  2

  y y y ∆ := X E Z

  1 Z 2 Ω = v 1 φ (v + e v )

  2

  3

  ❡ ❛ r❡str✐çã♦ ¯Ω

  2 ❞❛ ✶✲❢♦r♠❛

  

2

  1 φ y y Ω =

X E Z Ω = (v v dx v dx )

  2

  1

  1

  3

  2

  1

  2

  3

  − v ∆ ∆

  c

  ✽✹ ➱ ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r q✉❡

  1

  2

  

2

  2 −2x

  ¯ Ω = d ln c e + (c + c x ) .

  2

  2

  1

  3

  −

  1

  2c

  1

  2

  2 −2x

  2 2 := c e +(c 2 + c 1 x 3 ) E , Z 1 , Z 2 >

  ▲♦❣♦✱ I

  1 é ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❞❡ < X s♦❜r❡

  1

  y y Γ c c = c Ω =

X E Z Ω

  2

  3

  1

  1

  2

  ✳ ❙✉❜s❡q✉❡♥t❡♠❡♥t❡✱ s♦❜r❡ Γ ∩ {I }✱ ❛ r❡str✐çã♦ ¯ ❞❛ ❢♦r♠❛ Ω é

  ∆

  ❢❡❝❤❛❞❛ ❡ ✈❛❧❡

  1 y y dΩ

  1 = d

  X E Z

  2 Ω = dI 1 ,

  ∆ R

  c c c c ǫ

  1

  3

  1

  3

  = e arctg [(c + c x ) e ]+ dx

  1

  2

  1 3 √

  1

  ♦♥❞❡ I

  − ✳ ❊st❛ é ♦✉tr❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛

  2 2c1c3

φ c φ

−e

  E , Z , Z > c

  1

  2

  ❞❡ < X s♦❜r❡ Γ ✳ ❈♦♥❝❧✉✐♥❞♦✱ ❛s ❣❡♦❞és✐❝❛s sã♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ❞❡s❝r✐t❛s ♣❡❧♦ s✐st❡♠❛

  , f = c , f = c , I = c , I = c

  1

  1

  2

  

2

  2

  3

  3

  4 {E = c } .

  ❆❜♦r❞❛❣❡♠ ■■ ✿ ■♥t❡❣r❛çã♦ s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧ ❜✐❞✐♠❡♥s✐♦♥❛✐s ❆❣♦r❛✱ ❝♦♥s✐❞❡r❛r❡♠♦s ❛s ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧

  Γ c = , f

  1 = c

1 , f

2 = c 2 , f 3 = c

  3

  {E = c }

  E c

  1

  ❡ ♣r♦❝✉r❛♠♦s ✉♠❛ s✐♠❡tr✐❛ Z ❞❡ X s♦❜r❡ Γ ✳ P♦rt❛♥t♦✱ ❝♦♥str✉✐r❡♠♦s ✉♠❛ ❡str✉t✉r❛

  E > c

  s♦❧ú✈❡❧ ♣❛r❛ < X s♦❜r❡ ❡st❛s ✈❛r✐❡❞❛❞❡s ❞❡ ♥í✈❡❧✳ ❯♠❛ t❛❧ s✐♠❡tr✐❛ s♦❜r❡ Γ é ❞❛❞❛ ♣♦r

  ∂ ∂ ∂ ∂ ∂ ∂

  2

  2 2 2x

  2

  2

  • 2 Z = + φ v + φ v + φ e v v v .

  1

  2

  3

  2

  3 3 − 2φ

  ∂x ∂x ∂x ∂v ∂v ∂v

  1

  2

  3

  1

  2

  3

  , X E ] = 0 c ❊st❛ é ✉♠❛ s✐♠t❡r✐❛ ❞♦ t✐♣♦ ❈❛rt❛♥ t❛❧ q✉❡ [Z

  1 ❡ é t❛♥❣❡♥t❡ ❛ Γ ✳ ▲♦❣♦✱

  < X E , Z

  1 > c

  é ✉♠❛ ❡str✉t✉r❛ s♦❧ú✈❡❧ ❛❜❡❧✐❛♥❛ s♦❜r❡ Γ ✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ✉s❛r ❡st❛

  c

  ❡str✉t✉r❛ ♣❛r❛ ✐♥t❡❣r❛r ♦ ✢✉①♦ ❣❡♦❞és✐❝♦ s♦❜r❡ Γ ✳

  

4

  2

  2

  y Θ = φ (v + exp2x v )

