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A Curvatura de GaussKronecker de hipersuperfícies mínimas em formas espaciais 4dimensionais

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

❈❯❘❙❖ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❘❡♥❛t♦ ❖❧✐✈❡✐r❛ ❚❛r❣✐♥♦

❆ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s

♠í♥✐♠❛s ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s ✹✲❞✐♠❡♥s✐♦♥❛✐s

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❘❡♥❛t♦ ❖❧✐✈❡✐r❛ ❚❛r❣✐♥♦

❆ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s

♠í♥✐♠❛s ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s ✹✲❞✐♠❡♥s✐♦♥❛✐s

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

➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥✲ ❝✐❛❧✳

❖r✐❡♥t❛❞♦r✿

Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ❈❛♠✐♥❤❛ ▼✉♥✐③ ◆❡t♦✳

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❚❛r❣✐♥♦✱ ❘❡♥❛t♦ ❖❧✐✈❡✐r❛

❚✶✾✷❝ ❆ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s

♠í♥✐♠❛s ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s ✹✲❞✐♠❡♥s✐♦♥❛✐s ✴ ❘❡♥❛t♦ ❖❧✐✈❡✐r❛ ❚❛r❣✐♥♦✳✕❋♦rt❛❧❡③❛✱ ✷✵✶✶✳

✺✹ ❢✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ❈❛♠✐♥❤❛ ▼✉♥✐③ ◆❡t♦✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧

❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮✕❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❋♦rt❛❧❡③❛✱ ✷✵✶✶✳

✶✳●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧✳ ■✳▼✉♥✐③ ◆❡t♦✱ ❆♥t♦♥✐♦ ❈❛♠✐♥❤❛ ✭❖r✐❡♥t✳✮

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❆❣r❛❞❡❝✐♠❡♥t♦s

Ó ❉❡✉s✱ ❡✉ t❡ ❞♦✉ ❣r❛ç❛s ❡ t❡ ❧♦✉✈♦✱ ♣♦rq✉❡ ♠❡ ❞❡st❡ s❛❜❡❞♦r✐❛ ❡ ❢♦rç❛✳ ◆ã♦ ❛♣❡♥❛s ❛♦s q✉❡ ❝♦❧❛❜♦r❛r❛♠ ♣❛r❛ q✉❡ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦ ❢♦ss❡ ♣♦ssí✈❡❧✱ ♠❛s ❛ t♦❞♦s ❛q✉❡❧❡s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ❞✐r❡❝✐♦♥❛♠❡♥t♦ ♥❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✱ ❞❡✐①♦ ♠❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s✳

●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦ ♣r♦❢❡ss♦r ❆♥t♦♥✐♦ ❈❛♠✐♥❤❛ ▼✉♥✐③ ◆❡t♦✱ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ♣♦r s✉❛ ♣❛❝✐ê♥❝✐❛ ❞✉r❛♥t❡ ❡st❡ ♣❡rí♦❞♦ ❞❡ ♣r❡♣❛r❛çã♦ ❞❛ ❞✐ss❡rt❛çã♦ ❡ ❛♦s ♣r♦❢❡ss♦r❡s ❏♦sé ❘♦❜ér✐♦ ❘♦❣ér✐♦ ❡ ❏♦sé ❆❢♦♥s♦ ❞❡ ❖❧✐✈❡✐r❛ ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ❛❥✉❞❛r✳

❆❣r❛❞❡ç♦ ❛✐♥❞❛ à s❡❝r❡tár✐❛ ❞♦ ❉❡♣❛rt❛♠❡♥t♦✱ ❆♥❞r❡❛ ❈♦st❛ ❉❛♥t❛s✱ ♣♦r t♦❞♦ t❡♠♣♦ ❞❡❧❛ ❞❡❞✐❝❛❞♦ ❛ r❡s♦❧✈❡r ♦s ♣r♦❜❧❡♠❛s ❞♦s ❛❧✉♥♦s❀ ❡ s❡♠♣r❡ ❝♦♠ ❛t❡♥çã♦ ❡ ❡✜❝✐ê♥❝✐❛✳

❉❡✐①♦ ♠✐♥❤❛ ❡t❡r♥❛ ❣r❛t✐❞ã♦ ❛ ♠✐♥❤❛ ♠ã❡✱ ❋r❛♥❝✐s❝❛ ■✈❛♥❡✐❞❡ ❖❧✐✈❡✐r❛ ❚❛r❣✐♥♦✱ ♣♦r s❡r tã♦ ❜♦❛✳ ❆ ♠✐♥❤❛ ✐r♠ã✱ ❘✉t❤ ❚❛r❣✐♥♦✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s✲ ♠♦s✱ s♦✉ ❣r❛t♦✳

❆❣r❛❞❡❝✐♠❡♥t♦s sã♦ t❛♠❜é♠ ❞❡✈✐❞♦s ❛♦s ❛♠✐❣♦s ❘♦❞r✐❣♦✱ ▲♦❡st❡r✱ ❆❧❡✲ ①❛♥❞r❡ ❡ ❚✐❛r❧♦s✳

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❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❝♦♠♣❧❡t❛s ❡ ❝♦♠ ❝✉r✈❛✲ t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❝♦♥st❛♥t❡ ❡♠ ✉♠❛ ❢♦r♠❛ ❡s♣❛❝✐❛❧ Q4(c)✳ Pr♦✈❛♠♦s

q✉❡ ♦ í♥✜♠♦ ❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❛ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ♠í♥✐♠❛ ❝♦♠♣❧❡t❛ ❡♠ Q4(c), c 0, ♥❛ q✉❛❧ ❛ ❝✉r✈❛t✉r❛ ❞❡

❘✐❝❝✐ é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡✱ é ✐❣✉❛❧ ❛ ③❡r♦✳ ❆❧é♠ ❞✐ss♦✱ ❡st✉❞❛♠♦s ❤✐♣❡rs✉✲ ♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❝♦♥❡①❛s M3 ❡♠ ✉♠❛ ❢♦r♠❛ ❡s♣❛❝✐❛❧ Q4(c) ❝♦♠ ❝✉r✈❛t✉r❛

❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r K ❝♦♥st❛♥t❡✳ P❛r❛ ♦ ❝❛s♦ c 0✱ ♣r♦✈❛♠♦s✱ ♣♦r ✉♠

❛r❣✉♠❡♥t♦ ❧♦❝❛❧✱ q✉❡ s❡ K é ❝♦♥st❛♥t❡✱ ❡♥tã♦ K ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ ③❡r♦✳ ❚❛♠✲

❜é♠ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ♠í♥✐♠❛s

❡♠ Q4 ❝♦♠ K ❝♦♥st❛♥t❡✳ ❊①❡♠♣❧♦s ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s q✉❡ ♥ã♦

sã♦ t♦t❛❧♠❡♥t❡ ❣❡♦❞és✐❝❛s ♥♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ ❡ ♥♦ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦ ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ♥✉❧❛ sã♦ ❛♣r❡s❡♥t❛❞♦s✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② ❝♦♠♣❧❡t❡ ♠✐♥✐♠❛❧ ❤②♣❡rs✉r❢❛❝❡s ✇✐t❤ ❝♦♥st❛♥t ●❛✉ss✲ ❑r♦♥❡❝❦❡r ❝✉r✈❛t✉r❡ ✐♥ ❛ s♣❛❝❡ ❢♦r♠ Q4(c)✳ ❲❡ ♣r♦✈❡ t❤❛t t❤❡ ✐♥✜♠✉♠ ♦❢

t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ♦❢ t❤❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❝✉r✈❛t✉r❡ ♦❢ ❛ ❝♦♠♣❧❡t❡ ♠✐♥✐♠❛❧ ❤②♣❡rs✉r❢❛❝❡ ✐♥ Q4(c), c0, ✇❤♦s❡ ❘✐❝❝✐ ❝✉r✈❛t✉r❡ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✱ ✐s ❡q✉❛❧ t♦ ③❡r♦✳ ❋✉t❤❡r✱ ✇❡ st✉❞② t❤❡ ❝♦♥♥❡❝t❡❞ ♠✐♥✐♠❛❧ ❤②♣❡rs✉r❢❛❝❡s M3

♦❢ ❛ s♣❛❝❡ ❢♦r♠ Q4(c) ✇✐t❤ ❝♦♥st❛♥t ●❛✉ss✲❑r♦♥❡❝❦❡r ❝✉r✈❛t✉r❡ K✳ ❋♦r t❤❡

❝❛s❡c0✱ ✇❡ ♣r♦✈❡✱ ❜② ❛ ❧♦❝❛❧ ❛r❣✉♠❡♥t✱ t❤❛t ✐❢K✐s ❝♦♥st❛♥t✱ t❤❡♥ K♠✉st

❜❡ ❡q✉❛❧ t♦ ③❡r♦✳ ❲❡ ❛❧s♦ ♣r❡s❡♥t ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❝♦♠♣❧❡t❡ ♠✐♥✐♠❛❧ ❤②♣❡r✲ s✉r❢❛❝❡ ♦❢ Q4 ✇✐t❤K ❝♦♥st❛♥t✳ ❊①❛♠♣❧❡s ♦❢ ❝♦♠♣❧❡t❡ ♠✐♥✐♠❛❧ ❤②♣❡rs✉r❢❛❝❡s

✇❤✐❝❤ ❛r❡ ♥♦t t♦t❛❧❧② ❣❡♦❞❡s✐❝ ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ R4 ❛♥❞ t❤❡ ❤✐♣❡r❜♦❧✐❝ s♣❛❝❡ H4(c)✇✐t❤ ✈❛♥✐s❤✐♥❣ ●❛✉ss✲❑r♦♥❡❝❦❡r ❝✉r✈❛t✉r❡ ❛r❡ ❛❧s♦ ♣r❡s❡♥t❡❞✳

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❈♦♥t❡ú❞♦

✶ Pr❡❧✐♠✐♥❛r❡s ✶✵

✶✳✶ ❚❡♥s♦r❡s✱ ♠étr✐❝❛s ❘✐❡♠❛♥♥✐❛♥❛s ❡ ❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ✳ ✶✵ ✶✳✷ ❈✉r✈❛t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸ ❙♦❜r❡ r❡❢❡r❡♥❝✐❛✐s ♠ó✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷ ❍✐r♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s ✷✷

✷✳✶ ■♠❡rsõ❡s ✐s♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ▼❛✐s s♦❜r❡ r❡❢❡r❡♥❝✐❛✐s ♠ó✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸ ❖ ▲❛♣❧❛❝✐❛♥♦ ❞❛ ♥♦r♠❛ ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❛♦ q✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✷✽

✸ ❈✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ✸✻

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■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ ✐♥✈❡st✐❣❛♠♦s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❝♦♠♣❧❡t❛s ❝♦♠ ❝✉r✲ ✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❝♦♥st❛♥t❡ ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s Q4(c)✳ ❆♣❡s❛r ❞❡

♥♦ss♦s r❡s✉❧t❛❞♦s s❡r❡♠ ✈á❧✐❞♦s ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ❞❡ c✱ ♥❛s ♥♦ss❛s ❛✜r♠❛✲

çõ❡s ❡ ❞❡♠♦♥str❛çõ❡s t❡r❡♠♦s c = 0 ♦✉ c = ±1✱ ✐st♦ é✱ Q4(c) é ❛ ❡s❢❡r❛ Sn

✉♥✐tár✐❛ s❡ c >0✱ ♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦Rn s❡c= 0 ❡ ♦ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦ Hn

❞❡ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ −1✱ s❡ c <0✳

❆ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s r❡❝❡❜❡✉ ❛t❡♥çã♦ ❞❡ ✈ár✐♦s ❛✉t♦r❡s✳ P♦r ❡①❡♠♣❧♦✱ ❍❛❞❛♠❛r❞ ❬✼❪ ♠♦str♦✉ q✉❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❡♠Rn sã♦ ❞✐❢❡♦♠♦r❢❛s à ❡s❢❡r❛ s❡ ❡ s♦♠❡♥t❡ s❡ s✉❛s ❝✉r✈❛t✉✲ r❛s ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r sã♦ ♥ã♦ ♥✉❧❛s ❡♠ ❝❛❞❛ ♣♦♥t♦✱ ❡♥q✉❛♥t♦ ❞♦ ❈❛r♠♦ ❡ ❲❛r♥❡r ♦❜t✐✈❡r❛♠ ✉♠ r❡s✉❧t❛❞♦ s✐♠✐❧❛r ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠✲ ♣❛❝t❛s ❡ ♦r✐❡♥tá✈❡✐s ❞❛ ❡s❢❡r❛ Sn

