Universidade Federal do Espírito Santo Programa de Pós-Graduação em Matemática

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

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

❙♦❧✉çõ❡s ❞❡ ❱órt✐❝❡ ❞❛s ❊q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉

❖❧❡s②❛ ●❛❧❦✐♥❛

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

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

❖❧❡s②❛ ●❛❧❦✐♥❛

❙♦❧✉çõ❡s ❞❡ ❱órt✐❝❡ ❞❛s ❊q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉

❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ♣❡❧♦ Pr♦✲ ❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦✳

❖r✐❡♥t❛❞♦r✿ ▼❛❣♥♦ ❇r❛♥❝♦ ❆❧✈❡s

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❖❧❡s②❛ ●❛❧❦✐♥❛

❙♦❧✉çõ❡s ❞❡ ❱órt✐❝❡ ❞❛s ❊q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉

❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ♣❡❧♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦✳

❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆

Pr♦❢✳ ❉r✳ ▼❛❣♥♦ ❇r❛♥❝♦ ❆❧✈❡s ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ▼❛❣❛❧❤ã❡s ▼❛❝❛r✐♥✐ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ❞❡ ❏❛♥❡✐r♦

Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ▼❡✐r❡❧❡s ❈â♠❛r❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

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❘❡s✉♠♦

◆❡st❛ ❞✐ss❡rt❛çã♦ ❡st✉❞❛♠♦s ✉♠ t❡♦r❡♠❛ ❞❡ ❈✳❍✳ ❚❛✉❜❡s s♦❜r❡ s♦❧✉çõ❡s ❞❡ ✈órt✐❝❡ ❞❛s ❡q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉✱ q✉❡ ❞❡s❝r❡✈❡♠ ❛ s✉♣❡r❝♦♥❞✉t✐✈✐✲ ❞❛❞❡✳ P❛r❛ ♣r♦✈❛r ♦ t❡♦r❡♠❛✱ ♣r❡❝✐s❛♠♦s ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❛ s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❡❧í♣t✐❝❛ ♥ã♦✲❧✐♥❡❛r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ P❛r❛ ♦❜t❡r ❛ ❡①✐stê♥❝✐❛ ❞❛ s♦❧✉çã♦✱ ❡st✉❞❛♠♦s ✉♠ ❢✉♥❝✐♦♥❛❧ ♥ã♦✲❧✐♥❡❛r ❞❡✜♥✐❞♦ ♥✉♠ ❝❡rt♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈✱ ❡ ❞❡t❛❧❤❛♠♦s ❛s ❝♦♥t❛s ❞♦ ❛rt✐❣♦ ❞❡ ❚❛✉❜❡s✳ ❚❛♠❜é♠ ✐♥❝❧✉í♠♦s ❞♦✐s ❝❛♣ít✉❧♦s ❛✉①✐❧✐❛r❡s s♦❜r❡ ✜❜r❛❞♦s ❡♠ r❡t❛s ❝♦♠♣❧❡✲ ①♦s ❡ ♣r❡❧✐♠✐♥❛r❡s ❛♥❛❧ít✐❝♦s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡✱ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛s ❡❧í♣t✐❝❛s✱ ❡s♣❛✲ ç♦s ✜❜r❛❞♦s✱ ❡q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② ❛ t❤❡♦r❡♠ ♦❢ ❈✳❍✳ ❚❛✉❜❡s ❝♦♥❝❡r♥✐♥❣ ✈♦rt❡① s♦❧✉t✐♦♥ t♦ t❤❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❡q✉❛t✐♦♥s✱ ✇❤✐❝❤ ❞❡s❝r✐❜❡ s✉♣❡r❝♦♥❞✉❝t✐✈✐t②✳ ❚♦ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ s♦❧✉t✐♦♥ t♦ ❛ ♥♦♥✲❧✐♥❡❛r ❡❧✲ ❧✐♣t✐❝ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r✳ ❚♦ ♦❜t❛✐♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥ ✇❡ st✉❞② ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧ ❞❡✜♥❡❞ ♦♥ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❙♦❜♦❧❡✈ s♣❛❝❡✳ ❲❡ ❛❧s♦ ✐♥❝❧✉❞❡ t✇♦ ❛✉①✐❧✐❛r② ❝❤❛♣t❡rs ❝♦♥❝❡r♥✐♥❣ ❝♦♠♣❧❡① ❧✐♥❡ ❜✉♥✲ ❞❧❡s ❛♥❞ ❛♥❛❧②t✐❝❛❧ ♣r❡❧✐♠✐♥❛r✐❡s✳

❑❡②✲✇♦r❞s✿ s✉♣❡r❝♦♥❞✉❝t✐✈✐t②✱ ❡❧❧✐♣t✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❜✉♥❞❧❡ s♣❛❝❡s✱ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❡q✉❛t✐♦♥s✳

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❙✉♠ár✐♦

✶ ▲✐♥❡ ❇✉♥❞❧❡s ✼

✶✳✶ ▲✐♥❡ ❇✉♥❞❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❈♦♥❡①õ❡s ❡ ❈✉r✈❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸ ❈❧❛ss❡s ❞❡ ❈❤❡r♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷ Pr❡❧✐♠✐♥❛r❡s ❆♥❛❧ít✐❝❛s ✷✻

✷✳✶ ▼❡❞✐❞❛ ❡ ■♥t❡❣r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✸ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✸✺

✸✳✶ ❱♦rt❡① ◆✉♠❜❡r ❡ ❋ór♠✉❧❛ ❞❡ ❇♦❣♦♠♦❧✬♥②✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❢✉♥❝✐♦♥❛❧ G ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✸✳✹ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

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❈❛♣ít✉❧♦ ✶

▲✐♥❡ ❇✉♥❞❧❡s

✶✳✶ ▲✐♥❡ ❇✉♥❞❧❡s

❯♠ ❧✐♥❡ ❜✉♥❞❧❡ é ✉♠❛ tr✐♣❧❛ (L, M, π)❢♦r♠❛❞❛ ♣♦r ✈❛r✐❡❞❛❞❡sL❡ M ❡ ✉♠❛

♣r♦❥❡çã♦ s✉❛✈❡ π :LM t❛❧ q✉❡

✶✳ ❈❛❞❛ ✜❜r❛ π−1(m) é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠✲

♣❧❡①❛ ✶❀

✷✳ ❊①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛{Uα} ❞❡M ❡ ❞✐❢❡♦♠♦r✜s♠♦s

φα :π−1(Uα)×C

t❛✐s q✉❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦ m ❡♠ Uα t❡♠♦s q✉❡ φα(π−1(m))⊂ {m} ×C

❡ ❛ r❡str✐çã♦

φα |π−1

(m):π−1(m)→ {m} ×C

é ✉♠ ✐s♦♠♦r✜s♠♦ C✲❧✐♥❡❛r✳

❆ ❝♦❜❡rt✉r❛ {Uα}é ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛✳

❊①❡♠♣❧♦ ✶✳✶✳✶ ✭▲✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧✮✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✳ ❖ ❧✐♥❡

❜✉♥❞❧❡ tr✐✈✐❛❧ é ♦ ♣r♦❞✉t♦ L=M ×C ❝♦♠ ❛ ♣r♦❥❡çã♦π :LM ❞❛❞❛ ♣♦r

π(m, z) =m.

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❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✽ ❱❛♠♦s ♠♦str❛r q✉❡ π :LM é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳

✶✳ ❚❡♠♦s q✉❡π−1(m) = {mC✳ P♦❞❡♠♦s ✈❡r q✉❡π−1(m)é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶ ❝♦♠ ❛s ♦♣❡r❛çõ❡s

(m, z) + (m, w) = (m, z+w),

α(m, z) = (m, αz),

♦♥❞❡ z✱ w ❡α sã♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳

✷✳ ❚❡♠♦s q✉❡{M}é ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡M✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❛♣❧✐❝❛çã♦

✐❞❡♥t✐❞❛❞❡

Id :LM ×C

s❛t✐s❢❛③ Id(π−1(m)) = {m} ×C✱ ❡ ♣♦rt❛♥t♦

Id|π−1

(m):π−1(m)→ {m} ×C

é ✉♠ ✐s♦♠♦r✜s♠♦ C✲❧✐♥❡❛r✳

❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π : L M✳ ❯♠❛ ❛♣❧✐❝❛çã♦ s : U M L

é ✉♠❛ s❡çã♦ ❧♦❝❛❧ s❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦ m ❡♠ U✱ t❡♠♦s q✉❡ s(m) ❡stá ❡♠ π−1(m)✱ ♦✉ s❡❥❛✱

πs=IdU.

