Universidade Federal da Bahia - UFBA Instituto de Matemática - IM

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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

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▼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❛✐♦❧✐✳

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❞❛ ❈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✉❧♦✳

❈❉❯ ✿

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▼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✐♦ ❯❋❆▲

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✏❖ t❡♠♦r ❛ ❉❡✉s é ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝✐ê♥❝✐❛✳✑

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❘❡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✳

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❆❜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✳

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❙✉♠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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷✳✶ ❖ ♣r✐♠❡✐r♦ ❡s♣❛ç♦ ❞❡ ❥❛t♦s J1(M, n) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✷✳✷✳✷ 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❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

(10)

✷✳✹ ❙✐♠❡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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

✹✳✺✳✶ ❆♣❧✐❝❛çã♦ às ▼étr✐❝❛s ❞♦ ❚✐♣♦g =dx2

1+dx22+2x1dx2dx3+(1 +x21)dx23 ✼✾

✹✳✺✳✷ ❆♣❧✐❝❛çã♦ às ▼étr✐❝❛s ❞♦ t✐♣♦ g =dx2

1+φ(x1) (dx22+e2x2dx23) ✳ ✳ ✳ ✽✶

(11)

■♥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♦ ❝♦♠✉♠ ♥♦

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❡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✳

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

A={ϕU :U ⊂Rn−→ϕU(U)⊂M} t❛✐s q✉❡

✭✶✮ ▼ ❂ SϕU(U)❀

✭✷✮ ∀ϕU, ϕV ∈ A t❛✐s q✉❡ ϕU(U)∩ϕV(V)6=φ t❡♠♦s

ϕV−1ϕU :ϕU−1(ϕU(U)∩ϕV(V))−→ϕ−V1(ϕU(U)∩ϕV(V)) é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ Ck

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

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

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❉♦✐s ❛t❧❛s A1 ❡ A2 s♦❜r❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ M sã♦ ❡q✉✐✈❛❧❡♥t❡s

s❡✱ ❡ s♦♠❡♥t❡ s❡✱A1∪ A2 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❛❞♦✳

❙❡M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡xi :=Rn −→Rsã♦ ❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s ❡♠

Rn✱ ❝❛❞❛ ✈✐③✐♥❤❛♥ç❛ ϕU(U) ♣♦❞❡ s❡r ❡q✉✐♣❛❞❛ ❝♦♠ n✲❢✉♥çõ❡s xi : ϕU(U) → R ❞❡✜♥✐❞❛s ♣♦r xi := xi◦ϕU−1 ❡ ❝❤❛♠❛❞❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❡♠ ϕU(U)✳ ❯s❛♥❞♦ ❛s ❢✉♥çõ❡s ❝♦♦r✲ ❞❡♥❛❞❛s✱ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣♦❞❡ s❡r ❧♦❝❛❧♠❡♥t❡ ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠ ✉♠ ❛❜❡rt♦ ❞♦ Rn ❡ ❝❛❞❛ ♣♦♥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✈❡❧

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

U(f) = f ◦ϕU é ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ C∞ ❡♠

ϕ−U1(p)✱ ♣❛r❛ t♦❞❛ ϕU t❛❧ q✉❡ p∈ϕU(U)✳ ❆ ❢✉♥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

❞❡ R ❡ s❡rá ✐♥❞✐❝❛❞♦ ♣♦r C∞(M) ✭♦✉ C

A (M) 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❧❛sA1 ❡A2✳ ❯♠❛ ❢✉♥çã♦F :M →N é ❞✐❢❡r❡♥❝✐á✈❡❧

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

f C∞

A2(N)✳ ❆ ❛♣❧✐❝❛çã♦ F é ❞✐t❛ ❞✐❢❡r❡♥❝✐á✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ F∗(C∞A2(N)) ⊂

(15)

✶✳✶✳✷ ❊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✈❡❧ ❡ pM✳ ❯♠ ✈❡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✿

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

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

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

❖ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ M ❡♠ ✉♠ ♣♦♥t♦ p s❡rá ❞❡♥♦t❛❞♦ ♣♦r TpM ❡ ❞❡♥♦♠✐♥❛❞♦ ❡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)],

∀ξ, ηTpM✱ ∀f ∈C∞(M)❡ c∈R✳

❈♦♠ ❡st❛s ❞✉❛s ♦♣❡r❛çõ❡s✱ TpM 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❡✿

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

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

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

✭▲♦❝❛❧✐③❛❜✐❧✐❞❛❞❡✮

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

▲❡♠❛ ✶✳✶✳✼✳ ❙❡❥❛♠U ❡ V ❜♦❧❛s ❢❡❝❤❛❞❛s ❞❡ Rn t❛✐s q✉❡ U V V\U 6=✳ ❊①✐st❡ ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ϕ:Rn −→

R✱ ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦ ❜♦ss❛✱ t❛❧ q✉❡

(16)

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

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

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

✐♥❝✐❞❡♠ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ p✳ ❙❡ U1 ⊂ U2 ⊂ U sã♦ ❜♦❧❛s ❢❡❝❤❛❞❛s✱ ❝♦♠ p ∈ U1✱

❡♥tã♦✱ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❡①✐st❡hC∞(M)t❛❧ q✉❡h|

U1 = 1 ❡ h|M\U2 = 0✳ ❉❡ss❛ ❢♦r♠❛✱

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

ξ((f g)·h) =ξ(fg)·h(p) + (fg)(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♦✱ s❡❥❛ϕU ✉♠❛ ❝❛rt❛ ❝♦♠ ❞♦♠í♥✐♦U✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❜♦❧❛W ={(¯x1, ...,x¯n)∈Rn|

n

X

i=1

(¯xi−

¯

pi)2 < ǫ} ⊂U ❞❡ r❛✐♦ǫ ❡ ❝❡♥tr♦ p¯=ϕ−U1(p)✱ t❛❧ q✉❡ W ⊂U✳

P❡❧♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❝á❧❝✉❧♦✱ ♣❛r❛ t♦❞♦ (¯x1, ...,x¯n) ❡ t♦❞❛ f ∈C∞(M)✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

ϕ∗

U(f)(¯x1, ...,x¯n) = ϕ∗U(f)(¯p1, ...,p¯n) +

Z 1

0

d dt[ϕ

U(f)(¯p1+t(¯x1−p¯1), ...,p¯n+t(¯xn−

¯

pn))]dt✳

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

f(x) = f(p) +

n

X

i=1

(xi−pi)·gi(x), ♦♥❞❡

gi(x) =

Z 1

0

∂ϕ∗

U(f)

∂x¯i

(¯p1+t(¯x1−p¯1), ...,p¯n+t(¯xn−p¯n))dt ✱ i∈ {1, ..., n}.

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

❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡✜♥✐❞❛s ❡♠ t♦❞❛ ❛ ✈❛r✐❡❞❛❞❡✳ P❛r❛ ✐ss♦✱ ✉s❛♥❞♦ ♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❝♦♥s✐❞❡r❡ ✉♠❛ ❜♦❧❛ ❢❡❝❤❛❞❛U ⊂ W t❛❧ q✉❡ p ∈ U ❡ ✉♠❛ h C∞(M) t❛❧ q✉❡ h|

U = 1 ❡

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

e

xi =

(

(xi·h)(q) ✱s❡q ∈W;

(17)

e

gi =

(

(gi ·h)(q) ✱s❡ q∈W;

0 ✱s❡ q∈M\W .

❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛ s❡❣✉✐♥t❡ ❡①t❡♥sã♦ f˜❞❡ f¯

˜

f =f(p) +

" n X

i=1

(xei−pi)·gei

#

❈❧❛r❛♠❡♥t❡✱ f 6= ¯f✱ ♠❛s f|U = ˜f|U✳ P♦rt❛♥t♦✱ s❡ ξTpM✱ξ(f) = ξ( ¯f) ❆❣♦r❛✱ ✉s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s P✳✶✮ ❡ P✳✷✮✱ ♦❜t❡♠♦s✿

ξ(f) =

n

X

i=1

ξ(xei)·gei(p) = n

X

i=1

ξ(xi)

∂ϕ∗

U(f)

∂xi

(ϕ−U1(p))

=

" n X

i=1

ξ(xei)

∂ ∂xi|ϕ

−1 U (p)

#

(ϕ∗U(f)) =

" n X

i=1

ξ(xei)

∂ ∂xi|

p

#

(f),

♦♥❞❡ ∂

∂xi|

p :C∞(M)−→R é ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❞❡✜♥✐❞♦ ♣♦r ∂

∂xi|p(f) =

∂ ∂xi|ϕ

−1 U (p)(ϕ

U(f))✳

P♦rt❛♥t♦✱ t♦❞♦ ❡❧❡♠❡♥t♦ ξTpM t❡♠ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ❝♦♦r❞❡♥❛❞❛

ξ = n X i=1 αi ∂ ∂xi|p✱ ♦♥❞❡αi ∈R✳

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

∂x1|

p, ...,

∂ ∂xn|

p sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡✲

♣❡♥❞❡♥t❡s✳ ▲♦❣♦ ♦ ❝♦♥❥✉♥t♦

∂ ∂x1|p

, ..., ∂ ∂xn|

p

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

✶✳✶✳✸ ❉✐❢❡r❡♥❝✐❛❧

F

❞❡ ✉♠❛ ❆♣❧✐❝❛çã♦

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

(F)p :TpM −→TF(p)N

t❛❧ q✉❡✱ ❞❛❞❛f C∞(N) [(F

(18)

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

❡①♣r❡ss❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ s❡❥❛♠x1, ..., xn❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ϕU :U ⊂Rn t❛❧ q✉❡a ϕU(U) ❡ y1, ..., yn ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ψV : V ⊂Rm −→N t❛❧ q✉❡ F(a) ψV(V)✳ ◆❡ss❛s ❝♦♦r❞❡♥❛❞❛s✱ F ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❝♦♠♦ F(x1, ..., xn) =

(y1◦F(x1, ..., xn), ..., ym◦F(x1, ..., xn)) =: (F1(x1, ..., xn), ..., Fm(x1, ..., xn)) ❡ Xa ∈ TaM ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛Xa=

n

X

i=1

αi

∂ ∂xi|

a∈TaM✳ P♦rt❛♥t♦✱ s❡f ∈C∞(N) ❡♥tã♦

[(F)a(Xa)] (f) = n

X

i=1

αi

∂ ∂xi|

a(F∗(f))

= n X i=1 αi m X j=1 ∂f ∂yj

(F(a))∂Fj

∂xi

(a)

! = " m X j=1 n X i=1 αi ∂Fj ∂xi

(a)

!

∂ ∂yj|

F(a) #

(f).

▲♦❣♦✱ ❛ ❥✲és✐♠❛ ❝♦♠♣♦♥❡♥t❡ ❞♦ ✈❡t♦r(F)a(Xa)❝♦♠ r❡s♣❡✐t♦ à ❜❛s❡

n

∂ ∂yj|F(a)

o é n X i=1 αi ∂Fj ∂xi

(a)

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

          ∂F1 ∂x1

(a) ∂F1

∂x2

(a) · · · ∂F1

∂xn

(a)

∂F2

∂x1

(a) ∂F2

∂x2

(a) · · · ∂F2

∂xn

(a)

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

∂Fm

∂x1

(a) ∂Fm

∂x2

(a) · · · ∂Fm

∂xn

(a)

         

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

❡q✉✐♣❛❞♦✱ ❞❡ ♠❛♥❡✐r❛ ♥❛t✉r❛❧✱ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧✿ s❡ AM = {(ϕα, Uα)} ❡

(19)

(ϕα(u), ψβ(v)) ∈ M ×N✱ u ∈ Uα✱ v ∈ Vβ✳ ❯♠❛ ♣❡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❛❀

✷✮ ♣❛r❛ t♦❞♦ ♣♦♥t♦ x M✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ Ux ❝♦♥t❡♥❞♦ ♦ ♣♦♥t♦ x t❛❧ q✉❡ s✉❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ π−1(U

x) é ❞✐❢❡♦♠♦r❢❛ ❛♦ ♣r♦❞✉t♦ Ux ×N✱ ✐st♦ é✱ ❡①✐st❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ f|Ux : Ux×N −→ π−

1(U

x) t❛❧ q✉❡ f|Ux(x, y) ∈ π−

1(x)✱ ♣❛r❛ t♦❞♦s

x∈Ux ❡ y∈N✳

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

❆s s✉❜✈❛r✐❡❞❛❞❡s π−1(x) 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✳

❙❡❥❛ A={ϕUα}✉♠ ❛t❧❛s ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ M t❛❧ q✉❡ ❛ ✐♠❛❣❡♠ ✐♥✈❡rs❛ π−

1(V

α)✱

Vα =ϕα(Uα)✱ ❛❞♠✐t❡ ❛ ❛♣❧✐❝❛çã♦ tr✐✈✐❛❧✐③❛♥t❡fα =f|Vα :Vα×N −→π−

1(V

α)s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦2) ❞❡ ✉♠ ✜❜r❛❞♦✳ ❆ ❛♣❧✐❝❛çã♦

fβ−1fα : (Vα∩Vβ)×N −→(Vα∩Vβ)×N,

❛♣❧✐❝❛ ❞✐❢❡♦♠♦r✜❝❛♠❡♥t❡ t♦❞❛ ✜❜r❛ {x} ×N (Vα∩Vβ)×N✱x∈Vα∩Vβ✱ ♥❡❧❛ ♠❡s♠❛✳ ❉❡♥♦t❛r❡♠♦s ❡st❡ ❞✐❢❡♦♠♦r✜s♠♦✱ ♣♦st❡r✐♦r♠❡♥t❡ ❛ ✐❞❡♥t✐✜❝❛çã♦ ó❜✈✐❛ ❞❡ N ❡ {x} ×N✱

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

hαα(x) = id, hβα(x) =h−αβ1(x), hαβ(x)◦hβγ(x) =hαγ(x)

❡ q✉❡ ♦s ❞✐❢❡♦♠♦r✜s♠♦s hαβ(x) ❞❡♣❡♥❞❡♠ ❞✐❢❡r❡♥❝✐❛✈❡❧♠❡♥t❡ ❞❛ ✈❛r✐á✈❡❧ x ∈ Uα ∩Uβ✳ ❆♦ ❝♦♥trár✐♦✱ ❡s♣❡❝✐✜❝❛♥❞♦ ✉♠ ❛t❧❛s A = {ϕUα} ❡♠ M ❡ ✉♠ s✐st❡♠❛ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s

(20)

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

(E, M, π)✳ ❖❜✈✐❛♠❡♥t❡✱ σ é ✉♠❛ s❡çã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ σ(x) π−1(x)✳ ❆ ❝♦❧❡çã♦ ❞❡

t♦❞❛s ❛s s❡çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞♦ ✜❜r❛❞♦ ❡♠ q✉❡stã♦ é ❞❡♥♦t❛❞❛ ♣♦rΓ(π)✳

❙❡❥❛♠ (Ei, Mi, πi)✱ i ∈ {1,2}✱ ✜❜r❛❞♦s✳ ❖s ♣❛r❡s ❞❡ ❛♣❧✐❝❛çõ❡s F : E1 −→ E2✱

f :M1 −→M2sã♦ ❝❤❛♠❛❞♦s ♠♦r✜s♠♦s ❞♦ ♣r✐♠❡✐r♦ ✜❜r❛❞♦ ♥♦ s❡❣✉♥❞♦ s❡π2◦F =f◦π1✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠ ♠♦r✜s♠♦ ❛♣❧✐❝❛ ✜❜r❛s ❞❡π1 ❡♠ ✜❜r❛s ❞❡ π2✳ ❖ ♠♦r✜s♠♦ ✐♥✈❡rs♦

é ❝❤❛♠❛❞♦ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ✭♦✉ ✐s♦♠♦r✜s♠♦✮ ❞♦s ✜❜r❛❞♦s ❡♠ q✉❡stã♦✳