❘❡❝❡♥t❡♠❡♥t❡✱ ❤♦✉✈❡ ✉♠ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♥♦ ❡st✉❞♦ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ❡ ♠í♥✐♠❛s ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s Q4(c) ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲

❑r♦♥❡❝❦❡r ❝♦♥st❛♥t❡✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❈❤❡♥❣ ♣r♦✈♦✉✱ ❡♠ ❬✸❪✱ q✉❡ ✉♠❛ ❤✐✲ ♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛ ❡ ♠í♥✐♠❛ ❡♠ Q4(c)✱ c 0✱ ❝♦♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r

❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r K ❝♦♥st❛♥t❡✱ ❞❡✈❡

t❡r K = 0✳ ◆♦ t❡♦r❡♠❛ ✸✳✼ ♦❜t❡♠♦s ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦ r❡t✐r❛♥❞♦ ❛ ❤✐♣ó✲

t❡s❡ s♦❜r❡ ❛ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❡ s✉♣♦♥❞♦ q✉❡ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ é ❝♦♥❡①❛✳ ❊♠ ❬✽✱ ✾✱ ✶✵❪✱ ❍❛s❛♥✐s✱ ❙❛✈❛s✲❍❛❧✐❧❛❥ ❡ ❱❧❛❝❤♦s ❞❡r❛♠ ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❛s ❤✐♣❡r✲ s✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❝♦♠♣❧❡t❛s ❡♠Q4(c)❝♦♠K ✐❞❡♥t✐❝❛♠❡♥t❡ ③❡r♦✱ s❡❣✉♥❞❛

❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ♥ã♦ ♥✉❧❛ ❡ ❝♦♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❏✉♥t❛♥❞♦ ♦ t❡♦r❡♠❛ ✸✳✼ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❡♠ ❬✽✱ ✾❪ ❝❤❡❣❛♠♦s ❛♦ t❡♦r❡♠❛ ✸✳✶✷✳

❯♠ ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦ ❞❡ss❡ tr❛❜❛❧❤♦ é ♦ t❡♦r❡♠❛ ✸✳✾✱ ♦♥❞❡ sã♦ ❝❧❛ss✐✜❝❛❞❛s ❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❝♦♠♣❧❡t❛s ❞❛ ❡s❢❡r❛ S4 ❝♦♠ ❝✉r✈❛✲

t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❝♦♥st❛♥t❡K 6= 0✳ ❈♦♠❜✐♥❛♥❞♦ ❡ss❡ t❡♦r❡♠❛ ❝♦♠ ♦

r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡ ❬✶✵❪ ♦❜t✐✈❡♠♦s ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❝♦♠♣❧❡t❛s ❞❛ ❡s❢❡r❛ S4 ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❝♦♥s✲

t❛♥t❡✳

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P❛r❛ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛ ❡ ♠í♥✐♠❛ ❡♠ R4 (c = 0) ❝♦♠ K

❝♦♥st❛♥t❡✱ ❍❛s❛♥✐s✱ ❙❛✈❛s✲❍❛❧✐❧❛❥ ❡ ❱❧❛❝❤♦s ❬✽❪✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞❛ ❝✉r✈❛t✉r❛ ❞❡ ❙♠②t❤ ❡ ❳❛✈✐❡r✱ s❡♠ ❛ ❤✐♣ót❡s❡ s♦❜r❡ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r✱ ❝❤❡❣❛r❛♠ ❛♦ ♠❡s♠♦ r❡s✉❧t❛❞♦✳

❊①❡♠♣❧♦s ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉ss✲❑r♦♥❡❝❦❡r ♥✉❧❛ ❡♠ R4 ❡ H4 q✉❡ ♥ã♦ sã♦ t♦t❛❧♠❡♥t❡ ❣❡♦❞és✐❝❛ sã♦ ❛♣r❡s❡♥t❛❞❛s✳ P♦r

✐ss♦ ♥ã♦ ♣♦❞❡♠♦s ❡s♣❡r❛r ♣r♦✈❛r q✉❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s t♦t❛❧♠❡♥t❡ ❣❡♦❞és✐❝❛s sã♦ ❛s ú♥✐❝❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ♥✉❧❛ ❡♠Q4(c)✭c0✮✳ ❆❧é♠ ❞✐ss♦✱ ❛♣❧✐❝❛♠♦s ♦ ♠ét♦❞♦ ❞♦ r❡❢❡r❡♥❝✐❛❧ ♠ó✈❡❧

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❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐♠♦s ❛ ♥♦t❛çã♦ ✉s❛❞❛ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ ❜❡♠ ❝♦♠♦ ❛❧❣✉♥s ❢❛t♦s ❜ás✐❝♦s ❞❛ ❣❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✳ ❉✉r❛♥t❡ t♦❞♦ ♦ t❡①t♦✱ s✉❛✈❡ s❡rá ♣❛r❛ ♥ós s✐♥ô♥✐♠♦ ❞❡C∞✱ ✐♥❞✐❝❛r❡♠♦s ♣♦rX(M)♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s

✈❡t♦r✐❛✐s ❞❡ ❝❧❛ss❡ C∞❞❡✜♥✐❞❛s ❡♠M ❡ ♣♦rC(M)♦ ❛♥❡❧ ❞❛s ❢✉♥çõ❡s r❡❛✐s

❞❡ ❝❧❛ss❡C∞❡♠M✳ P❛r❛ ✉♠❛ ❛❜♦r❞❛❣❡♠ ♠❛✐s ❝♦♠♣❧❡t❛ s✉❣❡r✐♠♦s ❛♦ ❧❡✐t♦r

❬✶✷❪ ❡ ❬✺❪✳

✶✳✶ ❚❡♥s♦r❡s✱ ♠étr✐❝❛s ❘✐❡♠❛♥♥✐❛♥❛s ❡ ❛ ❝♦♥❡✲

①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛

❆ ♠❛✐♦r✐❛ ❞❛s ❢❡rr❛♠❡♥t❛s té❝♥✐❝❛s ❞❡ ❣❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ é ❝♦♥str✉í❞❛ ✉s❛♥❞♦ t❡♥s♦r❡s✳ ◆❛ ✈❡r❞❛❞❡✱ ❛ ♣ró♣r✐❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é ✉♠ t❡♥s♦r✳ ❆ss✐♠ ❝♦♠❡ç❛♠♦s ♣♦r r❡✈✐s❛r ❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞♦s t❡♥s♦✲ r❡s ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡✳

❯♠k✲t❡♥s♦r ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡n✲❞✐♠❡♥s✐♦♥❛✐sM é ✉♠❛ ❛♣❧✐❝❛çã♦ ♠✉❧✲

t✐❧✐♥❡❛r

F :X(M)×. . .×X(M)C(M).

■st♦ q✉❡r ❞✐③❡r q✉❡✱ ❞❛❞♦s X1, . . . , Xk ∈X(M), F(X1, . . . , Xk)é ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ❡♠ M✱ ❡ q✉❡ F é ❧✐♥❡❛r ❡♠ ❝❛❞❛ ❛r❣✉♠❡♥t♦✱ ✐st♦ é✱

F(X1, . . . , f X+gY, . . . , Xk) = f F(X1, . . . , X, . . . , Xk)

+gF(X1, . . . , Y, . . . , Xk)

♣❛r❛ t♦❞♦ X, Y X(M), f, gC(M)

❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s k✲t❡♥s♦r❡s ❡♠ M✱ ❞❡♥♦t❛❞♦ ♣♦r Tk(M)✱ é ✉♠

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✶✶

❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ ❛s s❡❣✉✐♥t❡s ♦♣❡r❛çõ❡s✿

(aF)(X1, . . . , Xk) = a(F(X1, ..., Xk));

(F +F′)(X1. . . , Xk) =F(X1, ..., Xk) +F′(X1, ..., Xk).

❙❡❥❛U ✉♠ ❛❜❡rt♦ ❞❡M ♦♥❞❡ é ♣♦ssí✈❡❧ ❞❡✜♥✐r ❝❛♠♣♦se1, . . . , en ∈X(M)✱ ❞❡ ♠♦❞♦ q✉❡ ❡♠ ❝❛❞❛ q U✱ ♦s ✈❡t♦r❡s {ei|q}, i = 1, . . . , n✱ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ❞❡ TqM❀ ❞✐r❡♠♦s ♥❡st❡ ❝❛s♦✱ q✉❡ {ei} é ✉♠ r❡❢❡r❡♥❝✐❛❧ ❡♠ U✳ ❆s ❢✉♥çõ❡s F(ei1, . . . , eik) =Fi1...ik ❡♠ U sã♦ ❝❤❛♠❛❞❛s ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡F

♥♦ r❡❢❡r❡♥❝✐❛❧ {ei}✳

❆❣♦r❛ ❞❡✜♥✐♠♦s ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛✳ ❯♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ▼ é ✉♠ ✷✲t❡♥s♦rgs✐♠étr✐❝♦ ✭✐✳❡✳✱ g(X, Y) = g(Y, X)✮ ❡ ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦ ✭✐✳❡✳✱ g(X, X) > 0 s❡ X 6= 0✮✳ ❯♠❛

♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦p❞❡M ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ♥♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡TpM q✉❡ é ❡s❝r✐t❛ ❝♦♠♦

hX, Yi=g(X, Y)♣❛r❛ X, Y TpM✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠ ✉♠❛ ❞❛❞❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ é ❝❤❛♠❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❆♣r❡s❡♥t❛♠♦s✱ ❛ s❡❣✉✐r✱ ♦ ❡①❡♠♣❧♦ ♠❛✐s s✐♠♣❧❡s ❡ ✐♠♣♦rt❛♥t❡ ❞❡ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❊①❡♠♣❧♦ ✶✳✶✳ ❖ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ Rn✱ ❝♦♠ ❛ ♠étr✐❝❛ ❞❡✜♥✐❞❛✱ ❡♠ ❝♦♦r❞❡✲ ♥❛❞❛s ❞❛❞❛s ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ idRn✱ ♣♦r

g =X

i

δijdxidxj =

X

i

dxidxi,

q✉❡ é ❛♣❡♥❛s ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ ❝❛❞❛ ❡s♣❛ç♦ t❛♥❣❡♥t❡ TpRn s♦❜r❡ ❛ ✐❞❡♥✲ t✐✜❝❛çã♦ ♥❛t✉r❛❧ TpRn =Rn✳

❆ss✐♠ ❝♦♠♦ ♥❛ ❣❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ s❡pé ✉♠ ♣♦♥t♦ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡

❘✐❡♠❛♥♥✐❛♥❛ (M, g)✱ ❞❡✜♥✐♠♦s ❛ ♥♦r♠❛ ❞❡ q✉❛❧q✉❡r ✈❡t♦r X TpM ♣♦r

|X|:=hX, Xi12✳ ❉✐③❡♠♦s q✉❡X ❡Y sã♦ ♦rt♦❣♦♥❛✐s s❡hX, Yi= 0✳ ❱❡t♦r❡s

X1, . . . , Xk sã♦ ❞✐t♦s ♦rt♦♥♦r♠❛✐s q✉❛♥❞♦ hXi, Xji=δij✱ ♦♥❞❡

δij =

(

1, s❡ i=j

0, s❡ i6=j.

❯♠❛ ❝♦♥❡①ã♦ ❧✐♥❡❛r ∇ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ✉♠❛ ❛♣❧✐✲

❝❛çã♦

∇:X(M)×X(M)X(M),

(13)

✶✷

✭❛✮ ∇XY é ❧✐♥❡❛r s♦❜r❡ C∞(M)❡♠ X✿

∇f X1+gX2Y =f∇X1Y +g∇X2Y ♣❛r❛ f, g ∈C

(M);

✭❜✮ ∇XY é ❧✐♥❡❛r s♦❜r❡ R❡♠ Y✿

∇X(aY1+bY2) = a∇XY1+b∇XY2 ♣❛r❛ a, b∈R; ✭❝✮ ∇ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ r❡❣r❛ ❞♦ ♣r♦❞✉t♦✿

∇X(f Y) = f∇XY + (Xf)Y ♣❛r❛ f ∈C∞(M).