◗✉❛♥❞♦ U =M✱ ❞✐③❡♠♦s q✉❡ s é ✉♠❛ s❡çã♦ ❣❧♦❜❛❧✳

❊①❡♠♣❧♦ ✶✳✶✳✷ ✭❙❡çõ❡s ❞♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧✮✳ ❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧ L = M ×C✳ P❛r❛ t♦❞❛ ❢✉♥çã♦ f : M C✱ t❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦

s :M L❞❛❞❛ ♣♦rs(m) = (m, f(m))é ✉♠❛ s❡çã♦ ❣❧♦❜❛❧✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱

♣❛r❛ t♦❞❛ s❡çã♦ ❣❧♦❜❛❧s:M L✱ t❡♠♦s q✉❡s(m)❡stá ❡♠{mC✱ ♣♦rt❛♥t♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ f :M C t❛❧ q✉❡ s(m) = (m, f(m))

Pr♦♣♦s✐çã♦ ✶✳✶✳✸✳ ❈♦♥s✐❞❡r❡ ✈❛r✐❡❞❛❞❡s L ❡ M ❡ ✉♠❛ ♣r♦❥❡çã♦ π : L M✱ s✉♣♦♥❤❛ q✉❡ ✈❛❧❡ ❛ ❝♦♥❞✐çã♦ ✶ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♥❡ ❜✉♥❞❧❡✳ ❚❡♠♦s q✉❡ π : L M é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ s❡ ❡ s♦♠❡♥t❡ s❡ ❡①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛

(9)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✾ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ π : L M é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳ ❚♦♠❡ ✉♠❛

❝♦❜❡rt✉r❛ ❛❜❡rt❛ {Uα} ❞❡M ❡ ❞✐❢❡♦♠♦r✜s♠♦s

φα :π−1(Uα)→Uα×C

t❛✐s q✉❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦ m ❡♠ Uα t❡♠♦s q✉❡ φα(π−1(m))⊂ {m} ×C ❡ ❛

r❡str✐çã♦

φα|π−1(m):π−1(m)→ {m} ×C

é ✉♠ ✐s♦♠♦r✜s♠♦C✲❧✐♥❡❛r✳ P❛r❛ ❝❛❞❛α✱ ❞❡✜♥✐♠♦s ❛ s❡çã♦ ❧♦❝❛❧sα :Uα L ♣♦r

sα(m) =φ−α1(m,1).

❚❡♠♦s q✉❡ sα(m)6= 0 ♣❛r❛ t♦❞♦m ❡♠ Uα✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛{Uα} ❞❡M

s❡çõ❡s ❧♦❝❛✐s sα :Uα L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ ❉❡✜♥✐♠♦s ❛ ❛♣❧✐❝❛çã♦

φα :π−1(Uα)×C ♣♦r

φα|π−1

(m)(z) =

m, z

sα(m)

✭✶✳✶✮ ♣❛r❛ ❝❛❞❛ π−1(m) ❡♠ π−1(Uα)✳ ❆q✉✐ ✉s❛♠♦s q✉❡ π−1(m) é ✉♠ ❡s♣❛ç♦

✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶ ❡ sα ♥ã♦ s❡ ❛♥✉❧❛✳ ❚❡♠♦s q✉❡ ❛

✐♥✈❡rs❛ é

φ−α1 :Uα×C→π−1(Uα)

❞❛❞❛ ♣♦r

φ−α1(m, λ) = λsα(m).

(10)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✵

m ❡♠ Uα✱ ♣♦rt❛♥t♦

φα−1φα(z) = φα−1◦φα|π−1(m)(z)

=φ−α1

m, z

sα(m)

= z

sα(m) sα(m)

=z;

✭✷✮ ❉❛❞♦ ✉♠(m, λ)❡♠Uα×C✱ t❡♠♦s q✉❡λsα(m)❡stá ❡♠π−1(m)✱ ♣♦rt❛♥t♦

φα◦φ−α1(m, λ) =φα(λsα(m)) =φα|π−1(m)(λsα(m))

=

m,λsα(m) sα(m)

= (m, λ).

❆ ❊q✉❛çã♦ ✭✶✳✶✮ ♠♦str❛ q✉❡ φα(π−1(m))⊂ {m} ×C ❡

φα|π−1(m)(z+w) = φα|π1(m)(z) +φα|π1(m)(w),

φα|π−1

(m)(zw) =φα|π−1

(m)(z)φα|π−1

(m)(w),

♣♦rt❛♥t♦

φα|π−1(m):π−1(m)→ {m} ×C é ✉♠ ✐s♦♠♦r✜s♠♦ C✲❧✐♥❡❛r✳

❊①❡♠♣❧♦ ✶✳✶✳✹ ✭❋✐❜r❛❞♦ t❛♥❣❡♥t❡ ❞❛ ❡s❢❡r❛✮✳ ▲❡♠❜r❡ q✉❡ ❛ ❡s❢❡r❛ S2 é ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s (x, y, z) ❡♠ R3 t❛✐s q✉❡

x2+y2+z2 = 1.

P♦❞❡♠♦s ✈❡r q✉❡ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ p❡♠ S2 é ❞❛❞♦ ♣♦r

(11)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✶ ▲❡♠❜r❡ q✉❡ ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ à ❡s❢❡r❛ S2 é ❞❡✜♥✐❞♦ ❝♦♠♦

TS2 = [

p∈S2

{p} ×TpS2

.

❈♦♥s✐❞❡r❡ ❛ ♣r♦❥❡çã♦ π:TS2 S2 ❞❛❞❛ ♣♦r

π(p, v) = p.

❱❛♠♦s ♠♦str❛r q✉❡ π : TS2 S2 é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳ ❚❡♠♦s q✉❡ π−1(p) =

TpS2 é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶ ❝♦♠ ❛s ♦♣❡r❛✲ çõ❡s

(p, v) + (p, w) = (p, v+w),

(α+iβ) (p, v) = (p, αv+βnp×v),

♦♥❞❡ np é ♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ ♥♦ ♣♦♥t♦p✳ ❖❜s❡r✈❡ q✉❡iv é ✉♠❛ r♦t❛çã♦

❞❡ ✉♠ â♥❣✉❧♦ π

2 ❞❛❞❛ ♣♦r

iv =np×v.

❱❛♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♠♦str❛r q✉❡

[(α+iβ) (γ+iδ)]v = (α+iβ) [(γ+iδ)v].

❚❡♠♦s q✉❡

[(α+iβ) (γ+iδ)]v = [αγ −βδ+i(αδ+βγ)]v

= (αγβδ)v+ (αδ+βγ)np×v.

◆❛ ❡s❢❡r❛ S2✱ t❡♠♦s q✉❡

i(iv) =i(np×v)

=np×(np×v)

(12)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✷ ♣♦rt❛♥t♦

(α+iβ) [(γ+iδ)v] = (α+iβ) (γv+δnp×v)

=α(γv+δnp×v) +βnp×(γv+δnp ×v)

= (aγβδ)v+ (aδ+βγ)np×v,

❧♦❣♦

[(α+iβ) (γ+iδ)]v = (α+iβ) [(γ+iδ)v].

➱ ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r q✉❡ ❛s ❞❡♠❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ✈❛❧❡♠ ❡♠ π−1(p)✳

➱ ❢á❝✐❧ ✈❡r q✉❡ ❛s s❡çõ❡s ❞❡ TS2 S2 sã♦ ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ♥❛ ❡s❢❡r❛ S2✳ P♦❞❡♠♦s ♦❜t❡r ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ♥❛ ❡s❢❡r❛ q✉❡ ❧♦❝❛❧♠❡♥t❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ ■st♦ ♣♦❞❡ ❢❡✐t♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✉s❛♥❞♦ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✸✱ ❝♦♥❝❧✉í♠♦s q✉❡ TS2 é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳

❊①❡♠♣❧♦ ✶✳✶✳✺ ✭❋✐❜r❛❞♦ ❞❡ ❍♦♣❢✮✳ ❆ r❡t❛ ♣r♦❥❡t✐✈❛ ❝♦♠♣❧❡①❛ CP1 é ❞❡✜✲

♥✐❞❛ ❝♦♠♦ ♦ ❡s♣❛ç♦ C2 \ {(0,0)} ❝♦♠ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ✱ ❞❡✜♥✐❞❛

♣♦r (z1, z2)∼(w1, w2) s❡ ❡①✐st❡ λ ❡♠ C∗ t❛❧ q✉❡

w1 =λz1,

w2 =λz2.

❆ ❝❧❛ss❡ ❞❡ (z1, z2) é ❞❡♥♦t❛❞❛ ♣♦r [z1, z2]✳ ❱❛♠♦s ♠♦str❛r q✉❡ CP1 é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❧❡①❛ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶✳ ❉❡✜♥✐♠♦s ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s

U1 =

[z1, z2]CP1 :z1 6= 0 ,

U2 =

[z1, z2]CP1 :z2 6= 0 ,

❡ ❞❡✜♥✐♠♦s ❞♦✐s ❞✐❢❡♦♠♦r✜s♠♦s ψ1 :U1 C ❡ ψ2 :U2 C ❞❛❞♦s ♣♦r

ψ1([z1, z2]) = z2 z1 ,

(13)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✸ ❘❡♣❛r❡ q✉❡

ψ1(U1∩U2) = C∗, ψ2(U1U2) = C∗,

ψ2 ◦ψ1−1 :C∗ →C∗ z 7→ 1

z.

■st♦ ♠♦str❛ q✉❡ CP1 é ✈❛r✐❡❞❛❞❡ ❝♦♠♣❧❡①❛ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶✳ ❖ ✜❜r❛❞♦ ❞❡ ❍♦♣❢ H é ❞❡✜♥✐❞♦ ♣♦r

H =

(z,[z])C2×CP1 :z C2\ {(0,0)} .

P♦❞❡♠♦s ✈❡r q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❍♦♣❢Hé ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❧❡①❛ ❞❡ ❞✐♠❡♥sã♦

❝♦♠♣❧❡①❛ ✷✳ ❉❡✜♥✐♠♦s ❛ ♣r♦❥❡çã♦ π : H CP1 ❞❛❞❛ ♣♦r π(z,[z]) = [z]✳ ❱❛♠♦s ♠♦str❛r q✉❡ π : H CP1 é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳ ❚❡♠♦s q✉❡ ❛ ✜❜r❛

π−1([z])é ❞❛❞❛ ♣♦r

π−1([z]) ={(λz,[z]) :λC∗}.

P♦❞❡♠♦s ✈❡r q✉❡ π−1([z])é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠✲

♣❧❡①❛ ✶ ❝♦♠ ❛s ♦♣❡r❛çõ❡s

(λz,[z]) + (µz,[z]) = ((λ+µ)z,[z]),

λ(µz,[z]) = ((λµ)z,[z]).

❉❡✜♥✐♠♦s ❛s s❡çõ❡s ❧♦❝❛✐s s1 :U1 →H ❡ s2 :U2 →H ❞❛❞❛s ♣♦r

s1([z1, z2]) =

1,z2 z1

,[z1, z2]

,

s2([z1, z2]) =

z1 z2,1

,[z1, z2]

(14)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✹ ❚❡♠♦s q✉❡ ❛s s❡çõ❡s ❧♦❝❛✐s s1 ❡ s2 ♥ã♦ s❡ ❛♥✉❧❛♠✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✸✱

❝♦♥❝❧✉í♠♦s q✉❡ H é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ s♦❜r❡CP1✳

❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π : L M ❝♦♠ ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛

{Uα}❞❡M✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✸✱ ♣♦❞❡♠♦s t♦♠❛r s❡çõ❡s ❧♦❝❛✐ssα :Uα L

q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ ❊♠ UαUβ ❡s❝r❡✈❛

sα =gαβsβ.

❆s ❢✉♥çõ❡s gαβ :UαC∗ sã♦ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡

❊①❡♠♣❧♦ ✶✳✶✳✻ ✭❋✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ✜❜r❛❞♦ ❞❡ ❍♦♣❢✮✳ ▲❡♠❜r❡ q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❍♦♣❢ é ❞❛❞♦ ♣♦r

H =

(z,[z]) :z C2\ {(0,0)} ,

❝♦♠ ❛ ♣r♦❥❡çã♦ π:H CP1 ❞❛❞❛ ♣♦r

π(z,[z]) = [z],

♦♥❞❡ [z] ={λz :λC∗}✳ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s

U1 =

[z1, z2]CP1 :z1 6= 0 ,

U2 =

[z1, z2]CP1 :z2 6= 0 ,

❡ ❛s s❡çõ❡s s1 :U1 H ❡s2 :U2 H ❞❛❞❛s ♣♦r

s1([z1, z2]) =

1,z2 z1

,[z1, z2]

,

s2([z1, z2]) =

z1 z2

,1

,[z1, z2]

.

(15)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✺

s1([z1, z2]) =

z2 z1

z1 z2,1

,[z1, z2]

= z2 z1

z1 z2,1

,[z1, z2]

= z2 z1

s2([z1, z2]).

❖❜t❡♠♦s ❛ ❢✉♥çã♦ ❞❡ tr❛♥s✐çã♦ g12 :U1 ∩U2 →C∗ ❞❛❞❛ ♣♦r

g12([z1, z2]) = z2 z1 .

Pr♦♣♦s✐çã♦ ✶✳✶✳✼✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π:LM ❝♦♠ ✉♠❛ ❝♦❜❡rt✉r❛

tr✐✈✐❛❧✐③❛❞♦r❛ {Uα} ❞❡ M ❡ s❡çõ❡s ❧♦❝❛✐s sα : Uα L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳

❈♦♥s✐❞❡r❡ ✉♠❛ s❡çã♦ ❣❧♦❜❛❧ ξ :M L ❡ ❡s❝r❡✈❛

ξ|Uα =ξαsα,

ξ |Uβ =ξβsβ

❡♠ Uα∩Uβ✳ ❚❡♠♦s q✉❡

ξβ =gαβξα,

♦♥❞❡ gαβ sã♦ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡ sα✳

❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

ξβsβ =ξ |Uα∩Uβ =ξαsα

=ξαgαβsβ.

❉✐✈✐❞✐♥❞♦ ♦s ❞♦✐s ❧❛❞♦s ♣♦r sβ✱ ♦❜t❡♠♦s

(16)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✻ Pr♦♣♦s✐çã♦ ✶✳✶✳✽✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π:LM ❝♦♠ ✉♠❛ ❝♦❜❡rt✉r❛

tr✐✈✐❛❧✐③❛❞♦r❛ {Uα} ❞❡ M ❡ s❡çõ❡s ❧♦❝❛✐s sα : Uα → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳

❉❡♥♦t❡ ♣♦r gαβ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡ sα✳ ❚❡♠♦s q✉❡

✶✳ gαα = 1 ❡♠ Uα❀

✷✳ gαβgβα= 1 ❡♠ UαUβ ✭s❡ ♥ã♦ ✈❛③✐♦✮❀

✸✳ gαβgβγgγα = 1 ❡♠ Uα∩Uβ ∩Uγ ✭s❡ ♥ã♦ ✈❛③✐♦✮✳

❉❡♠♦♥str❛çã♦✳ ✶✳ ❚❡♠♦s q✉❡ sα = gααsα. P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡ sα = 1·sα. P♦rt❛♥t♦gαα = 1.

✷✳ ❚❡♠♦s q✉❡

sα=gαβsβ

=gαβgβαsα.

P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡ sα = 1·sα. P♦rt❛♥t♦gαβgβα= 1.

✸✳ ❚❡♠♦s q✉❡

sα =gαβsβ

=gαβgβγsγ

=gαβgβγgγαsα.

P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡sα = 1·sα. P♦rt❛♥t♦gαβgβγgγα = 1.

❖❜s❡r✈❛çã♦ ✶✳✶✳✾✳ ➱ ❝♦♥❤❡❝✐❞♦ q✉❡ ❞❛❞❛ ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛{Uα}❞❡ ✉♠❛

✈❛r✐❡❞❛❞❡M✱ s❡ ❡①✐st❡♠ ❢✉♥çõ❡sgαβ :UαC∗s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ✶✱ ✷ ❡ ✸ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✽✱ ❡♥tã♦ ❡①✐st❡✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✱ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ LM ❡ s❡çõ❡s ❧♦❝❛✐s sα :Uα Lq✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠ t❛✐s q✉❡

sα =gαβsβ

(17)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✼

✶✳✷ ❈♦♥❡①õ❡s ❡ ❈✉r✈❛t✉r❛

❯♠❛ ❝♦♥❡①ã♦ ∇♥✉♠ ❧✐♥❡ ❜✉♥❞❧❡LM é ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r: Γ (L) Γ (T∗M L)t❛❧ q✉❡ ♣❛r❛ t♦❞❛ s❡çã♦ s:M L ❡ ❢✉♥çã♦f :M C ✈❛❧❡ ❛

r❡❣r❛ ❞❡ ▲❡✐❜♥✐③

∇(f s) = dfs+fs.

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠❛ ❝♦♥❡①ã♦ ∇ ♥✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L M é ✉♠❛

❛♣❧✐❝❛çã♦ q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ s❡çã♦ s : M L ❡ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❡♠ M

❝♦♠ ✉♠❛ s❡çã♦ ∇Xs :M →L✱ ❡ t❛❧ q✉❡

✶✳ P❛r❛ ✉♠❛ s❡çã♦s :M L❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s X1 ❡X2 ❡♠ M✱ t❡♠♦s

q✉❡

∇X1+X2s =∇X1s+∇X2s;

✷✳ P❛r❛ ✉♠❛ s❡çã♦ s : M L✱ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❡♠ M ❡ ✉♠❛

❢✉♥çã♦ f :M C✱ t❡♠♦s q✉❡

∇f Xs=f∇Xs;

✸✳ P❛r❛ s❡çõ❡s s1, s2 : M L ❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❡♠ M✱ t❡♠♦s

q✉❡

∇X(s1+s2) = ∇Xs1+∇Xs2;

✹✳ P❛r❛ ✉♠❛ s❡çã♦ s : M L✱ ✉♠❛ ❢✉♥çã♦ f : M C ❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❡♠ M✱ t❡♠♦s q✉❡

∇X(f s) =df(X)s+f∇Xs.

❊①❡♠♣❧♦ ✶✳✷✳✶ ✭❈♦♥❡①ã♦ ❞♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧✮✳ ❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧ L=M ×C✳ ▲❡♠❜r❡ q✉❡ ❛s s❡çõ❡s ❡♠ LM sã♦ ❞❛ ❢♦r♠❛

(18)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✽ ♦♥❞❡ f :M C✳ P♦❞❡♠♦s ❞❡✜♥✐r ❛ ❝♦♥❡①ã♦ ❞❛❞❛ ♣♦r

∇s=df.

▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱

(∇Xs) (m) = (m, dfm(Xm)).

❱❛♠♦s ♠♦str❛r q✉❡ ∇ é ✉♠❛ ❝♦♥❡①ã♦ ❡♠ LM✳ ❚♦♠❡ ❞✉❛s s❡çõ❡s

s1(m) = (m, f(m)),

s2(m) = (m, h(m)),

❡ ✉♠❛ ❢✉♥çã♦ φ:M C✳ ❚❡♠♦s q✉❡

(∇X(s1+s2)) (m) = (m, d(f +h)m(Xm))

= (m, dfm(Xm) +dhm(Xm))

= (m, dfm(Xm)) + (m, dhm(Xm))

= (Xs1) (m) + (∇Xs2) (m).

❆❧❡♠ ❞✐ss♦✱

(X(φs1)) (m) = (m, d(φf)m(Xm))

= (m, φ(m)dfm(Xm) +f(m)dφm(Xm))

=φ(m) (m, dfm(Xm)) +dφm(Xm) (m, f(m))

=dφm(Xm)s1(m) +φ(m) (∇Xs1) (m).

P♦rt❛♥t♦✱

∇(s1+s2) =∇s1+∇s2,

∇(φs1) =dφ⊗s1+φ∇s1.

Pr♦♣♦s✐çã♦ ✶✳✷✳✷✳ ❬✽✱ ♣✳ ✽❪ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π :LM ❝♦♠ ✉♠❛

❝♦♥❡①ã♦ ∇✳ ❈♦♥s✐❞❡r❡ ✉♠❛ s❡çã♦ ❧♦❝❛❧ s : U M L✳ P❛r❛ q✉❛✐sq✉❡r

(19)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✾

∇˜s|U =∇s¯|U.

P♦rt❛♥t♦ ♣♦❞❡♠♦s ❞❡✜♥✐r

∇s =|U.

❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L M ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ✱ ✉♠❛ ❝♦❜❡rt✉r❛

tr✐✈✐❛❧✐③❛❞♦r❛ {Uα} ❡ s❡çõ❡s ❧♦❝❛✐s sα : Uα L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ P❛r❛

❝❛❞❛ α✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ✶✲❢♦r♠❛Aα ❡♠ Uα t❛❧ q✉❡

∇sα =Aαsα.

▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱

(Xsα) (m) = (Aα)m(Xm)sα(m).

❆s ✶✲❢♦r♠❛s Aα sã♦ ❝❤❛♠❛❞❛s ✶✲❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ❞❡ sα✳

❖❜s❡r✈❡ q✉❡ ❛ ❞❡✜♥✐çã♦ é ❥✉st✐✜❝❛❞❛ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✸✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ LM ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦

✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛ {Uα} ❞❡ M ❡ s❡çõ❡s ❧♦❝❛✐ssα :Uα L q✉❡ ♥ã♦

s❡ ❛♥✉❧❛♠✳ ❚❡♠♦s q✉❡

Aα =Aβ +g

−1

αβdgαβ,

♦♥❞❡ gαβ sã♦ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡ sα ❡ Aα sã♦ ❛s ✶✲❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦

❞❡ sα✳

❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

∇sα =∇(gαβsβ)

=dgαβ sβ +gαβ

=dgαβ ⊗sβ +gαβAβ ⊗sβ

(20)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✵ P♦r ♦✉tr♦ ❧❛❞♦ t❡♠♦s q✉❡

∇sα =Aα⊗sα =gαβAαsβ.

P♦rt❛♥t♦

gαβAα =dgαβ +gαβAβ.

▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ❧❛❞♦s ♣♦r g−αβ1✱ ♦❜t❡♠♦s

Aα =Aβ+gαβ−1dgαβ.

Pr♦♣♦s✐çã♦ ✶✳✷✳✹✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ LM ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦

✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛ {Uα} ❡ s❡çõ❡s ❧♦❝❛✐s sα : Uα L q✉❡ ♥ã♦ s❡

❛♥✉❧❛♠✳ ❊①✐st❡ ✉♠❛ ✷✲❢♦r♠❛ F∇ ❡♠ M t❛❧ q✉❡

F∇|Uα =dAα,

♦♥❞❡ Aα sã♦ ❛s ✶✲❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ❞❡ sα✳

❆ ✷✲❢♦r♠❛ F∇ é ❛ ❝✉r✈❛t✉r❛ ❞❛ ❝♦♥❡①ã♦ ∇✳

❉❡♠♦♥str❛çã♦✳ ❇❛st❛ ♠♦str❛r q✉❡ dAα =dAβ ❡♠ UαUβ✳ ❚❡♠♦s q✉❡

dAα =d Aβ +g−αβ1dgαβ

=dAβ −gαβ−2dgαβ ∧dgαβ +gαβ−1d(dgαβ) =dAβ.

❆❝✐♠❛ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡ d2 = 0 ❡ ωω = 0 ♣❛r❛ t♦❞❛ ✶✲❢♦r♠❛ω✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✺✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡LM ❝♦♠ ❝♦♥❡①õ❡s ′✳

❊①✐st❡ ✉♠❛ ✶✲❢♦r♠❛ η ❡♠ M t❛❧ q✉❡

(21)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✶

F∇′ =F+dη.

❉❡♠♦♥str❛çã♦✳ ❚♦♠❡ ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛ {Uα} ❞❡ M ❡ s❡çõ❡s sα : Uα →L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ P♦❞❡♠♦s ❡s❝r❡✈❡r

∇sα =Aα⊗sα,

∇′sα =A′αsα.

❉❡✜♥✐♠♦s ❛ ✶✲❢♦r♠❛ η ❡♠ Uα ♣♦r

ηα =A′αAα.

❱❛♠♦s ♠♦str❛r q✉❡ ❡♠ UαUβ ✈❛❧❡ ηα =ηβ✳ ❉❡♥♦t❡ ♣♦r gαβ ❛s ❢✉♥çõ❡s ❞❡

tr❛♥s✐çã♦ ❞❡ sα✳ ❚❡♠♦s q✉❡

ηα =A′α

=A′β +g−αβ1dgαβAβ +g−αβ1dgαβ =A′β

=ηβ.

❚❡♠♦s q✉❡

∇′sα =A′αs

= (Aα+ηα)⊗sα,

❧♦❣♦

∇′s=s+ηs.

❚❛♠❜é♠ t❡♠♦s q✉❡

F∇′ =dA′α

=dAα+dηα

(22)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✷

Pr♦♣♦s✐çã♦ ✶✳✷✳✻✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ LM ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦

❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠❛ s❡çã♦ ξ:M L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛ ❡ t❛❧ q✉❡

∇ξ = 0.

❚❡♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❛ ❝♦♥❡①ã♦ é ③❡r♦✱ ♦✉ s❡❥❛

F∇= 0.

❉❡♠♦♥str❛çã♦✳ ❚♦♠❡ ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛{Uα}❞❡M ❡ s❡çõ❡s ❧♦❝❛✐s sα:Uα →L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✱ ❡ ❡s❝r❡✈❛

ξ|Uα =ξαsα.

❚❡♠♦s q✉❡

∇ξ|Uα =∇(ξαsα)

=dξα⊗sα+ξα∇sα

=dξα⊗sα+ξαAα⊗sα

= (dξα+ξαAα)sα.

❈♦♠♦ ∇ξ = 0✱ t❡♠♦s q✉❡

ξαAα =dξα,

❧♦❣♦

(23)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✸ P♦rt❛♥t♦

F∇=ddAα = 0.

✶✳✸ ❈❧❛ss❡s ❞❡ ❈❤❡r♥

❚❡♦r❡♠❛ ✶✳✸✳✶✳ ❬✽✱ ♣✳ ✶✷❪ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L Σ s♦❜r❡ ✉♠❛

s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ s❡♠ ❜♦r❞♦ Σ ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ∇✳ ❚❡♠♦s q✉❡

1 2πi

ˆ

Σ F∇

é ✉♠ ✐♥t❡✐r♦ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ ∇✳

❖ ♥ú♠❡r♦c1(L) = 1 2πi

´

ΣF∇ é ❛ ❝❧❛ss❡ ❞❡ ❈❤❡r♥✳

❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧ L = Σ × C s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ Σ✳ ❙❛❜❡♠♦s q✉❡

∇=d

é ✉♠❛ ❝♦♥❡①ã♦ ❡♠ LΣ✳ ❙❛❜❡♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛F∇ é ♥✉❧❛✳ P♦rt❛♥t♦ ❛

❝❧❛ss❡ ❞❡ ❈❤❡r♥ é ❞❛❞❛ ♣♦r

c1(Σ×C) = 1 2πi

ˆ

Σ F∇

= 0.

(24)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✹ ♥❛❞❛s ❡s❢ér✐❝❛s ❞❡ S2 sã♦

    

   

x(θ, φ) = sinφcosθ

y(θ, φ) = sinφsinθ

z(θ, φ) = cosφ

.

❆ ❜❛s❡ ❞♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ é ❢♦r♠❛❞❛ ♣❡❧♦s ✈❡t♦r❡s

∂θ = (−sinθsinφ,cosθsinφ,0),

∂φ = (cosθcosφ,sinθcosφ,−sinφ).

❖ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ n é ❞❛❞♦ ♣♦r

n = ∂ ∂φ ×

∂ ∂θ

∂ ∂φ ×

∂ ∂θ

= (cosθsinφ,sinθsinφ,cosφ).

▲❡♠❜r❡ q✉❡ ❛s s❡çõ❡s ❞♦ ❧✐♥❡ ❜✉♥❞❧❡TS2 S2 sã♦ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❡♠S2

❈♦♥s✐❞❡ ❛ s❡çã♦ s :S2 TS2 ❞❛❞❛ ♣♦r

s= (−sinθ,cosθ,0).

❚❡♠♦s q✉❡

∇R3

s=dθ(−cosθ,sinθ,0).

(25)

❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✺ ♣♦rt❛♥t♦

∇s=∇R3

s<R3s, n > n

=R3

s+dθsinφn

=dθ⊗ −cosθcos2φ,sinθcos2φ,sinφcosφ

=dθcosφ(n×s)

=icosφdθs.

❘❡s✉♠✐♥❞♦✱ t❡♠♦s q✉❡

∇s=icosφdθs.

P♦rt❛♥t♦ ❛ ✶✲❢♦r♠❛ ❞❡ ❝♦♥❡①ã♦ ❞❛ s❡çã♦ s é ❞❛❞❛ ♣♦r

A=icosφdθ,

❡ ❝✉r✈❛t✉r❛ é ❞❛❞❛ ♣♦r

F∇=d(icosφdθ)

=isinφdφdθ.

P♦rt❛♥t♦ ❛ ❝❧❛ss❡ ❞❡ ❈❤❡r♥ é ❞❛❞❛ ♣♦r

c1 TS2

= 1 2πi

ˆ

S2

F∇

= −1 2π

ˆ 2π

0

ˆ π

0

sinφdφ

(26)

❈❛♣ít✉❧♦ ✷

Pr❡❧✐♠✐♥❛r❡s ❆♥❛❧ít✐❝❛s

✷✳✶ ▼❡❞✐❞❛ ❡ ■♥t❡❣r❛çã♦

❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ X✳ ❯♠❛ ❢❛♠✐❧✐❛ A ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X é ✉♠❛ σ✲á❧❣❡❜r❛ s❡✿

✶✳ ∅ ❡ X ❡stã♦ ❡♠ A

✷✳ ❙❡ E ∈ A✱ ❡♥tã♦ ♦ ❝♦♠♣❧❡♠❡♥t❛r X\E ∈ A

✸✳ ❙❡ E1, ..., En, ...∈ A✱ ❡♥tã♦ ❛ ✉♥✐ã♦ Sn

i=1Ei ∈ A✳

❖s ❡❧❡♠❡♥t♦s ❞❡ A sã♦ ❝♦♥❥✉♥t♦s ♠❡♥s✉rá✈❡✐s✳

❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ X ❡ ✉♠❛ ❢❛♠í❧✐❛ S ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X✳ ➱

♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❛ ✐♥t❡rs❡çã♦ ❞❡ t♦❞❛s ❛s σ✲á❧❣❡❜r❛s ❝♦♥t❡♥❞♦ S é ✉♠❛ σ✲á❧❣❡❜r❛✱ ❝❤❛♠❛❞❛σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r S✳ ❆σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧❛ ❢❛♠í❧✐❛

❞♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❡♠ Rn é ❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❬✺✱ ✾❪✳

❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ X ❝♦♠ ✉♠❛ σ✲á❧❣❡❜r❛ A✳ ❯♠❛ ♠❡❞✐❞❛ é ✉♠❛

❢✉♥çã♦

µ:A →[0,],

t❛❧ q✉❡ ❞❛❞♦s ♦s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s E1, E2, ... ∈ A✱ t❡♠♦s q✉❡

µ

[

i=1 Ei

!

=

X

i=1

µ(Ei).

(27)

❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✷✼ ❚❡♦r❡♠❛ ✷✳✶✳✶✳ ❬✺✱ ✾❪ ❊①✐st❡ ú♥✐❝❛ ♠❡❞✐❞❛ µ ♥❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❞❡ Rn t❛❧ q✉❡

µ([a1, b1]×...×[an, bn]) = (bnan)...(b1a1).

❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦X❝♦♠ ✉♠❛σ✲á❧❣❡❜r❛A✳ ❯♠❛ ❢✉♥çã♦f :X R é ♠❡♥s✉rá✈❡❧ s❡ ♣❛r❛ t♦❞♦ a ❡♠ R t❡♠♦s q✉❡

{xX :f(x)> a} ∈ A.

❯♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ é ✉♠❛ tr✐♣❧❛ (X,A, µ) ❢♦r♠❛❞❛ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ X ❝♦♠ ✉♠❛ σ✲á❧❣❡❜r❛ A ❡ ✉♠❛ ♠❡❞✐❞❛ µ✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✷✳ ❬✺✱ ✾❪ ❚❡♠♦s q✉❡

✶✳ ❙❡ f ❡ g sã♦ ❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s✱ ❡♥tã♦ f +g ❡ f g sã♦ ❢✉♥çõ❡s

♠❡♥s✉rá✈❡✐s✳

✷✳ ❙❡ f ❡g sã♦ ❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s✱ ❡♥tã♦ max{f, g}❡ min{f, g}sã♦

♠❡♥s✉rá✈❡✐s✳

✸✳ ❙❡ ✉♠❛ s❡q✉❡♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s fn ❝♦♥✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡

♣❛r❛ ✉♠❛ ❢✉♥çã♦ f✱ ❡♥tã♦ f é ♠❡♥s✉rá✈❡❧✳

❖❜s❡r✈❛çã♦ ✷✳✶✳✸✳ ❚♦❞❛ ❢✉♥çã♦ ❝♦♥t✐♥✉❛ f : Rn R é ♠❡♥s✉rá✈❡❧ ✭❇♦r❡❧✮✳ ❆ r❡❝í♣r♦❝❛ é ❢❛❧s❛ ❬✺❪✳

❈♦♥s✐❞❡r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ (X,A, µ). ❯♠❛ ❢✉♥çã♦ s : X R é s✐♠♣❧❡s s❡ ❡❧❛ ❛ss✉♠❡ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈❛❧♦r❡s {a1, ..., an}✳ ❆ ✐♥t❡❣r❛❧ ❞❡s é

❞❡✜♥✐❞❛ ♣♦r

ˆ

X

sdµ= n

X

i=1 aiµ

s−1(ai) ,

♦♥❞❡ s−1(ai) ={xX :s(x) =a i}.

❈♦♥s✐❞❡r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ (X,A, µ) ❡ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ f : X [0,)✳ ❆ ✐♥t❡❣r❛❧ ❞❡ f é ❞❡✜♥✐❞❛ ♣♦r

ˆ

X

f dµ := sup

ˆ

X

sdµ: ❢✉♥çõ❡s s✐♠♣❧❡s sf

(28)

❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✷✽ ❈♦♥s✐❞❡r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ (X,A, µ) ❡ ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ f : X R✳ P♦❞❡♠♦s ❡s❝r❡✈❡r

f =f+f−,

♦♥❞❡ f+ = max{f,0} ❡ f− = max{−f,0}✳ ❆ ❢✉♥çã♦ f é ✐♥t❡❣rá✈❡❧ s❡

´

Xf

+dµ < ´

Xf

dµ <✳ ❆ ✐♥t❡❣r❛❧ ❞❡ f é ❞❡✜♥✐❞❛ ♣♦r

ˆ

X

f dµ=

ˆ

X

f+dµ

ˆ

X

f−dµ.

❖❜s❡r✈❡ q✉❡ ✈❛❧❡ ´

X f

+dµ < ´

Xf

dµ < s❡✱ ❡ s♦♠❡♥t❡ s❡✱

´

X|f|dµ < ∞✳

❚❡♦r❡♠❛ ✷✳✶✳✹ ✭❚❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♠♦♥ót♦♥❛✮✳ ❬✺✱ ✾❪ ❈♦♥s✐❞❡r❡ ✉♠❛ s❡q✉❡♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s fn✳ ❙✉♣♦♥❤❛ q✉❡

0≤f1(x)≤f2(x)≤...

♣❛r❛ t♦❞♦ x✳ ❆ss✉♠❛ q✉❡ fn ❝♦♥✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f✳

❊♥tã♦ ˆ

f = lim n

ˆ

fn.

❚❡♦r❡♠❛ ✷✳✶✳✺ ✭❚❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦♠✐♥❛❞❛✮✳ ❬✺✱ ✾❪ ❈♦♥s✐❞❡r❡ ✉♠❛ s❡q✉❡♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐sfn✳ ❙✉♣♦♥❤❛ q✉❡fn❝♦♥✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡

♣❛r❛ ✉♠❛ ❢✉♥çã♦ f✳ ❆ss✉♠❛ q✉❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ✐♥t❡❣rá✈❡❧ g t❛❧ q✉❡

|fn(x)| ≤g(x)

♣❛r❛ t♦❞♦ n ❡ x✳ ❊♥tã♦

ˆ

f = lim n

ˆ

fn.

✷✳✷ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈

❖ ❡s♣❛ç♦ L2(Rn) é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f ❡♠ Rn t❛✐s q✉❡

ˆ

Rn

(29)

❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✷✾ ❆ ♥♦r♠❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❡♠ L2(Rn) é ❞❡✜♥✐❞❛ ♣♦r

kf k2L2(Rn)=

ˆ

Rn f2.

❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ❞✉❛s ❢✉♥çõ❡sf ❡ g ❡♠ L2(R2)é ❞❡✜♥✐❞♦ ♣♦r

hf, giL2

(Rn) =

ˆ

Rn f g.

❯♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ f ❡♠ Rn é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡ ´

K|f| < ∞

♣❛r❛ t♦❞♦ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ K ❡♠ Rn✳

❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧f ❡♠ Rn✳ ❯♠❛ ❢✉♥çã♦ ❧♦❝❛❧✲ ♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ∂if ❡♠ Rn é ❛ ✐✲❡s✐♠❛ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❞❡ f s❡

ˆ

Rn

f ∂iφ=−

ˆ

Rn φ∂if

♣❛r❛ t♦❞❛ ❢✉♥çã♦ φ ❡♠ Cc∞(Rn)✳ ❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛ é ❥✉st✐✜❝❛❞❛ ♣❡❧❛s s❡✲ ❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

✶✳ ❙❡ ❡①✐st❡♠ ❢✉♥çõ❡s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s g ❡h t❛✐s q✉❡

ˆ

Rn gφ=

ˆ

Rn hφ

♣❛r❛ t♦❞❛ ❢✉♥çã♦φ ❡♠C∞

c (Rn)✱ ❡♥tã♦g =h✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ✐✲és✐♠❛

❞❡r✐✈❛❞❛ ❢r❛❝❛✱ s❡ ❡①✐st✐r✱ é ú♥✐❝❛✳

✷✳ ❙❡ ❛ ❢✉♥çã♦f é s✉❛✈❡✱ ✐♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s✱ ♦❜t❡♠♦s

ˆ

Rn

f ∂iφ=

ˆ

Rn

φ(∂if)

♣❛r❛ t♦❞❛φ❡♠Cc∞(Rn)✳ P❡❧♦ ✐t❡♠ ✶✱ ❞❡❞✉③✐♠♦s q✉❡ ❛ ✐✲és✐♠❛ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❞❡ f é ✐❣✉❛❧ ❛ ✐✲és✐♠❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ✭❝❧áss✐❝❛✮ ❞❡ f✳

(30)

❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✵ ❖ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H1(Rn) é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡✲ ❣rá✈❡✐s f ❡♠ Rn t❛✐s q✉❡ ❛s ❞❡r✐✈❛❞❛s ❢r❛❝❛s 1f, . . . , ∂nf ❡①✐st❡♠✱ ❡ ❛❧❡♠

❞✐ss♦ ˆ

Rn

f2 <,

ˆ

Rn|∇

f|2 <,

♦♥❞❡ ∇f = (∂1f, . . . , ∂nf)✳

❆ ♥♦r♠❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❡♠ H1(Rn) é ❞❡✜♥✐❞❛ ♣♦r

kf k2H1(Rn)=

ˆ

Rn f2+

ˆ

Rn|∇ f|2.

❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ❢✉♥çõ❡sf ❡g ❡♠ H1(Rn) é ❞❡✜♥✐❞♦ ♣♦r

hf, giH1

(Rn) =

ˆ

Rn f g+

ˆ

Rnh∇

f,gi.

❚❡♦r❡♠❛ ✷✳✷✳✶✳ ❬✶✱ ✹✱ ✼❪ ❖ ❡s♣❛ç♦H1(Rn) ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦hf, gi

H1

(Rn)

é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳

▼❛✐s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ H1(Rn) ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✼✱ ❝❛♣ ✼❪ ❡

❬✹✱ ❝❛♣ ✺❪✳

❖ ❡s♣❛ç♦ Lp(Rn)é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐sf ❡♠ Rn t❛✐s q✉❡

ˆ

Rn|

f|p <.

❆ ♥♦r♠❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❡♠ Lp(Rn) é ❞❡✜♥✐❞❛ ♣♦r

kf kpLp(Rn)=

ˆ

Rn| f|p.

❊①✐st❡♠ ✈ár✐♦s t❡♦r❡♠❛s s♦❜r❡ ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈✳ ❱❛♠♦s ✉s❛r ♦ s❡✲ ❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✸✳

(31)

❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✶ ❡♠ H1(R2)✱ t❡♠♦s q✉❡

kf kkLk(R2)≤2

k

2+2kk kf kk

H1

(R2

)

♣❛r❛ t♦❞♦ 2≤k <

❉❡♠♦♥str❛çã♦✳ ❚✐✈❡♠♦s ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❡♥❝♦♥tr❛r ❛ ❝♦♥st❛♥t❡ ❡①♣❧í❝✐t❛2k2+2kk ♥❛ r❡❢❡rê♥❝✐❛ ✉s❛❞❛ ♣❡❧♦ ❛rt✐❣♦ ❞♦ ❚❛✉❜❡s✳ ❊st❛ ❝♦♥st❛♥t❡ é ✐♠♣♦rt❛♥t❡ ♥♦ ❈❛♣ít✉❧♦ ✸✳ ❱❛♠♦s ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ✉s❛♥❞♦ ♦✉tr❛ r❡❢❡rê♥❝✐❛✳ P❡❧♦ ❚❡♦r❡♠❛ ✽✳✺ ✐t❡♠ ✭✐✐✮ ❡♠ ❬✼✱ ♣✳ ✷✵✻❪✱ s❛❜❡♠♦s q✉❡

kf kLk(R2)

1

p

S2,k kf kH

1

(R2

),

♦♥❞❡ S2,k >

n

k1−2k (k−1)−1+

1

k k−2 8π 1 2− 1 k

o−2

✳ ❚❡♠♦s q✉❡

1

p

S2,k

< k1−k2 1 (k1)1−1k

k2 8π

12− 1

k

≤k1−2k 1 k 2

1−1k

k 8π

12− 1

k

≤k12− 2

k21−

1 k 1 16 1 2− 1 k

=k12− 2

k2−1+

3

k

≤k212+ 2

k.

P♦rt❛♥t♦

kf kkLk(R2)≤2

k

2+2kk kf kk

H1

(R2

) .

❊①✐st❡ ✉♠❛ t❡♦r✐❛ q✉❡ ❣❡♥❡r❛❧✐③❛ ❛ t❡♦r✐❛ ❝❧áss✐❝❛ ❞❛s ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈✳ ❖ ❡s♣❛ç♦ LA(Rn)é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f ❡♠ Rn t❛✐s q✉❡

ˆ

Rn

(32)

❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✷ ♦♥❞❡ A é ✉♠❛ ❢✉♥çã♦ s❛t✐s❢❛③❡♥❞♦ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s ❬✶✱ ♣✳ ✷✻✷❪✳ ◆♦ ♥♦ss♦

❝❛s♦ A(t) =et2 1✳ ❱❛♠♦s ✉s❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

❚❡♦r❡♠❛ ✷✳✷✳✸✳ ❬✶✱ ♣✳ ✷✼✼✲✷✽✵❪ P❛r❛ t♦❞❛ ❢✉♥çã♦f ❡♠ H1(R2)✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ρ ✭❞❡♣❡♥❞❡♥❞♦ ❞❡ f✮ t❛❧ q✉❡

ˆ

R2

eρ2f2 1<.

❉❡♠♦♥str❛çã♦✳ ●♦st❛rí❛♠♦s ❞❡ ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ✽✳✷✼ ❡♠ ❬✶✱ ♣✳ ✷✼✼❪ ❝♦♠

n = 2✱ p = 2 ❡ m = 1✳ ▼❛s ❡❧❡ só ✈❛❧❡ ♣❛r❛ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s✳ ◆❛ ♣✳

✷✽✵ ♦ ❛✉t♦r ❡①♣❧✐❝❛ ❛ ❛❞❛♣t❛çã♦ ♣❛r❛ ❞♦♠í♥✐♦s ♥ã♦ ❧✐♠✐t❛❞♦s✱ ❡ ♠♦str❛ ❝♦♠♦ ❡s❝♦❧❤❡r ❛ ❝♦♥st❛♥t❡ ρ✳

✷✳✸ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧

▲❡♠❜r❡ q✉❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ H ❝♦♠ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ,·iH é ✉♠

❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt s❡ H ❝♦♠ ❛ ♥♦r♠❛ kvk2H = hv, viH é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦

❝♦♠♣❧❡t♦✳ ▲❡♠❜r❡ q✉❡ ✉♠ ❢✉♥❝✐♦♥❛❧ G ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H é ✉♠❛

❢✉♥çã♦ G:H R✳

❈♦♥s✐❞❡r❡ ✉♠ ❢✉♥❝✐♦♥❛❧G ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❉❛❞♦s v ❡ h ❡♠ H✱ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① G′(v, h) é ❞❡✜♥✐❞❛ ♣♦r

G′(v, h) = lim t→0

G(v+th)G(v)

t ,

s❡ ♦ ❧✐♠✐t❡ ❡①✐st✐r✳

❯♠❛ s❡q✉ê♥❝✐❛ vn ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H ❝♦♥✈❡r❣❡ ❢r❛❝❛♠❡♥t❡ ♣❛r❛

✉♠ v ❡♠ H s❡

hvn, wiH → hv, wiH

♣❛r❛ t♦❞♦ w ❡♠ H✳

❯♠ ❢✉♥❝✐♦♥❛❧G ❞❡✜♥✐❞♦ ❡♠ H é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦

❢r❛❝♦ s❡ ♣❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛vn❡♠H❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛ ✉♠v❡♠H♥♦ s❡♥t✐❞♦

❢r❛❝♦✱ t❡♠♦s q✉❡

G(v)≤lim inf

(33)

❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✸ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ✈❛♠♦s ✉s❛r ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♣❛r❛ ♠♦str❛r q✉❡ ✉♠ ❝❡rt♦ ❢✉♥❝✐♦♥❛❧ G❞❡✜♥✐❞♦ ❡♠ H1(R2) ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✶✳ ❬✶✶✱ ♣✳ ✶✵✵❪ ❈♦♥s✐❞❡r❡ ✉♠ ❢✉♥❝✐♦♥❛❧ G ❞❡✜♥✐❞♦ ♥✉♠

❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① G′(x, h) ❡①✐st❡

♣❛r❛ t♦❞♦ x ❡ h ❡♠ H✳ ❆ss✉♠❛ q✉❡ G é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦

s❡♥t✐❞♦ ❢r❛❝♦✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ R t❛❧ q✉❡

G′(v, v)>0

♣❛r❛ t♦❞♦ v ❡♠ H ❝♦♠ kvkH =R✳ ❊♥tã♦ ❡①✐st❡ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ G

♥♦ ✐♥t❡r✐♦r ❞❛ ❜♦❧❛ |x| ≤R✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ v0 ❡♠ H ❝♦♠ kv0kH < R t❛❧

q✉❡

G(v0)G(v)

♣❛r❛ t♦❞♦ v s✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦ ❞❡ v0✳ ❊♠ ♣❛rt✐❝✉❧❛r✱

G′(v0) = 0.

❯♠ ❢✉♥❝✐♦♥❛❧ G ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rtH é ❝♦♥✈❡①♦ s❡

G(tv1+ (1−t)v2)≤tG(v1) + (1−t)G(v2)

♣❛r❛ t♦❞♦ 0≤t 1❡ v1 ❡ v2 ❡♠ H✳

❯♠ ❢✉♥❝✐♦♥❛❧G❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rtH é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦

s❡

G(tv1+ (1−t)v2)< tG(v1) + (1−t)G(v2)

♣❛r❛ t♦❞♦ 0< t <1 ❡v1 ❡ v2 ❡♠ H✳

❱❛♠♦s ✉s❛r ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♣❛r❛ ♠♦str❛r ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❞♦ ❢✉♥❝✐♦♥❛❧ G♠❡♥❝✐♦♥❛❞♦ ❛❝✐♠❛✳

Pr♦♣♦s✐çã♦ ✷✳✸✳✷✳ ❬✶✶✱ ♣✳ ✾✻❪ ❈♦♥s✐❞❡r❡ ✉♠ ❢✉♥❝✐♦♥❛❧ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦

G ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❆ss✉♠❛ q✉❡ G ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ❞❡

♠í♥✐♠♦✳ ❊♥tã♦ ❡st❡ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ é ú♥✐❝♦✳

(34)

❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✹ ♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✳ ❱❛♠♦s ✉s❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♣❛r❛ ♠♦str❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧Gé s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✳

Pr♦♣♦s✐çã♦ ✷✳✸✳✸✳ ❬✶✶✱ ♣✳ ✽✷❪ ❈♦♥s✐❞❡r❡ ✉♠ ❢✉♥❝✐♦♥❛❧ ❝♦♥✈❡①♦ G ♥✉♠

❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① G′(v, h) ❡①✐st❡

♣❛r❛ t♦❞♦ v ❡ h ❡♠ H✳ ❆ss✉♠❛ q✉❡ G′(v,·) é ❝♦♥tí♥✉♦ ♣❛r❛ t♦❞♦ v ❡♠ H✳

❊♥tã♦ G é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✳

❋✐①❛❞♦ ✉♠ v ❡♠ H✱ ❧❡♠❜r❡ q✉❡ G′(v,·) é ❝♦♥tí♥✉♦ ❡♠ h H s❡ ♣❛r❛

t♦❞❛ s❡q✉ê♥❝✐❛ hn ❡♠ H t❛❧ q✉❡ kh−hnkH →0✱ t❡♠♦s q✉❡

|G′(v, hn)−G′(v, h)| →0,

(35)

❈❛♣ít✉❧♦ ✸

❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

✸✳✶ ❱♦rt❡① ◆✉♠❜❡r ❡ ❋ór♠✉❧❛ ❞❡ ❇♦❣♦♠♦❧✬♥②✐

❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧R2×CR2✳ P❡❧♦ ❊①❡♠♣❧♦ ✶✳✷✳✶ ❡ Pr♦♣♦s✐çã♦

✶✳✷✳✺✱ s❛❜❡♠♦s q✉❡ t♦❞❛ ❝♦♥❡①ã♦ ❡♠ R2×CR2 é ❞❛ ❢♦r♠❛

dA =d−iA

♣❛r❛ ❛❧❣✉♠❛ ✶✲❢♦r♠❛ A ❡♠ R2✳ ▲❡♠❜r❡ q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❛ ❝♦♥❡①ã♦ dA é ❞❛❞❛ ♣♦r

FA=dA.

P❡❧♦ ❊①❡♠♣❧♦ ✶✳✶✳✷✱ s❛❜❡♠♦s q✉❡ ❛s s❡çõ❡s ❞❡ R2 ×C R2 ♣♦❞❡♠ s❡r ✐❞❡♥t✐✜❝❛❞❛s ❝♦♠ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s ❡♠ R2✳ ◆❡st❡ ❝❛♣ít✉❧♦ ✐❞❡♥t✐✜❝❛♠♦s s❡çõ❡s ❝♦♠ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s ❡ ❝♦♥❡①õ❡s ❝♦♠ ✶✲❢♦r♠❛s✳

❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ φ ❡♠ R2 ❡ ✉♠❛ ✶✲❢♦r♠❛ A ❡♠ R2✳ ❙✉✲ ♣♦♥❤❛ q✉❡ φ ❡ A ♣♦ss✉❡♠ ❞❡r✐✈❛❞❛s ❢r❛❝❛s✳ ❆ ❡♥❡r❣✐❛ é ❞❡✜♥✐❞❛ ♣♦r

E(φ, A) =

ˆ

R2

1 2|dAφ|

2 +1

2|FA| 2+ λ

8 |φ| 2

−12

,

♦♥❞❡ λ é ✉♠❛ ❝♦♥st❛♥t❡✳

❖ ❡st✉❞♦ ❞❛ ❡♥❡r❣✐❛ é ✐♠♣♦rt❛♥t❡ ♥❛ t❡♦r✐❛ ❞❡ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡✳ ◗✉❛♥❞♦

λ <1✱ ❛ ❡♥❡r❣✐❛ ❞❡s❝r❡✈❡ s✉♣❡r❝♦♥❞✉t♦r❡s ❞♦ t✐♣♦I✱ ❡ q✉❛♥❞♦λ >1✱ ❛ ❡♥❡r✲

(36)

❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✻ ❣✐❛ ❞❡s❝r❡✈❡ s✉♣❡r❝♦♥❞✉t♦r❡s ❞♦ t✐♣♦ II ❬✸✱ ✻✱ ✶✷❪✳ ◆❡st❡ ❝❛♣ít✉❧♦ ❡st❛♠♦s

✐♥t❡r❡ss❛❞♦s ♥♦ ❝❛s♦λ= 1✭✈❛❧♦r ❝rít✐❝♦✮✳ ◗✉❛♥❞♦λ= 1✱ ❛ ❡♥❡r❣✐❛ é ❧✐♠✐t❛❞❛

✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✈♦rt❡① ♥✉♠❜❡r ✭❚❡♦r❡♠❛ ✸✳✶✳✷✮✳ ◗✉❛❧q✉❡r ♣❛r (φ, A) q✉❡ ❛t✐♥❣❡ ♦ ♠í♥✐♠♦ ❞❡ ❡♥❡r❣✐❛ é ✉♠❛ s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡

●✐♥③❜✉r❣✲▲❛♥❞❛✉ ✭❚❡♦r❡♠❛ ✸✳✷✳✶✮✳

❚❡♦r❡♠❛ ✸✳✶✳✶✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛φ ❡♠ R2 ❡ ✉♠❛ ✶✲❢♦r♠❛A ❡♠ R2✳ ❆ss✉♠❛ q✉❡φA ♣♦ss✉❡♠ ❞❡r✐✈❛❞❛s ❢r❛❝❛s✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡♥❡r❣✐❛

E(φ, A)<✳ ❊♥tã♦ ❛ ❡♥❡r❣✐❛ E é ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝❛❧✐❜r❡ φ

AAig−1dg,

♦♥❞❡ g é ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❝♦♠ |g|= 1✱ ♦✉ s❡❥❛✱

E gφ, Aig−1dg

=E(φ, A).

❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

dA−ig−1dg(gφ) = d(gφ)−i A−ig−1dg

(gφ)

=φdg+gdφiAgφφdg

=g(dφ−iAφ)

=gdAφ,

FA−ig−1

dg =d A−ig−1dg

=dAi(−1)g−2dgdgig−1ddg =dA

(37)

❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✼ ♣♦rt❛♥t♦

E gφ, Aig−1dg

=

ˆ

R2

1

2|dA−ig−1dg(gφ)| 2

+ 1

2|FA−ig−1dg| 2+λ

8 |gφ| 2

−12 = ˆ R2 1

2|gdAφ| 2

+ 1 2|FA|

2+λ 8 |gφ|

2

−12 = ˆ R2 1 2|dAφ|

2 + 1

2|FA| 2+λ

8 |φ| 2

−12

=E(φ, A).

❆♥t❡s ❞❡ ❡♥✉♥❝✐❛r ♦ ♣ró①✐♠♦ t❡♦r❡♠❛✱ ✈❛♠♦s ✐♥tr♦❞✉③✐r ✉♠❛ ❢✉♥çã♦ ❝✉t✲ ♦✛ χR t❛❧ q✉❡

χR=

 

1 ❡♠ BR, 0 ❡♠ R2\B2R,

0≤χR≤1,

|dχR| ≤ C1

R,

♦♥❞❡ BR é ❛ ❜♦❧❛ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❛✐♦ R✳

❚♦♠❡ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡χ˜❡♠ [0,) t❛❧ q✉❡

˜ χ=

 

1 ❡♠ [0,1], 0 ❡♠ [2,), 0≤χ˜1.

❚❡♠♦s q✉❡

sup|χ˜′|= sup [1,2]|

˜ χ′|

(38)

❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✽ ❉❡✜♥✐♠♦s

χR(x) = ˜χ

|x| R

.

❚❡♠♦s q✉❡

χR=

 

1 ❡♠ BR, 0 ❡♠ R2\B2R,

0χR1.

❆❧é♠ ❞✐ss♦✱

dχR(x) = ˜χ′

|x| R

1 Rd|x|,

❧♦❣♦

|dχR(x)| ≤ C1 R.

❚❡♦r❡♠❛ ✸✳✶✳✷✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛φ ❡♠ R2 ❡ ✉♠❛ ✶✲❢♦r♠❛A ❡♠ R2✳ ❆ss✉♠❛ q✉❡φA ♣♦ss✉❡♠ ❞❡r✐✈❛❞❛s ❢r❛❝❛s✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡♥❡r❣✐❛

E(φ, A)<✳ ❖ ❧✐♠✐t❡

vort (φ, A) = lim R→∞

1 2π

ˆ

R2

χRFA

❡①✐st❡✳ ❆❧é♠ ❞✐ss♦✱ vort (φ, A) é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❡ ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s✲

❢♦r♠❛çõ❡s ❞❡ ❝❛❧✐❜r❡✱ ❝❤❛♠❛❞♦ ✈♦rt❡① ♥✉♠❜❡r✳ ▼❛✐s ❛✐♥❞❛✱ s❡ ❛ ❢✉♥çã♦ φ

é s✉❛✈❡ ❡ |φ| → 1 ♥♦ ✐♥✜♥✐t♦✱ ❡♥tã♦ vort (φ, A) é ✐❣✉❛❧ ❛♦ í♥❞✐❝❡ ❞❡ φ ♥♦

✐♥✜♥✐t♦✳

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ❛ss✉♠✐r q✉❡ ❛ ❢✉♥çã♦ φ é s✉❛✈❡ ❡ |φ| → 1 q✉❛♥❞♦

|x| → ∞✳ ❖ ❝❛s♦ ❣❡r❛❧ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✷❪✳

✶✳❛✳ ❱❛♠♦s ♠♦str❛r q✉❡

A=d(❛r❣φ) 1 2i|φ|

−2 ¯

φdAφφdAφ. ✭✸✳✶✮

▲❡♠❜r❛♥❞♦ q✉❡ dAφ=dφiAφ✱ t❡♠♦s q✉❡ ¯

(39)

❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✾ ❡

φdAφ=φdφ+iA|φ|2. ✭✸✳✸✮

❙✉❜tr❛✐♥❞♦ ✭✸✳✷✮ ❞❡ ✭✸✳✸✮✱ ♦❜t❡♠♦s

2i|φ|2A= ¯φdφφdφ

− φd¯ Aφ−φdAφ

.

❈♦♠♦ |φ| → 1 q✉❛♥❞♦ x → ∞✱ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♣♦r |φ| ♣❛r❛ |x| s✉✜❝✐❡♥t❡✲

♠❡♥t❡ ❣r❛♥❞❡✳ ❚❡♠♦s q✉❡

A= 1

2i|φ|

−2 ¯

φdφφdφ 1 2i|φ|

−2 ¯

φdAφφdAφ.

❊s❝r❡✈❛

φ=ef

=ef1

eif2

.

❚❡♠♦s q✉❡

dφ=ef(df1+idf2),

dφ=ef¯(df1−idf2),

♣♦rt❛♥t♦

1 2i|φ|

−2 ¯

φdφφdφ

= 1 2i

1 e2f1

ef¯dφefdφ

= 1 2i

1 e2f1

ef+f¯ (df1+idf2)−ef+ ¯f(df1−idf2)

= 1

2i(df1+idf2−df1+idf2) =d(❛r❣φ).

▲♦❣♦

A=d(❛r❣φ) 1 2i|φ|

−2 ¯

φdAφ−φdAφ

(40)

❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✵ ✶✳❜✳ ❯s❛♥❞♦ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s✱ ♦❜t❡♠♦s

1 2π

ˆ

R2

χRFA= 1

2π ˆ R2 χRdA = 1 2π ˆ R2

d(χRA)− 1 2π

ˆ

R2

dχR∧A.

❈♦♠♦ χR t❡♠ ♦ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ✭❛ ❢✉♥çã♦ s❡ ❛♥✉❧❛ ❢♦r❛ ❞❡ B2R✮✱ ♣❡❧♦

t❡♦r❡♠❛ ❞❡ ❙t♦❦❡s✱ t❡♠♦s q✉❡

1 2π

ˆ

R2

d(χRA) = 0,

♣♦rt❛♥t♦

1 2π

ˆ

R2

χRFA= 1 2π

ˆ

R2

dχRA. ✭✸✳✹✮

❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✉❛çã♦ ✭✸✳✶✮ ♥❛ ❊q✉❛çã♦ ✭✸✳✹✮✱ ♦❜t❡♠♦s

1 2π

ˆ

R2

χRFA=− 1 2π

ˆ

R2

dχR∧d(❛r❣φ) ✭✸✳✺✮ + 1

ˆ

R2

dχR∧ 1 2i|φ|

−2 ¯

φdAφ−φdAφ

.

P❡❧♦ t❡♦r❡♠❛ ❞❡ ❙t♦❦❡s✱ t❡♠♦s q✉❡

− 1

ˆ

R2

dχRd(❛r❣φ) = 1 2π

ˆ

B2R\BR

d(χRd(❛r❣φ)) =− 1

ˆ

∂B2R

χRd(❛r❣φ) + 1 2π

ˆ

∂BR

χRd(❛r❣φ) = 1

ˆ

∂BR

d(❛r❣φ) = vort (φ, A).

P♦❞❡♠♦s ✈❡r q✉❡ vort (φ, A)é ♦ í♥❞✐❝❡ ❞❡ φ✳ ▲❡♠❜r❡ q✉❡ ♦ í♥❞✐❝❡ ❞❡φ é ✉♠

♠ú❧t✐♣❧♦ ✐♥t❡✐r♦ ❞♦ í♥❞✐❝❡ ❞❛ ❝✉r✈❛ ∂BR✱ ♣♦rt❛♥t♦ é ✉♠ ♠ú❧t✐♣❧♦ ✐♥t❡✐r♦ ❞❡

✶✳ P♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❊q✉❛çã♦ ✭✸✳✺✮ ❝♦♠♦

1 2π

ˆ

R2

χRFA = vort (φ, A) + 1 2π

ˆ

R2

dχR 1 2i|φ|

−2 ¯

Figure

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