  1

  2

  ❖❜s❡r✈❛çã♦ ✹✳✺✳✶✳ ❖❜s❡r✈❡ q✉❡ Z

  2 3 ❞❡♣❡♥❞❡ ❢✉♥❝✐♦♥❛❧♠❡♥t❡ ❞❡

  f , f , f

  1

  2

  3

  ❡ f ✳ , f = c , f = c , f = c

  1

  

1

  2

  2

  3

  3

  ❈♦♠♦ ❛s ❡q✉❛çõ❡s {E = c } ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞❛s ♥❛ ❢♦r♠❛

  2

  x + 2x c

  1

  

3

  2

  3

  −c

  

3 − 2c

  v

  3 =

  

2

  φ x c + c

  3

  1

  2

  v =

  2

  2

  φ p

  2

  

2

  c φ c

  1

  3

  − c

  2 − 2c

  v = ǫ , ǫ =

  1

  ±1 φ c

  1 x = ln .

  2

  −

  2

  • c

  3

  2

  2

  − c

  3

  c

  1

  φ p −2c

  c

  dx

  1

  2

  2

  c

  

2

  

2

  3 +c

  1 c

  2

  1

  √

  = c

  = c

  1

  , I

  3

  = c

  3

  , f

  

2

  2

     

  , f

  1

  = c

  1

  , f

  ▲♦❣♦✱ ❛s ❣❡♦❞és✐❝❛s ✭♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❛s ♣ré✲❣❡♦❞és✐❝❛s✮ sã♦ ❞❡s❝r✐t❛s ♣❡❧♦ s✐st❡♠❛ {E = c

  1 .

  = dI

  2c

  

2

  4 } .

  3

  X E y Ω =

  1 ∆

  =

  1

  ❡ q✉❡ ❛ 1✲❢♦r♠❛ Ω

  2

  φ

  v

  (v

  1

  ✳ ❯s❛♥❞♦ ❡ss❛ ❢♦r♠❛✱ é ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r q✉❡ ∆ = −v

  3

  ∧ dx

  1

  é Ω = dx

  c

  ✽✺ ✉♠❛ ❢♦r♠❛ ❞❡ ✈♦❧✉♠❡ s♦❜r❡ Γ

  1 ∆

  3

  3

  1

  1 x

  c

  −arctg

  = d    

  1

  = ¯ Ω

  Γ c

  |

  Ω

  dx

  ✳ ❉❡ ❢❛t♦✱ t❡♠♦s q✉❡

  c

  ) é ❢❡❝❤❛❞❛ s♦❜r❡ Γ

  3

  dx

  1

  − v

  1

  • Z ǫ
  • 2c
  • c φ

  ❘❡❢❡rê♥❝✐❛s

  ❬✶❪ ❆✳ ❱✳ ❆♠✐♥♦✈❛ ✷✵✵✸ Pr♦❥❡❝t✐✈❡ ❚r❛♥s❢♦r♠❛t✐♦♥s ♦❢ Ps❡✉❞♦✲❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙❝✐❡♥❝❡s ✶✶✸ ✸✻✼✲✹✼✵

  ❬✷❪ ■✳ ❆♥❞❡rs♦♥ ❛♥❞ ▼✳ ❋❡❧s ✷✵✵✺ ❊①t❡r✐♦r ❉✐✛❡r❡♥t✐❛❧ ❙②st❡♠s ✇✐t❤ ❙②♠♠❡tr② ❆❝t❛ ❆♣♣❧✳ ▼❛t❤✳ ✽✼ ✸✲✸✶

  ❬✸❪ ■✳ ❆♥❞❡rs♦♥✱ ◆✳ ❑❛♠r❛♥ ❛♥❞ P✳ ❖❧✈❡r ✶✾✾✸ ■♥t❡r♥❛❧✱ ❊①t❡r♥❛❧ ❛♥❞ ●❡♥❡r❛❧✐③❡❞ ❙②♠✲ ♠❡tr✐❡s ❆❞✈✳ ▼❛t❤✳ ✶✵✵ ✺✸✲✶✵✵

  ❬✹❪ ❱✳ ■✳ ❆r♥♦❧❞✱ ❆✳ ❇✳ ●✐✈❡♥t❛❧ ✷✵✵✶ ❙②♠♣❧❡t✐❝ ●❡♦♠❡tr② ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ■❱ ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✮