❖ ✜❜r❛❞♦ π1 é ❝❤❛♠❛❞♦ s✉❜✜❜r❛❞♦ s❡ ♦ ♠♦r✜s♠♦ (F, f)❞❡t❡r♠✐♥❛ ♠❡r❣✉❧❤♦s ❞❡

s✉❜✈❛r✐❡❞❛❞❡sF :E1 ֒→E2✱f :M1 ֒→M2✳

◗✉❛❧q✉❡r ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ f : M1 −→M ♥❛ ❜❛s❡ ❞♦ ✜❜r❛❞♦ π :E −→M

❣❡r❛ ✉♠ ✜❜r❛❞♦ s♦❜r❡M1✱ ❝❤❛♠❛❞♦ ✜❜r❛❞♦ ✐♥❞✉③✐❞♦✱ ❡ ❞❡♥♦t❛❞♦ ♣♦rf∗(x) :f∗(E)−→

M1✳

❯♠❛ ✜❜r❛✱ ❞♦ ✜❜r❛❞♦ ✐♥❞✉③✐❞♦✱ ❡♠ ✉♠ ♣♦♥t♦ y M1 ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ✜❜r❛ ❞♦

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

❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ M1 ×E ❝♦♥s✐st✐♥❞♦ ❞❡ ♣❛r❡s (y, e) t❛✐s q✉❡ f(y) = π(e) ❡ ❛ ♣r♦❥❡çã♦

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

❯♠ ✜❜r❛❞♦ (E, M, π) é ❝❤❛♠❛❞♦ ✜❜r❛❞♦ ✈❡t♦r✐❛❧ s❡ s✉❛s ✜❜r❛s π−1(x) 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❛♥✲

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

❙❡❥❛♠ Si✱ i ∈ {1, ..., k}✱ s❡çõ❡s ❞♦ ✜❜r❛❞♦ ✈❡t♦r✐❛❧ π ❡ s❡❥❛♠ fi ∈ C∞(M) ❢✉♥çõ❡s s♦❜r❡ ❛ ❜❛s❡ M✳ ❊♥tã♦ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ s❡çã♦ s = Pifisi ❞❛❞❛ ♣♦r

s(x) = Pifi(x)si(x)✱ ❥á q✉❡ ❛ ✜❜r❛ π−1(x) 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 ✜❜r❛❞♦s sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ♣❡❧♦ ❢❛t♦ q✉❡ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦hαβ(x)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 = [

pM

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

t♦❞❛ϕU ∈ A ❝♦♠ x1, ..., xn ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ϕU✱ ❞❡✜♥❛ ❛s ❛♣❧✐❝❛çõ❡s

e

(21)

e

ϕUe(ξp) = (x1(p), ..., xn(p), ξp(x1), ..., ξp(xn))✱

♦♥❞❡U˜ =π−1(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❛

❞✐❢❡r❡♥❝✐á✈❡❧Ae= {ϕeUe : Ue −→ R

2n} ❞❡ ❝❧❛ss❡ C♣♦✐s ❛s ❛♣❧✐❝❛çõ❡sϕe

e U ◦ϕe−

1

e

V sã♦ ❞✐❢❡♦✲ ♠♦r✜s♠♦s ❞❡ ❝❧❛ss❡C∞ ♣❛r❛ t♦❞❛s ϕe

e

U,ϕeVe ∈Ae✳

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

π :T M −→M

é ✉♠ ✜❜r❛❞♦ ✈❡t♦r✐❛❧ q✉❡ é ❞❡♥♦♠✐♥❛❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❛ ▼✳ ❖s ❡s♣❛ç♦sTpM =π−1(p) sã♦ ❛s ✜❜r❛s✳

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

q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ ❝❛❞❛ ♣♦♥t♦ x M ❡stá ♥❛ ✜❜r❛ π−1(x) s♦❜r❡ x✱ ♦✉ s❡❥❛✱ ❛ ✐♠❛❣❡♠ ❞❡

❝❛❞❛ ♣♦♥t♦ x M ♣❡rt❡♥❝❡ ❛♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ TpM✳ 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✐❡❞❛❞❡ ❝♦♠♦ ❞❡r✐✈❛çõ❡s ❞❛ á❧❣❡❜r❛ C∞(M)✳ ❉❡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)−→C∞(M) q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

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

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

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

❞❡r✐✈❛çõ❡s s❡rá ❞❡♥♦t❛❞♦ ♣♦rD(M)✳ ❈♦♥s✐❞❡r❛r❡♠♦s s❡♠♣r❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s C∞

(22)

✶✵

(X1+X2)(f) :=X1(f) +X2(f)✱ ∀X1, X2 ∈ D(M)✱∀f ∈C∞(M)

(f X)(g) := f X(g)✱ X ∈ D(M)✱ f, g C∞(M)

❈♦♠ ❡st❛s ♦♣❡r❛çõ❡s✱ D(M) é ✉♠ C∞(M)✲♠ó❞✉❧♦✳

❙❡❥❛♠ M ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ X ∈ D(M) ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s✳ ❊♠ ❝❛❞❛

♣♦♥t♦pM✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ✈❡t♦r t❛♥❣❡♥t❡Xp ♣♦r

Xp(f) :=X(f)(p)✱ ∀f ∈C∞(M)✳

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

M −→ T M ♣♦r φX(p) = Xp✱ ♦✉ 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❡❥❛♠

x1, ..., xn❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ϕU✱X ∈ D(M)❡f ∈C∞(M)✳ P❡❧❛ ♦❜s❡r✈❛çã♦ ❛♥t❡r✐♦r✱ ❡♠ ❝❛❞❛ ♣♦♥t♦pM✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ✉♠ ✈❡t♦r t❛♥❣❡♥t❡Xp t❛❧ q✉❡Xp(f) =

X(f)(p)✳ ❆❧é♠ ❞✐ss♦✱ ❥á ❢♦✐ ✈✐st♦ q✉❡ s❡ p ∈ ϕU(U) ❡♥tã♦ ♣♦❞❡♠♦s ❡①♣r❡ss❛r Xp ❡♠ ❝♦♦r❞❡♥❛❞❛s ♣♦rXp =

n

X

i=1

Xp(xi)

∂xi|p✳ ▲♦❣♦✱ ♦❜t❡♠♦s q✉❡

X =

n

X

i=1

X(xi)

∂ ∂xi✳

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

❡♠ ▼ ❡ ◆✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞✐③❡♠♦s q✉❡ ❳ ❡ ❨ sã♦ ❋✲r❡❧❛❝✐♦♥❛❞♦s s❡ Fp(Xp) =YF(p)✱

♣❛r❛ t♦❞♦pM✳

❉❡❝♦rr❡ ❞❡st❛ ❞❡✜♥✐çã♦ q✉❡ s❡ X ❡Y sã♦ F relacionados ❡♥tã♦

YF(p)(g) = F∗p(Xp)(g) =Xp(g◦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

(23)

✶✶

➱ ❢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❛r❛ ✐ss♦✱ s❡❥❛♠ x1, ..., xn ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s t❛✐s q✉❡ X =

P

iai ∂

∂xi ❡ Y =

P

ibi ∂ ∂xi✳

❊♥tã♦✱

[X, Y] =

n

X

i=1

(X(bi)−Y(ai)).

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✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✺✳ ❙❡ Xi ∈ D(M) ❡ Yi ∈ D(N) sã♦ ❋✲r❡❧❛❝✐♦♥❛❞♦s✱ ♣❛r❛ i ∈ {1,2}✱ ❡♥tã♦ [X1, X2] ❡ [Y1, Y2] sã♦ ❋✲r❡❧❛❝✐♦♥❛❞♦s✳

Pr♦✈❛✿ ❙❡ g :N −→R éC∞✱ ❡♥tã♦✱ ♣♦r ✶✳✶✱

(Yi(g))◦f =Xi(g◦f),∀i∈ {1,2}✳ ▲♦❣♦✱

([Y1, Y2](g))◦f = Y1(Y2(g))◦f −Y2(Y1(g))◦f

= X1(Y2(g)◦f)−X2(Y1(g)◦f)

= X1(X2(g◦f))−X2(X1(g ◦f))

= [X1, X2](g◦f).