❙❡❥❛g ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡M✳ ❯♠❛ ❝♦♥❡①ã♦

é ❞✐t❛ ❝♦♠♣❛tí✈❡❧ ❝♦♠ g s❡ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ r❡❣r❛ ❞♦ ♣r♦❞✉t♦

XhY, Zi=h∇XY, Zi+hY,∇XZi, X, Y, Z ∈X(M).

❯♠❛ ❝♦♥❡①ã♦ ❧✐♥❡❛r ∇ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ❞✐t❛ s✐♠é✲

tr✐❝❛ q✉❛♥❞♦

∇XY − ∇YX = [X, Y]♣❛r❛ t♦❞♦ X, Y ∈X(M).

❊①❡♠♣❧♦ ✶✳✷✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ ❛ ❝♦♥❡①ã♦ ❊✉❝❧✐❞✐❛♥❛ ❞❛❞❛ ♣♦r

∇XY =

X

j

∇X(Yj∂j) =

X

j

(XYj)∂j

é s✐♠étr✐❝❛ ❡ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ♠étr✐❝❛ ❊✉❝❧✐❞✐❛♥❛✳

❚❡♦r❡♠❛ ✶✳✸ ✭▲❡✈✐✲❈✐✈✐t❛✮✳ ❙❡❥❛(M, g)✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳ ❊①✐s✲

t❡ ✉♠❛ ú♥✐❝❛ ❝♦♥❡①ã♦ ❧✐♥❡❛r ∇ ❡♠ M q✉❡ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ g ❡ s✐♠étr✐❝❛✳

❉❡♠♦♥str❛çã♦✳ ❱❡r t❡♦r❡♠❛ ✸✳✻ ❞♦ ❝❛♣ít✉❧♦ ✷ ❞❡ ❬✶✸❪✳

✶✳✷ ❈✉r✈❛t✉r❛s

◆❡st❛ s❡çã♦✱ ❞❡✜♥✐♠♦s ❛s ❝✉r✈❛t✉r❛s ♠❛✐s ❝♦♥❤❡❝✐❞❛s ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡✲ ♠❛♥♥✐❛♥❛✳ ❆s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝✉r✈❛t✉r❛ sã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ♥❛ ✐♥✈❡st✐✲ ❣❛çã♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧✳

❆ ❝✉r✈❛t✉r❛ R ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ M é ❛ ❛♣❧✐❝❛çã♦ R :

X(M)×X(M)×X(M)X(M) ❞❡✜♥✐❞❛ ♣♦r

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✶✸

❊①❡♠♣❧♦ ✶✳✹✳ ❊♠ Rn ❝♦♠ ❛ ♠étr✐❝❛ ❡ ❝♦♥❡①ã♦ ❊✉❝❧✐❞✐❛♥❛✱ t❡♠♦s

∇X∇YZ =

X

j

∇X(Y Zj)∂j =

X

j

XY Zj∂j,

∇Y∇XZ =

X

j

XY Zj∂j.

❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❛s ❞✉❛s ❡①♣r❡ssõ❡s éPk(XY Z

kY XZk)

k=∇[X,Y]Z✳ P♦rt❛♥t♦✱ ♥❡ss❡ ❝❛s♦ R= 0✳

Pr♦♣♦s✐çã♦ ✶✳✺✳ ❆ ❝✉r✈❛t✉r❛ R ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❣♦③❛ ❞❛s

s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✭✐✮ R é ❜✐❧✐♥❡❛r ❡♠ X(M)×X(M)✱ ✐st♦ é✱

R(f X1+gX2, Y1)Z =f R(X1, Y1)Z +gR(X2, Y1),

R(X1, f Y1+gY2)Z =f R(X1, Y1)Z +gR(X1, Y2);

✭✐✐✮ P❛r❛ t♦❞♦ X, Y X(M)✱ ♦ ♦♣❡r❛❞♦r ❝✉r✈❛t✉r❛ R(X, Y) : X(M) X(M) é ❧✐♥❡❛r✱ ✐st♦ é✱

R(X, Y)(Z+W) =R(X, Y)Z +R(X, Y)W, R(X, Y)f Z =f R(X, Y)Z,

♦♥❞❡ f, g C∞(M) X

1, X2, Y1, Y2, Z, W ∈X(M)✳

❉❡♠♦♥str❛çã♦✳ ➱ ❡✈✐❞❡♥t❡ q✉❡ ❘ é ♠✉❧t✐❧✐♥❡❛r s♦❜r❡ R✳ P❛r❛f ∈C∞(M)✱

R(X, f Y)Z =X∇f YZ− ∇f Y∇XZ− ∇[X,f Y]Z

=X(f∇YZ)−f∇Y∇XZ− ∇f[X,Y]+(Xf)YZ

= (Xf)∇YZ +f∇X∇YZ−f∇Y∇XZ −f∇[X,Y]Z −(Xf)∇YZ

=f R(X, Y)Z.

P❡❧❛ ❞❡✜♥✐çã♦R(X, Y)Z =R(Y, X)Z✱ ❡♥tã♦Ré ♠✉❧t✐❧✐♥❡❛r ❡♠X✳ ◗✉❛♥t♦

❛ s❡❣✉♥❞❛ ♣❛rt❡✱ t❡♠♦s

R(X, Y)f Z = X∇Yf Z− ∇Y∇Xf Z− ∇[X,Y]f Z

= X(f∇YZ+ (Y f)Z)− ∇Y(f∇XZ+ (Xf)Z)

−[X, Y]f Zf[X,Y]Z

= fX∇YZ + (Xf)∇YZ + (Y f)∇XZ+X(Y f)Z

−(Y f)∇XZ −(Xf)∇YZ−Y(Xf)Z−[X, Y]f Z−f∇[X,Y]Z

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✶✹

❚❛♠❜é♠✱ ❞❡✜♥✐♠♦s ♦ t❡♥s♦r ❝✉r✈❛t✉r❛ ♣♦r

Rm(X, Y, Z, W) =hR(X, Y)Z, Wi.

❆s ❝♦♠♣♦♥❡♥t❡s ❞❡ Rm ❡♠ ✉♠ r❡❢❡r❡♥❝✐❛❧ {ei} s❡rã♦ r❡♣r❡s❡♥t❛❞❛s ♣♦r

Rm(ei, ej, ek, el) = Rijkl.

❖ t❡♥s♦r ❝✉r✈❛t✉r❛ ❛♣r❡s❡♥t❛ ❛s s❡❣✉✐♥t❡s s✐♠❡tr✐❛s✿ Pr♦♣♦s✐çã♦ ✶✳✻✳ ✭❛✮ Rm(X, Y, Z, W) =−Rm(Y, X, Z, W)❀

✭❜✮ Rm(X, Y, Z, W) =−Rm(X, Y, W, Z)❀

✭❝✮ Rm(X, Y, Z, W) =Rm(Z, W, X, Y)❀

✭❞✮ Rm(X, Y, Z, W) +Rm(Y, Z, X, W) +Rm(Z, X, Y, W) = 0✳

❉❡♠♦♥str❛çã♦✳ ✭❛✮ s❡❣✉❡ ❞❡ R(X, Y)Z = R(Y, X)Z✳ P❛r❛ ♣r♦✈❛r ✭❜✮ é

s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ hR(X, Y)Z, Zi = 0 ♣❛r❛ t♦❞♦ Z✳ ❯s❛♥❞♦ ❛ ❝♦♠♣❛t✐✲

❜✐❧✐❞❛❞❡ ❞❛ ♠étr✐❝❛✱ t❡♠♦s

XY|Z|2 =X(2h∇YZ, Zi) = 2h∇X∇YZ, Zi+ 2h∇XZ,∇YZi ✭✶✳✶✮

Y X|Z|2 =Y(2h∇XZ, Zi) = 2h∇Y∇XZ, Zi+ 2h∇XZ,∇YZi ✭✶✳✷✮

[X, Y]|Z|2 = 2h∇[X,Y]Z, Zi. ✭✶✳✸✮

◗✉❛♥❞♦ s✉❜tr❛í♠♦s ❞❡ ✭✶✳✶✮ ❛s ❡q✉❛çõ❡s ✭✶✳✷✮ ❡ ✭✶✳✸✮✱ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ é ③❡r♦ ❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ ♦❜t❡♠♦s

0 = 2(h∇X∇YZ, Zi − h∇Y∇XZ, Zi − h∇[X,Y]Z, Zi) = 2hR(X, Y)Z, Zi.

❆❣♦r❛ ♣r♦✈❡♠♦s ✭❞✮✱ q✉❡ s❡❣✉❡ ❞✐r❡t♦ ❞❡

R(X, Y)Z +R(Y, Z)X+R(Z, X)Y = 0.

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ R ❡ ❛ s✐♠❡tr✐❛ ❞❛ ❝♦♥❡①ã♦✱ t❡♠♦s

(X∇YZ− ∇Y∇XZ − ∇[X,Y]Z)

+ (Y∇ZX− ∇Z∇YX− ∇[Y,Z]X)

+ (Z∇XY − ∇X∇ZY − ∇[Z,X]Y)

=X(∇YZ− ∇ZY) +∇Y(∇ZX− ∇XZ) +∇Z(∇XY − ∇YX)

− ∇[X,Y]Z − ∇[Y,Z]X− ∇[Z,X]Y

=∇X[Y, Z]− ∇[Y,Z]X+∇Y[Z, X]− ∇[Z,X]Y +∇Z[X, Y]− ∇[X,Y]Z

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✶✺

◆❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✱ ✉s❛♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✳ ❋✐♥❛❧♠❡♥t❡✱ ♠♦str❛r❡✲ ♠♦s ✭❝✮✱ ♣❡❧♦ ✐t❡♠ ✭❞✮✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

hR(X, Y)Z, Wi+hR(Y, Z)X, Wi+hR(Z, X)Y, Wi= 0

hR(Y, Z)W, Xi+hR(Z, W)Y, Xi+hR(W, Y)Z, Xi= 0

hR(Z, W)X, Yi+hR(W, X)Z, Yi+hR(X, Z)W, Yi= 0

hR(W, X)Y, Zi+hR(X, Y)W, Zi+hR(Y, W)X, Zi= 0.

❙♦♠❛♥❞♦ ❛s q✉❛tr♦ ❡q✉❛çõ❡s ❛❝✐♠❛ ❡ ✉s❛♥❞♦ ♦ ✐t❡♠ ✭❜✮ ♦s t❡r♠♦s ❞❛s ❞✉❛s ♣r✐♠❡✐r❛s ❝♦❧✉♥❛s s❡ ❝❛♥❝❡❧❛♠✳ ❆♣❧✐❝❛♥❞♦ ♦ ✐t❡♠ ✭❛✮ ❡ ✭❜✮ ♥♦s t❡r♠♦s r❡s✲ t❛♥t❡s ♦❜t❡♠♦s 2hR(X, Z)W, Yi −2hR(W, Y)X, Zi = 0 q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛

✭❝✮✳

❖❜s❡r✈❛çã♦ ✶✳✼✳ ❆ s✐♠❡tr✐❛ ❡①♣r❡ss❛ ❡♠ ✭❞✮ é ❝❤❛♠❛❞❛ ❞❡ ♣r✐♠❡✐r❛ ✐❞❡♥t✐✲ ❞❛❞❡ ❞❡ ❇✐❛♥❝❤✐✳ ❯s❛♥❞♦ ✭❛✮✲✭❞✮✱ é ❢á❝✐❧ ♠♦str❛r q✉❡ ❛ s♦♠❛ ♦❜t✐❞❛ ♣♦r ✉♠❛ ♣❡r♠✉t❛çã♦ ❝í❝❧✐❝❛ ❞❡ q✉❛✐sq✉❡r três ❡♥tr❛❞❛s ❞❡ Rm t❛♠❜é♠ é ③❡r♦✳ ❊♠

t❡r♠♦s ❞❛s ❝♦♠♣♦♥❡♥t❡s ❛ ♣r♦♣♦s✐çã♦ s❡ tr❛❞✉③ ❡♠✿ ✭❛✮ Rijkl=−Rjikl❀

✭❜✮ Rijkl=−Rijlk❀ ✭❝✮ Rijkl=Rklij❀

✭❞✮ Rijkl+Rjkil+Rkijl = 0✳

❯♠❛ ✈❡③ q✉❡ ✹✲t❡♥s♦r❡s sã♦ ❜❛st❛♥t❡ ❝♦♠♣❧✐❝❛❞♦s✳ ▼✉✐t❛ ✈❡③❡s é út✐❧ ❝♦♥str✉✐r t❡♥s♦r❡s ♠❛✐s s✐♠♣❧❡s q✉❡ r❡s✉♠❡♠ ❛❧❣✉♠❛s ❞❛s ✐♥❢♦r♠❛çõ❡s ❝♦♥✲ t✐❞❛ ♥♦ t❡♥s♦r ❝✉r✈❛t✉r❛✳ ❖ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❡ss❡s t❡♥s♦r❡s é ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐✱ ❞❡♥♦t❛❞♦ ♣♦r Ric✱ ❞❡✜♥✐❞♦ ♣♦r

Ric(X, Y) :=tr❛ç♦(Z R(X, Y)Z).

❖✉ s❡❥❛✱ t♦♠❛♥❞♦ ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❧♦❝❛❧ {ei}✱ t❡♠♦s

Ric(X, Y) =XhR(ei, X)Y, eii.

❆s ❝♦♠♣♦♥❡♥t❡s ❞❡ Ric s❡rã♦ ❞❡♥♦t❛❞❛s ♣♦rRij✱ ✐st♦ é✱

Rij =

X

k

(17)

✶✻

◆♦t❡ q✉❡ ❛s s✐♠❡tr✐❛s ❞♦ t❡♥s♦r ❝✉r✈❛t✉r❛ ❣❛r❛♥t❡♠ q✉❡Ric(X, Y)é s✐♠étr✐❝♦

❝♦♠ r❡s♣❡✐t♦ ❛ X, Y✳ ❙❡ {ei} é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ TpM✱ ❞❡✜♥✐♠♦s ❛

❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❞❡ M ❡♠ p❝♦♠♦

❘(p) = X

i

Ric(ei, ei) =

X

ij

hR(ei, ej)ej, eji.

❆ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦ ♣❡r♠✐t❡ ❞❡✜♥✐r ❛ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧✳

Pr♦♣♦s✐çã♦ ✶✳✽✳ ❙❡❥❛ σTpM ✉♠ s✉❜❡s♣❛ç♦ ❜✐✲❞✐♠❡♥s✐♦♥❛❧ ❡ s❡❥❛♠ X,

Y σ ❞♦✐s ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❊♥tã♦

K(X, Y) = Rm(X, Y, Y, X)

|X|2|Y|2− hX, Yi2 ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦s ✈❡t♦r❡s X, Y σ✳

❉❡♠♦♥str❛çã♦✳ ❖❧❤❛r ♣r♦♣♦s✐çã♦ ✸✳✶ ♥♦ ❝❛♣ít✉❧♦ ✹ ❞❡ ❬✺❪✳

❆ss✐♠✱ ♦ ♥ú♠❡r♦K(X, Y)❡stá ✐♥t✐♠❛♠❡♥t❡ ❛ss♦❝✐❛❞♦ ❛♦ ♣❧❛♥♦σ❞❡TpM✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐rK(p, σ) = K(X, Y)❝♦♠♦ s❡♥❞♦ ❛ ❝✉r✈❛t✉r❛ s❡❝✲

❝✐♦♥❛❧ ❞❡M ❡♠ p✱ ❛♦ ❧♦♥❣♦ ❞❡σ✳ ❖ ❧❡♠❛ ❛❜❛✐①♦ ♠♦str❛ q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦

❞❡ K(σ)✱ ♣❛r❛ t♦❞♦ σ✱ ❞❡t❡r♠✐♥❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ❛ ❝✉r✈❛t✉r❛✳

▲❡♠❛ ✶✳✾✳ ❙❡❥❛♠ R1 ❡ R2 4✲t❡♥s♦r❡s ❞❡✜♥✐❞♦s ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V ❝♦♠ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ❡ ❛♠❜♦s s❛t✐s❢❛③❡♥❞♦ ❛s s✐♠❡tr✐❛s ❞♦ t❡♥s♦r ❝✉r✈❛✲ t✉r❛ ✭❞❡s❝r✐t❛s ♥❛ ♣r♦♣♦s✐çã♦ ✶✳✻✮✳ ❆❧é♠ ❞✐ss♦✱ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s X, Y V✱

R1(X, Y, Y, X)

|X|2|Y|2− hX, Yi2 =

R2(X, Y, Y, X)

|X|2|Y|2− hX, Yi2, ❡♥tã♦ R1 =R2✳

❉❡♠♦♥str❛çã♦✳ ❱❡❥❛ ❧❡♠❛ ✸✳✸ ❞♦ ❝❛♣ít✉❧♦ ✹ ❞❡ ❬✺❪✳

▲❡♠❛ ✶✳✶✵✳ ❙❡❥❛ (M, g) ✉♠❛ n✲✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠ ❝✉r✈❛t✉r❛ s❡❝✲

❝✐♦♥❛❧ ❝♦♥st❛♥t❡ ❝✳ ❖ t❡♥s♦r ❝✉r✈❛t✉r❛ é ❞❛❞♦ ♣❡❧❛ ❢ór♠✉❧❛

Rm(X, Y, Z, W) =c{hX, WihY, Zi − hX, ZihY, Wi}. ✭✶✳✹✮

❊♠ ❝♦♠♣♦♥❡♥t❡s✱

Rijkl =c(gilgjk−gikgjl). ❉❡♠♦♥str❛çã♦✳ ❊♥❝♦♥tr❛✲s❡ ♥❛ ♣á❣✐♥❛ ✶✵✻ ❞❡ ❬✺❪✳

(18)

✶✼

✶✳✸ ❙♦❜r❡ r❡❢❡r❡♥❝✐❛✐s ♠ó✈❡✐s

❆❣♦r❛ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❢❛t♦s s♦❜r❡ r❡❢❡r❡♥❝✐❛✐s ♠ó✈❡✐s✳ ■♥✐❝✐❛r❡♠♦s ❝♦♠ ✉♠ ❢❛t♦ ♣✉r❛♠❡♥t❡ ❛❧❣é❜r✐❝♦✳

▲❡♠❛ ✶✳✶✶ ✭❈❛rt❛♥✮✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ n✳ ❙❡❥❛♠

w1, . . . , wr :V →R, r≤n✱ ❢♦r♠❛s ❧✐♥❡❛r❡s ❞❡ V ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛♠ ❢♦r♠❛s ❧✐♥❡❛r❡s θ1, . . . , θr : V → R s❛t✐s❢❛③❡♥❞♦ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çõ❡s✿

r

X

i=1

wi∧θi = 0.

❊♥tã♦ ❡①✐t❡♠ ♥ú♠❡r♦s r❡❛✐s aij t❛✐s q✉❡

θi =

X

j

aijwj, i, j = 1, . . . r, aij =aji.

❉❡♠♦♥str❛çã♦✳ ❙❡♥❞♦ w1, . . . , wr ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ♣♦❞❡♠♦s ❝♦♠✲ ♣❧❡t❛r ❡ss❛s ❢♦r♠❛s ❡♠ ✉♠❛ ❜❛s❡ {w1, . . . , wr, wr+1, . . . , wn}❞❡V∗ ❡ ❡s❝r❡✈❡r

θi =

X

j

aijwj +

X

l

bilwl, l=r+ 1, . . . , n.

❊♥tã♦

0 =X

i

wi∧θi =

X

i

wi∧

X

j

aijwj+

X

i

wi∧

X

l

bilwl

=X

ij

aijwi∧wj+

X

il

bilwi∧wl

=X

i<j

aijwi∧wj+

X

i

aiiwi∧wi+

X

i>j

aijwi∧wj +

X

il

bilwi∧wl

=X

i<j

aijwi∧wj−

X

i>j

aijwj∧wi+

X

il

bilwi∧wl

=X

i<j

aijwi∧wj−

X

i<j

ajiwi∧wj +

X

il

bilwi∧wl

=X

i<j

(aij −aji)wi∧wj +

X

il

bilwi∧wl.

(19)

✶✽

❙❡❥❛ Mn ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❞✐♠❡♥sã♦ n ✭❞❡ ❛❣♦r❛ ❡♠ ❞✐✲ ❛♥t❡✱ ✉s❛r❡♠♦s ✉♠ í♥❞✐❝❡ s✉♣❡r✐♦r q✉❛♥❞♦ q✉✐s❡r♠♦s ✐♥❞✐❝❛r ❛ ❞✐♠❡♥sã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡✮ ❡ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ∇✳ ❯♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ {e1, . . . , en} s♦❜r❡ ✉♠ ❛❜❡rt♦ U ⊂ M é ❝❤❛♠❛❞♦ ✉♠ r❡❢❡r❡♥❝✐❛❧ ♠ó✈❡❧✳ ❊①✐st❡ ✉♠ ú♥✐❝♦ ❝♦rr❡❢❡r❡♥❝✐❛❧ {w1, . . . , wn}s❛t✐s❢❛③❡♥❞♦ wi(ej) =δij✳

❆s ❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ wij ♥♦ r❡❢❡r❡♥❝✐❛❧ {e1, . . . , en} sã♦ ❛s ✶✲❢♦r♠❛s s✉❛✈❡s ❡♠ U ❞❛❞❛s ♣❛r❛X X(U) ♣♦r

wij(X) =h∇Xei, eji.

❙❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ q✉❡wij sã♦✱ ❞❡ ❢❛t♦✱ ✶✲❢♦r♠❛s s✉❛✈❡s ❡

wij +wji = 0

♣❛r❛ t♦❞♦s 1≤i, j n✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ wii = 0 ♣❛r❛ t♦❞♦ i✳

❆s ❢♦r♠❛s ❞❡ ❝✉r✈❛t✉r❛ ♥♦ r❡❢❡r❡♥❝✐❛❧{e1, . . . , en}sã♦ ❛s ✷✲❢♦r♠❛s Ωij ❡♠ U ❞❛❞❛s✱ ♣❛r❛ X, Y X(M)✱ ♣♦r

Ωij(X, Y) =hR(ei, ej)X, Yi.

❆s s✐♠❡tr✐❛s ❞♦ ♦♣❡r❛❞♦r ❞❡ ❝✉r✈❛t✉r❛ ❣❛r❛♥t❡♠✱ ✐♠❡❞✐❛t❛♠❡♥t❡✱ q✉❡ ❛s Ωij sã♦ ❞❡ ❢❛t♦ ✷✲❢♦r♠❛s ❡♠ U✱ t❛✐s q✉❡Ωij + Ωji = 0✳ ◆♦t❡ ❛✐♥❞❛ q✉❡

Ωij(ek, el) = hR(ei, ej)ek, eli=Rijkl.

▲♦❣♦✱ s❡❣✉❡ q✉❡

Ωij =

1 2

X

l,k

Rijklwk∧wl.

❖ ♣♦♥t♦ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ♠ét♦❞♦ ❞♦ r❡❢❡r❡♥❝✐❛❧ ♠ó✈❡❧ é q✉❡ ❛s ❢♦r♠❛s

wi, wij ❡ Ωij s❛t✐s❢❛③❡♠ ❛s ❝❤❛♠❛❞❛s ❡q✉❛çõ❡s ❞❡ ❡str✉t✉r❛s ❞❡ ❊❧✐❡ ❈❛rt❛♥✳ ❆s ♣r♦♣♦s✐çõ❡s ✶✳✶✷ ❡ ✶✳✶✸ ❛♣r❡s❡♥t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ❡q✉❛çõ❡s ❞❡ ❡str✉t✉r❛s✳

Pr♦♣♦s✐çã♦ ✶✳✶✷✳ ