❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
❙♦❧✉çõ❡s ❞❡ ❱órt✐❝❡ ❞❛s ❊q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉
❖❧❡s②❛ ●❛❧❦✐♥❛
❯♥✐✈❡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
❖❧❡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♦
❘❡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❣✲▲❛♥❞❛✉✳
❆❜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✳
❙✉♠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✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
❈❛♣ít✉❧♦ ✶
▲✐♥❡ ❇✉♥❞❧❡s
✶✳✶ ▲✐♥❡ ❇✉♥❞❧❡s
❯♠ ❧✐♥❡ ❜✉♥❞❧❡ é ✉♠❛ tr✐♣❧❛ (L, M, π)❢♦r♠❛❞❛ ♣♦r ✈❛r✐❡❞❛❞❡sL❡ M ❡ ✉♠❛
♣r♦❥❡çã♦ s✉❛✈❡ π :L→M 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)→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 :L→M ❞❛❞❛ ♣♦r
π(m, z) =m.
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✽ ❱❛♠♦s ♠♦str❛r q✉❡ π :L→M é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳
✶✳ ❚❡♠♦s q✉❡π−1(m) = {m}×C✳ 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 :L→M ×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❡❥❛✱
π◦s=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 ❡♠{m}×C✱ ♣♦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❛
❈❆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)→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).
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✵
m ❡♠ Uα✱ ♣♦rt❛♥t♦
φα−1◦φα(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|π−1(m)(z) +φα|π−1(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
❈❆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
= (αγ−βδ)v+ (αδ+βγ)np×v.
◆❛ ❡s❢❡r❛ S2✱ t❡♠♦s q✉❡
i(iv) =i(np×v)
=np×(np×v)
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✷ ♣♦rt❛♥t♦
(α+iβ) [(γ+iδ)v] = (α+iβ) (γv+δnp×v)
=α(γv+δnp×v) +βnp×(γv+δnp ×v)
= (aγ−βδ)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 ,
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✸ ❘❡♣❛r❡ q✉❡
ψ1(U1∩U2) = C∗, ψ2(U1∩U2) = 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]) :λ∈C∗}.
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]
❈❆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❡ ❛♥✉❧❛♠✳ ❊♠ Uα∩Uβ ❡s❝r❡✈❛
sα =gαβsβ.
❆s ❢✉♥çõ❡s gαβ :Uα∩Uβ →C∗ sã♦ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡sα✳
❊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 :λ∈C∗}✳ ❈♦♥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]
.
❈❆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:L→M ❝♦♠ ✉♠❛ ❝♦❜❡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
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✻ Pr♦♣♦s✐çã♦ ✶✳✶✳✽✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π:L→M ❝♦♠ ✉♠❛ ❝♦❜❡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 ❡♠ Uα∩Uβ ✭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 :Uα∩Uβ →C∗s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ✶✱ ✷ ❡ ✸ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✽✱ ❡♥tã♦ ❡①✐st❡✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✱ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L→M ❡ s❡çõ❡s ❧♦❝❛✐s sα :Uα →Lq✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠ t❛✐s q✉❡
sα =gαβsβ
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✼
✶✳✷ ❈♦♥❡①õ❡s ❡ ❈✉r✈❛t✉r❛
❯♠❛ ❝♦♥❡①ã♦ ∇♥✉♠ ❧✐♥❡ ❜✉♥❞❧❡L→M 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) = df⊗s+f∇s.
❊♠ ♦✉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 ❡♠ L→M sã♦ ❞❛ ❢♦r♠❛
❈❆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♦ ❡♠ L→M✳ ❚♦♠❡ ❞✉❛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 :L→M ❝♦♠ ✉♠❛
❝♦♥❡①ã♦ ∇✳ ❈♦♥s✐❞❡r❡ ✉♠❛ s❡çã♦ ❧♦❝❛❧ s : U ⊂ M → L✳ P❛r❛ q✉❛✐sq✉❡r
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✾
∇˜s|U =∇s¯|U.
P♦rt❛♥t♦ ♣♦❞❡♠♦s ❞❡✜♥✐r
∇s =∇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 =Aα⊗sα.
▼❛✐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❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L→M ❝♦♠ ✉♠❛ ❝♦♥❡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∇sβ
=dgαβ ⊗sβ +gαβAβ ⊗sβ
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✵ P♦r ♦✉tr♦ ❧❛❞♦ t❡♠♦s q✉❡
∇sα =Aα⊗sα =gαβAα⊗sβ.
P♦rt❛♥t♦
gαβAα =dgαβ +gαβAβ.
▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ❧❛❞♦s ♣♦r g−αβ1✱ ♦❜t❡♠♦s
Aα =Aβ+gαβ−1dgαβ.
Pr♦♣♦s✐çã♦ ✶✳✷✳✹✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L→M ❝♦♠ ✉♠❛ ❝♦♥❡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 ❡♠ Uα∩Uβ✳ ❚❡♠♦s q✉❡
dAα =d Aβ +g−αβ1dgαβ
=dAβ −gαβ−2dgαβ ∧dgαβ +gαβ−1d(dgαβ) =dAβ.
❆❝✐♠❛ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡ d2 = 0 ❡ ω∧ω = 0 ♣❛r❛ t♦❞❛ ✶✲❢♦r♠❛ω✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✺✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡L→M ❝♦♠ ❝♦♥❡①õ❡s ∇❡ ∇′✳
❊①✐st❡ ✉♠❛ ✶✲❢♦r♠❛ η ❡♠ M t❛❧ q✉❡
❈❆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′α⊗sα.
❉❡✜♥✐♠♦s ❛ ✶✲❢♦r♠❛ η ❡♠ Uα ♣♦r
ηα =A′α−Aα.
❱❛♠♦s ♠♦str❛r q✉❡ ❡♠ Uα∩Uβ ✈❛❧❡ ηα =ηβ✳ ❉❡♥♦t❡ ♣♦r gαβ ❛s ❢✉♥çõ❡s ❞❡
tr❛♥s✐çã♦ ❞❡ sα✳ ❚❡♠♦s q✉❡
ηα =A′α−Aα
=A′β +g−αβ1dgαβ−Aβ +g−αβ1dgαβ =A′β−Aβ
=ηβ.
❚❡♠♦s q✉❡
∇′sα =A′α⊗s
= (Aα+ηα)⊗sα,
❧♦❣♦
∇′s=∇s+η⊗s.
❚❛♠❜é♠ t❡♠♦s q✉❡
F∇′ =dA′α
=dAα+dηα
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✷
Pr♦♣♦s✐çã♦ ✶✳✷✳✻✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L→M ❝♦♠ ✉♠❛ ❝♦♥❡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,
❧♦❣♦
❈❆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♦ ❡♠ L→Σ✳ ❙❛❜❡♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛F∇ é ♥✉❧❛✳ P♦rt❛♥t♦ ❛
❝❧❛ss❡ ❞❡ ❈❤❡r♥ é ❞❛❞❛ ♣♦r
c1(Σ×C) = 1 2πi
ˆ
Σ F∇
= 0.
❈❆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).
❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✺ ♣♦rt❛♥t♦
∇s=∇R3
s−<∇R3s, n > n
=∇R3
s+dθ⊗sinφn
=dθ⊗ −cosθcos2φ,−sinθcos2φ,sinφcosφ
=dθ⊗cosφ(n×s)
=icosφdθ⊗s.
❘❡s✉♠✐♥❞♦✱ t❡♠♦s q✉❡
∇s=icosφdθ⊗s.
P♦rt❛♥t♦ ❛ ✶✲❢♦r♠❛ ❞❡ ❝♦♥❡①ã♦ ❞❛ s❡çã♦ s é ❞❛❞❛ ♣♦r
A=icosφdθ,
❡ ❝✉r✈❛t✉r❛ é ❞❛❞❛ ♣♦r
F∇=d(icosφdθ)
=−isinφdφ∧dθ.
P♦rt❛♥t♦ ❛ ❝❧❛ss❡ ❞❡ ❈❤❡r♥ é ❞❛❞❛ ♣♦r
c1 TS2
= 1 2πi
ˆ
S2
F∇
= −1 2π
ˆ 2π
0
ˆ π
0
sinφdφ
dθ
❈❛♣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).
❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✷✼ ❚❡♦r❡♠❛ ✷✳✶✳✶✳ ❬✺✱ ✾❪ ❊①✐st❡ ú♥✐❝❛ ♠❡❞✐❞❛ µ ♥❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❞❡ Rn t❛❧ q✉❡
µ([a1, b1]×...×[an, bn]) = (bn−an)...(b1−a1).
❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦X❝♦♠ ✉♠❛σ✲á❧❣❡❜r❛A✳ ❯♠❛ ❢✉♥çã♦f :X →R é ♠❡♥s✉rá✈❡❧ s❡ ♣❛r❛ t♦❞♦ a ❡♠ R t❡♠♦s q✉❡
{x∈X :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) ={x∈X :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 s≤f
❈❆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
❈❆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✳
❈❆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✉❧♦ ✸✳
❈❆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 (k−1)1−1k
k−2 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
❈❆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♥♦ h·,·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
❈❆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♥✐❝♦✳
❈❆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,
❈❛♣ít✉❧♦ ✸
❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧
✸✳✶ ❱♦rt❡① ◆✉♠❜❡r ❡ ❋ór♠✉❧❛ ❞❡ ❇♦❣♦♠♦❧✬♥②✐
❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧R2×C→R2✳ P❡❧♦ ❊①❡♠♣❧♦ ✶✳✷✳✶ ❡ Pr♦♣♦s✐çã♦✶✳✷✳✺✱ s❛❜❡♠♦s q✉❡ t♦❞❛ ❝♦♥❡①ã♦ ❡♠ R2×C→R2 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✲
❈❆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✉❡φ ❡ A ♣♦ss✉❡♠ ❞❡r✐✈❛❞❛s ❢r❛❝❛s✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡♥❡r❣✐❛
E(φ, A)<∞✳ ❊♥tã♦ ❛ ❡♥❡r❣✐❛ E é ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝❛❧✐❜r❡ φ→gφ
A→A−ig−1dg,
♦♥❞❡ g é ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❝♦♠ |g|= 1✱ ♦✉ s❡❥❛✱
E gφ, A−ig−1dg
=E(φ, A).
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡
dA−ig−1dg(gφ) = d(gφ)−i A−ig−1dg
(gφ)
=φdg+gdφ−iAgφ−φdg
=g(dφ−iAφ)
=gdAφ,
❡
FA−ig−1
dg =d A−ig−1dg
=dA−i(−1)g−2dg∧dg−ig−1ddg =dA
❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✼ ♣♦rt❛♥t♦
E gφ, A−ig−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′|
❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✽ ❉❡✜♥✐♠♦s
χR(x) = ˜χ
|x| R
.
❚❡♠♦s q✉❡
χR=
1 ❡♠ BR, 0 ❡♠ R2\B2R,
0≤χR≤1.
❆❧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✉❡φ ❡ A ♣♦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 ¯
φdAφ−φdAφ. ✭✸✳✶✮
▲❡♠❜r❛♥❞♦ q✉❡ dAφ=dφ−iAφ✱ t❡♠♦s q✉❡ ¯
❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✾ ❡
φdAφ=φdφ+iA|φ|2. ✭✸✳✸✮
❙✉❜tr❛✐♥❞♦ ✭✸✳✷✮ ❞❡ ✭✸✳✸✮✱ ♦❜t❡♠♦s
2i|φ|2A= ¯φdφ−φdφ
− φd¯ Aφ−φdAφ
.
❈♦♠♦ |φ| → 1 q✉❛♥❞♦ x → ∞✱ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♣♦r |φ| ♣❛r❛ |x| s✉✜❝✐❡♥t❡✲
♠❡♥t❡ ❣r❛♥❞❡✳ ❚❡♠♦s q✉❡
A= 1
2i|φ|
−2 ¯
φdφ−φdφ− 1 2i|φ|
−2 ¯
φdAφ−φdAφ.
❊s❝r❡✈❛
φ=ef
=ef1
eif2
.
❚❡♠♦s q✉❡
dφ=ef(df1+idf2),
❡
dφ=ef¯(df1−idf2),
♣♦rt❛♥t♦
1 2i|φ|
−2 ¯
φdφ−φdφ
= 1 2i
1 e2f1
ef¯dφ−efdφ
= 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
❈❆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
dχR∧A. ✭✸✳✹✮
❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✉❛çã♦ ✭✸✳✶✮ ♥❛ ❊q✉❛çã♦ ✭✸✳✹✮✱ ♦❜t❡♠♦s
1 2π
ˆ
R2
χRFA=− 1 2π
ˆ
R2
dχR∧d(❛r❣φ) ✭✸✳✺✮ + 1
2π
ˆ
R2
dχR∧ 1 2i|φ|
−2 ¯
φdAφ−φdAφ
.
P❡❧♦ t❡♦r❡♠❛ ❞❡ ❙t♦❦❡s✱ t❡♠♦s q✉❡
− 1
2π
ˆ
R2
dχR∧d(❛r❣φ) =− 1 2π
ˆ
B2R\BR
d(χRd(❛r❣φ)) =− 1
2π
ˆ
∂B2R
χRd(❛r❣φ) + 1 2π
ˆ
∂BR
χRd(❛r❣φ) = 1
2π
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 ¯