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

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

c(t)✱ ♦♥❞❡ t ❞❡♥♦t❛ ♦

♣❛râ♠❡tr♦ ❞❛ ❝✉r✈❛ ❡ c′(t) = c ∗ dtd

✳ ❙❡ ✉♠❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ c t❡♠ ❞♦♠í♥✐♦ R✱ ❞✐③❡♠♦s

q✉❡c é ✉♠❛ tr❛❥❡tór✐❛ ❣❧♦❜❛❧✳

◗✉❡r❡♠♦s s❛❜❡r s♦❜r❡ q✉❛✐s ❝♦♥❞✐çõ❡s ✉♠❛ ❝✉r✈❛ c : I R −→ M é ✉♠❛

tr❛❥❡tór✐❛ ❞❡ ✉♠ ❝❛♠♣♦ X✳ P❛r❛ ✐ss♦ s❡❥❛♠ τ I✱ c(τ) = a ❡ s❡❥❛♠ x1, ..., xn ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ϕU ❝♦♠ a ∈ϕU(U)✳ ❈♦♥s✐❞❡r❡ t❛♠❜é♠ q✉❡ X é r❡♣r❡s❡♥t❛❞♦ ❡♠ ϕU(U)♣♦r X =

n

X

i=1

αi

∂ ∂xi

(24)

✶✷

d

dt|t=τc(t) = n

X

i=1

dci

dt (τ) ∂ ∂xi|a✳

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

s❡❣✉✐♥t❡✿

n

X

i=1

αi(a)

∂ ∂xi a

=Xa=

d

dt|t=τc(t) =

n

X

i=1

dxi

dt (τ) ∂ ∂xi|a✱ ♦✉ s❡❥❛✱

dci

dt(t) =αi(c(t))✱ i∈ {1, ..., n}✳ ■ss♦ ❥✉st✐✜❝❛ ♦ s❡❣✉✐♥t❡

❚❡♦r❡♠❛ ✶✳✷✳✼✳ ❙❡❥❛♠ ▼ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ x1, ..., xn ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ❛ ❝❛rt❛ ϕU : U ⊂ Rn −→ M ❡ X ∈ D(M) ❡①♣r❡ss♦ ❧♦❝❛❧♠❡♥t❡ ♣♦r

n

X

i=1

αi

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

♣❡❧❛s ❢✉♥çõ❡s c1(t), ..., cn(t)✱ s❡❥❛ ✉♠❛ tr❛❥❡tór✐❛ ❧♦❝❛❧ ❞♦ ❝❛♠♣♦ X é q✉❡ ❛s ❢✉♥çõ❡s xi(t) s❛t✐s❢❛çã♠ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s

dci

dt(t) = αi(c(t))✱ i∈ {1, ..., n}✳

❚❡♠♦s✱ ♣♦r t❡♦r❡♠❛s ❞❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ❧♦❝❛❧ ✭❬✺✷❪✱❬✹✼❪✱❬✺✵❪✱❬✷✻❪✮✱ q✉❡✱ ♣❛r❛ t♦❞♦ aM✱ ❡①✐st❡ ✉♠❛ tr❛❥❡tór✐❛ ❧♦❝❛❧ ♣❛ss❛♥❞♦ ♣♦r a✳

❉❡✜♥✐çã♦ ✶✳✷✳✽✳ ❙❡ t♦❞❛ tr❛❥❡tór✐❛ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡sX 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✐❛sci :Ii −→M✱i∈ {1,2}✱ ♣❛ss❛♥❞♦ ♣♦r ✉♠ ♣♦♥t♦a∈M ❡τi ∈c−i 1(a)✱i∈ {1,2}✱ ❡♥tã♦c1(t) =c2(t+τ2−τ1)✱ ♣❛r❛ t♦❞♦ts✉✜❝✐❡♥t❡♠❡♥t❡ ♣r♦ó①✐♠♦ ❞❡τ1 ✭❬✺✷❪✱❬✹✼❪✱❬✺✵❪✮✳

❉❡✜♥✐çã♦ ✶✳✷✳✾✳ ❙❡❥❛♠ X ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s C∞ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M I

x0 ⊂R

♦ ✐♥t❡r✈❛❧♦ ♠á①✐♠♦ ♣❛r❛ ♦ q✉❛❧ ✉♠❛ tr❛❥❡tór✐❛ ♣❛ss❛♥❞♦ ♣♦r x ∈ M ❡stá ❞❡✜♥✐❞❛ ❡ D={(t, y)∈R×M;t∈Iy e y ∈M}✳ ❖ ✢✉①♦ ❞♦ ❝❛♠♣♦ X é ❛ ❛♣❧✐❝❛çã♦

A:D−→M A(t, y) = c(c−1(y) +t)

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✶✸

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

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

s(t) = c(t+c−1(y)s−1(y))

▲♦❣♦✱

s(s−1(y) +t) = c(s−1(y) +t+c−1(y)−s−1(y)) =c(t+c−1(y))✳

❋✐①❛❞♦s ✉♠ ♣♦♥t♦ x ∈ M ❡ t ∈ Ix t❛✐s q✉❡ A(−t, x) ❡ A(t, x) ❡stã♦ ❞❡✜♥✐❞♦s✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ⊂ M ❞♦ ♣♦♥t♦ x t❛❧ q✉❡ ❛ ❢❛♠í❧✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s {As : U −→

M;As(y) = A(s, y)}|s|<|t| ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ✭✈✳ ❬✺✷❪✱❬✹✼❪✱❬✺✵❪✮✳ ❚❛❧ ❢❛♠í❧✐❛ é ❞❡♥♦♠✐♥❛❞❛

❣r✉♣♦ ❧♦❝❛❧ ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ▼✱ ♣♦✐s

As1 ◦As2 =As1+s2✱ ♣❛r❛ t♦❞♦ s1, s2 t❛✐s q✉❡ |s1|,|s2|,|s1+s2|<|t|✳

❙❡ ♦ ❝❛♠♣♦ ❢♦r ❝♦♠♣❧❡t♦ ❡♥tã♦ ❛ ❢❛♠í❧✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s{As :M −→M}sR ❝♦♥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 ❞❡xt❛❧ q✉❡ ❛ ❛♣❧✐❝❛çã♦ At : U −→ M✱At(y) = A(t, y)✱ é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧✳ ❚❛❧ ❛♣❧✐❝❛çã♦ é ❞❡♥♦♠✐♥❛❞❛ ♦♣❡r❛❞♦r ❞❡ tr❛♥s❧❛çã♦ ❧♦❝❛❧ ❛♦ ❧♦♥❣♦ ❞❛s tr❛❥❡tór✐❛s ❞❡X✳ ❉❡ss❛ ❢♦r♠❛✱ ♥ã♦ é ❞✐❢í❝✐❧

✈❡r q✉❡ d(A∗t)

dt |t=0 =X ✭✈✳ ❬✺✷❪✮✳

P♦rt❛♥t♦✱ ❞❛❞♦ ✉♠ ❝❛♠♣♦ X✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ❢❛♠í❧✐❛ ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡

❞✐❢❡♦♠♦r✜s♠♦s ❧♦❝❛✐s ❞❡M✳ ❖ ❝❛♠✐♥❤♦ ❝♦♥trár✐♦ é ♣♦ssí✈❡❧ ♣❡❧♦ s❡❣✉✐♥t❡

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

❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❞❡✜♥✐❞♦ ♣❡❧❛ ❢ór♠✉❧❛

X= lim

t0

A∗

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

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

d dtA

t✳

Pr♦✈❛✿ ❙❛❜❡♠♦s q✉❡

d dtA

t = lims

→0

A∗

t+s−A∗t

s =A

t ◦

lim

s0

A∗

s−A∗0

s

=A∗

Figure

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