  ❬✺❪ ❱✳ ■✳ ❆r♥♦❧❞ ✶✾✽✾ ▼❛t❤❡♠❛t✐❝❛❧ ▼❡t❤♦❞s ♦❢ ❈❧❛ss✐❝❛❧ ▼❡❝❤❛♥✐❝s ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✮ ❬✻❪ P✳ ❇❛s❛r❛❜✲❍♦r✇❛t❤ ✶✾✾✷ ■♥t❡❣r❛❜✐❧✐t② ❜② q✉❛❞r❛t✉r❡s ❢♦r s②st❡♠s ♦❢ ✐♥✈♦❧✉t✐✈❡ ✈❡❝t♦r

  ✜❡❧❞s ❯❦r❛✐♥✳ ▼❛t✳ ❩❤✳ ✹✸ ✶✸✸✵✲✶✸✸✼❀ tr❛♥s❧❛t✐♦♥ ✐♥ ❯❦r❛✐♥✳ ▼❛t❤✳ ❏✳ ✹✸ ✶✷✸✻✲✶✷✹✷ ❬✼❪ ▼✳❆✳ ❇❛r❝♦ ❛♥❞ ●✳❊✳ Pr✐♥❝❡ ✷✵✵✶ ❙♦❧✈❛❜❧❡ s②♠♠❡tr② str✉❝t✉r❡s ✐♥ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠

  ❛♣♣❧✐❝❛t✐♦♥s ❆❝t❛ ❆♣♣❧✳ ▼❛t❤✳ ✻✻ ✽✾✲✶✷✶ ❬✽❪ ▼✳❆✳ ❇❛r❝♦ ❛♥❞ ●✳❊✳ Pr✐♥❝❡ ✷✵✵✶ ◆❡✇ s②♠♠❡tr② s♦❧✉t✐♦♥ t❡❝❤♥✐q✉❡s ❢♦r ✜rst✲♦r❞❡r

  ♥♦♥✲❧✐♥❡❛r P❉❊s ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳ ✶✷✹ ✶✻✾✲✶✾✻ ❬✾❪ ▲✳ ❇✐❛♥❝❤✐ ✶✽✾✽ ❙✉❣❧✐ s♣❛③✐ ❛ tr❡ ❞✐♠❡♥s✐♦♥✐ ❝❤❡ ❛♠♠❡tt♦♥♦ ✉♥ ❣r✉♣♣♦ ❝♦♥t✐♥✉♦ ❞✐

  ♠♦✈✐♠❡♥t✐ ▼❡♠♦r✐❡ ❞✐ ▼❛t❡♠❛t✐❝❛ ❡ ❞✐ ❋✐s✐❝❛ ❞❡❧❧❛ ❙♦❝✐❡t❛ ■t❛❧✐❛♥❛ ❞❡❧❧❡ ❙❝✐❡♥③❡✱ ❙❡r✐❡ ❚❡r③❛ ❚♦♠♦ ❳■ ✷✻✼✲✸✺✷ ✭❘❡✐♠♣r❡ssã♦✿ ●❡♥❡r❛❧ ❘❡❧❛t✐✈✐t② ❛♥❞ ●r❛✈✐t❛t✐♦♥ ✷✵✵✶✱ ✈✳ ✸✸✱ ◆♦ ✶✷✮

  ❬✶✵❪ ●✳ ❲✳ ❇❧✉♠❛♥ ❛♥❞ ❙✳ ❈✳ ❆♥❝♦ ✷✵✵✷ ❙②♠♠❡tr② ❛♥❞ ■♥t❡❣r❛t✐♦♥ ▼❡t❤♦❞s ❢♦r ❉✐✛❡r✲ ❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✮

  ❬✶✶❪ ●✳ ❲✳ ❇❧✉♠❛♥ ❛♥❞ ❙✳ ❑✉♠❡✐ ✶✾✽✾ ❙②♠♠❡tr✐❡s ❛♥❞ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✮

  ✽✼ ❬✶✷❪ ●✳ ❲✳ ❇❧✉♠❛♥ ❛♥❞ ●✳ ❏✳ ❘❡✐❞ ✶✾✽✽ ◆❡✇ s②♠♠❡tr✐❡s ❢♦r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛✲ t✐♦♥s ■▼❆ ❏✳ ❆♣♣❧✳ ▼❛t❤✳ ✹✵ ✽✼✲✾✹ ❬✶✸❪ ❆✳ ❱✳ ❇♦❧s✐♥♦✈ ❛♥❞ ❆✳ ❚✳ ❋♦♠❡♥❦♦ ✷✵✵✹ ■♥t❡❣r❛❜❧❡ ❍❛♠✐❧t♦♥✐❛♥ ❙②st❡♠s✱ ●❡♦♠❡tr②✱

  ❚♦♣♦❧♦❣②✱ ❈❧❛ss✐✜❝❛t✐♦♥ ✭❈❤❛♣♠❛♥✲❍❛❧❧✴❈❘❈✮ ❬✶✹❪ ❆✳ ❱✳ ❇♦❧s✐♥♦✈ ❛♥❞ ❇✳ ❏♦✈❛♥♦✈✐➣ ✷✵✵✷ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ■♥t❡❣r❛❜✐❧✐t②✱ ▼♦♠❡♥t ▼❛♣

  ❛♥❞ ●❡♦❞❡s✐❝ ❋❧♦✇s ❆♥♥✳ ●❧♦❜✳ ❆♥❛❧✳ ❛♥❞ ●❡♦♠✳ ✷✸ ✸✵✺✲✸✷✷ ❬✶✺❪ ❋✳ ❇r✐❝❦❡❧❧ ❛♥❞ ❘✳ ❙✳ ❈❧❛r❦ ✶✾✼✵ ❉✐✛❡r❡♥t✐❛❜❧❡ ▼❛♥✐❢♦❧❞s ✭❱❛♥ ◆♦str❛♥❞ ❘❡✐♥❤♦❧❞

  ❈♦✳✮ ❬✶✻❪ ❘✳ ❇r②❛♥t✱ ❙✳❙✳ ❈❤❡r♥✱ ❘✳ ●❛r❞♥❡r✱ ❍✳ ●♦❧❞s❝❤♠✐❞t✱ P✳ ●r✐✣t❤s ✶✾✾✵ ❊①t❡r✐♦r ❉✐❢✲

  ❢❡r❡♥t✐❛❧ ❙②st❡♠s ✭▼❙❘■ P✉❜❧✐❝❛t✐♦♥s✮ ❬✶✼❪ ❆✳ ❇r✉❝❡✱ ❘✳ ▼❝▲❡♥❛❣❤❛♥✱ ❘✳ ❙♠✐r♥♦✈ ✷✵✵✶ ❆ ❣❡♦♠❡tr✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ t❤❡ ♣r♦❜❧❡♠

  ♦❢ ✐♥t❡❣r❛❜✐❧✐t② ♦❢ ❍❛♠✐❧t♦♥♥✐❛♥ s②st❡♠s ❜② s❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ❏✳ ●❡♦♠✳ P❤②s✳ ✸✾ ✸✵✶✲✸✷✷

  ❬✶✽❪ ▼✳ P✳ ❞♦ ❈❛r♠♦ ✶✾✾✷ ❘✐❡♠❛♥♥✐❛♥ ●❡♦♠❡tr② ✭❇✐r❦❛✉s❡r✮ ❬✶✾❪ ❉✳ ❈❛t❛❧❛♥♦ ❋❡rr❛✐♦❧✐✱ ❙♦❧✈❛❜❧❡ ❙tr✉❝t✉r❡s ❆♣♣❧✐❡❞ t♦ ■♥t❡❣r❛t✐♦♥ ♦❢ ❱❛r✐❛t✐♦♥❛❧ ❖❉❊s

  ✭❡♠ ♣r❡♣❛r❛çã♦✮ ❬✷✵❪ ❉✳ ❈❛t❛❧❛♥♦ ❋❡rr❛✐♦❧✐✱ P✳ ▼♦r❛♥❞♦ ✷✵✵✾ ❆♣♣❧✐❝❛t✐♦♥s ♦♥ ❙♦❧✈❛❜❧❡s ❙tr✉❝t✉r❡s t♦ t❤❡

  ◆♦♥❧♦❝❛❧ ❙②♠♠❡tr②✲❘❡❞✉❝t✐♦♥ ♦❢ ❖❉❊s ❏✳ ◆♦♥❧✐♥❡❛r ▼❛t❤✳ P❤②s✳ ✶✻ ✷✼✲✹✷ ❬✷✶❪ ❉✳ ❈❛t❛❧❛♥♦ ❋❡rr❛✐♦❧✐✱ P✳ ▼♦r❛♥❞♦ ✷✵✵✾ ▲♦❝❛❧ ❛♥❞ ◆♦♥❧♦❝❛❧ ❙♦❧✈❛❜❧❡ ❙tr✉❝t✉r❡s ✐♥ t❤❡ ❘❡❞✉❝t✐♦♥ ♦❢ ❖❉❊s ❏✳ P❤②s✳ ❆ ✹✷ ✵✸✺✷✶✵ ❬✷✷❪ ❉✳ ❈❛t❛❧❛♥♦ ❋❡rr❛✐♦❧✐ ✷✵✵✼ ◆♦♥❧♦❝❛❧ ❛s♣❡❝ts ♦❢ λ✲s②♠♠❡tr✐❡s ❛♥❞ ❖❉❊s r❡❞✉❝t✐♦♥ ❏✳

  P❤②s✳ ❆✿ ▼❛t❤ ❚❤❡♦r ✹✵ ✺✹✼✾✲✺✹✽✾ ❬✷✸❪ ❉✳ ❈❛t❛❧❛♥♦ ❋❡rr❛✐♦❧✐ ❛♥❞ ❆✳ ▼✳ ❱✐♥♦❣r❛❞♦✈ ✷✵✵✻ ❘✐❝❝✐ ❋❧❛t ✹✲▼❡tr✐❝s ✇✐t❤ ❇✐❞✐✲

  ♠❡♥s✐♦♥❛❧ ◆✉❧❧ ❖r❜✐ts P❛rt ■✳ ●❡♥❡r❛❧ ❆s♣❡❝ts ❛♥❞ ◆♦♥❛❜❡❧✐❛♥ ❈❛s❡ ❆❝t❛ ❆♣❧✳ ▼❛t❤✳ ✾✷ ✷✵✾✲✷✷✺

  ❬✷✹❪ ❉✳ ❈❛t❛❧❛♥♦ ❋❡rr❛✐♦❧✐ ❛♥❞ ❆✳ ▼✳ ❱✐♥♦❣r❛❞♦✈ ✷✵✵✻ ❘✐❝❝✐ ❋❧❛t ✹✲▼❡tr✐❝s ✇✐t❤ ❇✐❞✐✲ ♠❡♥s✐♦♥❛❧ ◆✉❧❧ ❖r❜✐ts P❛rt ■■✳ ❚❤❡ ❆❜❡❧✐❛♥ ❈❛s❡ ❆❝t❛ ❆♣❧✳ ▼❛t❤✳ ✾✷ ✷✷✻✲✷✸✾

  ❬✷✺❪ ▼✳ ❈r❛♠♣✐♥ ✷✵✵✺ ❖♥ t❤❡ ♦rt❤♦❣♦♥❛❧ s❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❍❛♠✐❧t♦♥✲❏❛❝♦❜✐ ❡q✉❛t✐♦♥ ❢♦r ❣❡♦❞❡s✐❝s ✐♥ ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞✱ ❉✐✛✳ ●❡♦♠✳ ❆♣♣❧✳✱ Pr♦❝✳ ❈♦♥❢✳ Pr❛❣✉❡✱ ❆✉❣✉st ✸✵✱ ✷✵✵✹✱ ✹✺✸✲✹✾✻

  ✽✽ ❬✷✻❪ ❘✐❝❤❛r❞✱ ▲✳ ✱ ❇✐s❤♦♣✱ ❘✐❝❤❛r❞ ❏✳ ❈r✐tt❡♥❞❡♥ ✶✾✻✹ ●❡♦♠❡tr② ♦❢ ▼❛♥✐❢♦❧❞s✱ ✭❆▼❙✮ ❬✷✼❪ ❇✳ ❆✳ ❉✉❜r♦✈✐♥✱ ❆✳ ❚✳ ❋♦♠❡♥❦♦✱ ❙✳ P✳ ◆♦✈✐❦♦✈ ✶✾✽✺ ▼♦❞❡r♥ ●❡♦♠❡tr②✕♠❡t❤♦❞s ❛♥❞

  ❆♣♣❧✐❝❛t✐♦♥s✿ ❚❤❡ ❣❡♦♠❡tr② ❛♥❞ t♦♣♦❧♦❣② ♦❢ ♠❛♥✐❢♦❧❞s ✭❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✮ ❬✷✽❪ ❆✳ ❋❛s❛♥♦✱ ❙✳ ▼❛r♠✐ ✷✵✵✻ ❆♥❛❧②t✐❝❛❧ ▼❡❝❤❛♥✐❝s✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ ✭❖①❢♦r❞ ❯♥✐✈❡rs✐t②

  Pr❡ss✮ ❬✷✾❪ ▼✳ ❊✳ ❋❡❧s ✷✵✵✼ ■♥t❡❣r❛t✐♥❣ s❝❛❧❛r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ s②♠♠❡tr② r❡✈✐s✐t❡❞ ❋♦✉♥❞✳ ❈♦♠♣✉t✳ ▼❛t❤✳ ✼ ✹✶✼✲✹✺✹ ❬✸✵❪ ❍✳ ●♦❧❞s❤♠✐❞t ❛♥❞ ❙✳ ❙t❡r♥❜❡r❣ ✶✾✼✸ ❍❛♠✐❧t♦♥✲❈❛rt❛♥ ❢♦r♠❛❧✐s♠ ✐♥ t❤❡ ❝❛❧❝✉❧✉s ♦❢

  ✈❛r✐❛t✐♦♥s ❆♥♥❛❧❡s ❞❡ ▲✬✐♥st✐t✉t ❋♦✉r✐❡r ✶ ✷✵✸✲✷✻✼ ❬✸✶❪ ❚✳ ❍❛rt❧ ❛♥❞ ❈✳ ❆t❤♦r♥❡ ✶✾✾✹ ❙♦❧✈❛❜❧❡ str✉❝t✉r❡s ❛♥❞ ❤✐❞❞❡♥ s②♠♠❡tr✐❡s ❏✳ P❤②s✳

  ❆✿ ▼❛t❤ ●❡♥ ✷✼ ✸✹✻✸✲✸✹✼✶ ❬✸✷❪ ❏✳ ❍♦r✇♦♦❞✱ ❘✳ ▼❝▲❡♥❛❣❤❛♥✱ ❘✳ ❙♠✐r♥♦✈ ✷✵✵✺ ■♥✈❛r✐❛♥t ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♦rt❤♦❣♦♥❛❧❧② s❡♣❛r❛❜❧❡ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠s ✐♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s ❈♦♠♠✉♠✳ ▼❛t❤✳ P❤②s✳ ✷✺✾ ✻✼✾✲

  ✼✵✾✳ ❬✸✸❪ ◆✳ ❑❛♠r❛♥ ✷✵✵✷ ❙❡❧❡❝t❡❞ t♦♣✐❝s ✐♥ t❤❡ ❣❡♦♠❡tr✐❝❛❧ st✉❞② ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

  ✭❆▼❙✮ ❬✸✹❪ ●✳ ❍✳ ❑❛t③✐♥ ❛♥❞ ❏✳ ▲❡✈✐♥❡ ✶✾✼✷ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ▲✐❡ ❉❡r✐✈❛t✐✈❡ t♦ ❙②♠❡tr✐❡s✱

  ●❡♦❞❡s✐❝s✱ ❆♥❞ ✜rst ■♥t❡❣r❛❧s ✐♥ ❘✐❡♠❛♥♥✐❛♥ ❙♣❛❝❡s ❈♦❧❧♦q✳ ▼❛t❤✳ ✭❲r♦❝❧❛✇✮ ✷✻ ✷✶

  ❬✸✺❪ ■✳ ❙✳ ❑r❛s✐❧✬s❤❝❤✐❦ ❛♥❞ ❆✳ ▼✳ ❱✐♥♦❣r❛❞♦✈ ✶✾✽✾ ◆♦♥❧♦❝❛❧ ❚r❡♥❞s ✐♥ t❤❡ ●❡♦♠❡tr② ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✿ ❙②♠♠❡tr✐❡s✱ ❈♦♥s❡r✈❛t✐♦♥ ▲❛✇s✱ ❛♥❞ ❇ä❝❦❧✉♥❞ ❚r❛♥s❢♦r♠❛✲ t✐♦♥s ❆❝t❛ ❆♣♣❧✳ ▼❛t❤✳ ✶✺ ✶✻✶✲✷✵✾

  ❬✸✻❪ ❆✳ ❑✉s❤♥❡r✱ ❱✳ ▲②❝❤❛❣✐♥✱ ❱✳ ❘✉❜st♦✈ ✷✵✵✼ ❈♦♥t❛❝t ●❡♦♠❡tr② ❛♥❞ ◆♦♥✲❧✐♥❡❛r ❉✐❢✲ ❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✮

  ❬✸✼❪ ❏✳ ▼✳ ▲❡❡ ✶✾✾✼ ❘✐❡♠❛♥♥✐❛♥ ▼❛♥✐❢♦❧❞s✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈✉r✈❛t✉r❡ ✭❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✮

  ❬✸✽❪ ❆✳ ❙✳ ▼✐s❤❝❤❡♥❦♦ ❛♥❞ ❆✳ ❚✳ ❋♦♠❡♥❦♦ ✶✾✼✽ ●❡♥❡r❛❧✐③❡❞ ▲✐♦✉✈✐❧❧❡ ▼❡t❤♦❞ ♦❢ ■♥t❡❣r❛✲ t✐♦♥ ♦❢ ❍❛♠✐❧t♦♥✐❛♥ ❙②st❡♠s ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧✳ ❆♣♣❧✳ ✶✷ ✶✶✸✲✶✷✶ ❬✸✾❪ ◆✳ ◆✳ ◆❡❦❤♦r♦s❤❡✈ ✶✾✼✷ ❆❝t✐♦♥✲❆♥❣❧❡ ❱❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r ●❡♥❡r❛❧✐③❛t✐♦♥ ❚r❛♥s✳

  ▼♦s❝♦✇ ▼❛t❤✳ ❙♦❝✳ ✷✻ ✶✽✵✲✶✾✽✳

  ✽✾ ❬✹✵❪ P✳❏✳ ❖❧✈❡r ✶✾✾✸ ❆♣♣❧✐❝❛t✐♦♥ ♦❢ ▲✐❡ ❣r♦✉♣s t♦ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✮ ❬✹✶❪ P✳❏✳ ❖❧✈❡r ✶✾✾✺ ❊q✉✐✈❛❧❡♥❝❡✱ ✐♥✈❛r✐❛♥ts✱ ❛♥❞ s②♠♠❡tr② ✭❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✮ ❬✹✷❪ ●✳ Pr✐♥❝❡ ✶✾✽✸ ❚♦✇❛r❞ ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❞②♥❛♠✐❝❛❧ s②♠♠❡tr✐❡s ✐♥ ❝❧❛ss✐❝❛❧ ♠❡❝❤❛♥✐❝s

  ❇✉❧❧✳ ❆✉str❛❧✳ ▼❛t❤✳ ❙♦❝✳ ✷✼ ✺✸✲✼✶ ❬✹✸❪ ●✳ ❙♣❛r❛♥♦✱ ●✳ ❱✐❧❛s✐ ❛♥❞ ❆✳ ▼✳ ❱✐♥♦❣r❛❞♦✈ ✷✵✵✶ ●r❛✈✐t❛t✐♦♥❛❧ ❋✐❡❧❞s ✇✐t❤ ❛ ♥♦♥✲

  ❆❜❡❧✐❛♥✱ ❜✐❞✐♠❡♥s✐♦♥❛❧ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ s②♠♠❡tr✐❡s P❤②s✳ ▲❡tt✳✱ ❇ ✺✶✸ ✶✹✷✲✶✹✻ ❬✹✹❪ ●✳ ❙♣❛r❛♥♦✱ ●✳ ❱✐❧❛s✐ ❛♥❞ ❆✳ ▼✳ ❱✐♥♦❣r❛❞♦✈ ✷✵✵✷ ❱❛❝✉✉♠ ❊✐♥st❡✐♥ ♠❡tr✐❝s ✇✐t❤

  ❜✐❞✐♠❡♥s✐♦♥❛❧ ❑✐❧❧✐♥❣ ❧❡❛✈❡s✳ ■✳ ▲♦❝❛❧ ❛s♣❡❝ts✳ ❉✐✛❡r✳ ●❡♦♠✳ ❆♣♣❧✳ ✶✻ ✾✺✲✶✷✵ ❬✹✺❪ ●✳ ❙♣❛r❛♥♦✱ ●✳ ❱✐❧❛s✐ ❛♥❞ ❆✳ ▼✳ ❱✐♥♦❣r❛❞♦✈ ✷✵✵✷ ❱❛❝✉✉♠ ❊✐♥st❡✐♥ ♠❡tr✐❝s ✇✐t❤

  ❜✐❞✐♠❡♥s✐♦♥❛❧ ❑✐❧❧✐♥❣ ❧❡❛✈❡s✳ ■■✳ ●❧♦❜❛❧ ❛s♣❡❝ts✳ ❉✐✛❡r✳ ●❡♦♠✳ ❆♣♣❧✳ ✶✼ ✶✺✲✸✺ ❬✹✻❪ ❏✳ ❙❤❡rr✐♥❣ ❛♥❞ ●✳ Pr✐♥❝❡ ✶✾✾✷ ●❡♦♠❡tr✐❝ ❛s♣❡❝ts ♦❢ r❡❞✉❝t✐♦♥ ♦❢ ♦r❞❡r ❚r❛♥s✳ ❆♠❡r✳

  ▼❛t❤✳ ❙♦❝✳ ✸✸✹ ✹✸✸✲✹✺✸ ❬✹✼❪ ▼✳ ❙♣✐✈❛❦ ✶✾✾✾ ❆ ❈♦♠♣r❡❤❡♥s✐✈❡ ■♥tr♦❞✉❝t✐♦♥ ❚♦ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr② ✭P✉❜❧✐s❤

  ❖r P❡r✐s❤✮ ❬✹✽❪ ❍✳ ❙t❡♣❤❛♥✐ ✶✾✽✾ ❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❚❤❡✐r s♦❧✉t✐♦♥ ✉s✐♥❣ s②♠♠❡tr✐❡s ✭❈❛♠❜r✐❞❣❡

  ❯♥✐✈❡rs✐t② Pr❡ss✮ ❬✹✾❪ ❍✳ ❙t❡♣❤❛♥✐✱ ❉✳ ❑r❛♠❡r✱ ▼✳ ▼❛❝❈❛❧❧✉♠✱ ❈✳ ❍♦❡♥s❡❧❛❡rs✱ ❊✳ ❍❡r❧t ✷✵✵✸ ❊①❛❝t ❙♦❧✉✲ t✐♦♥s ♦❢ ❊✐♥st❡✐♥ ❋✐❡❧❞ ❊q✉❛t✐♦♥s ✷♥❞ ❡❞✳ ✭❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✮ ❬✺✵❪ ❙✳ ❙t❡r♥❜❡r❣ ✶✾✻✹ ▲❡❝t✉r❡s ♦♥ ❉✐✛❡r❡♥t✐❛❧ ●❡♦♠❡tr② ✭Pr❡♥t✐❝❡✲❍❛❧❧✮ ❬✺✶❪ ▼✳ ❚s❛♠♣❛r❧✐s✱ ❆✳ P❛❧✐❛t❤❛♥❛s✐s ✷✵✶✵ ▲✐❡ ❛♥❞ ◆♦❡t❤❡r ❙②♠♠❡tr✐❡s ♦❢ ❣❡♦❞❡s✐❝ ❡q✉❛✲ t✐♦♥s ❛♥❞ ❝♦❧❧✐♥❡❛t✐♦♥s ●❡♥ ❘❡❧❛t✐✈ ●r❛✈✐t ✹✷ ✷✾✺✼✲✷✾✽✵ ❬✺✷❪ ▲✳ ❚✉ ✷✵✵✼ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❛♥✐❢♦❧❞s ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✮ ❬✺✸❪ ●✳ ❱✐❧❛s✐ ✷✵✵✽ ❊✐♥st❡✐♥ ▼❡tr✐❝s ✇✐t❤ 2✲❞✐♠❡♥s✐♦♥❛❧ ❑✐❧❧✐♥❣ ▲❡❛✈❡s ❛♥❞ t❤❡✐r P❤②s✐❝❛❧

  ■♥t❡r♣r❡t❛t✐♦♥ ❘❡❝❡♥t ❉❡✈❡❧♦♣♠❡♥t ✐♥ Ps❡✉❞♦✲❘✐❡♠❛♥♥✐❛♥❛♥ ●❡♦♠❡tr② ✹✾✺ ✺✷✻✳ ❬✺✹❪ ❆✳ ▼✳ ❱✐♥♦❣r❛❞♦✈ ❡t ❛❧✳ ✶✾✾✾ ❙②♠♠❡tr✐❡s ❛♥❞ ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s ❢♦r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛✲ t✐♦♥s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s ✭❆▼❙✮

  ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ✲ ❯❋❇❆ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✴ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

  ❆✈✳ ❆❞❤❡♠❛r ❞❡ ❇❛rr♦s✱ s✴♥✱ ❈❛♠♣✉s ❯♥✐✈❡rs✐tár✐♦ ❞❡ ❖♥❞✐♥❛✱ ❙❛❧✈❛❞♦r ✲ ❇❆ ❈❊P✿ ✹✵✶✼✵ ✲✶✶✵

  < ❤tt♣✿✴✴✇✇✇✳♣❣♠❛t✳✉❢❜❛✳❜r>

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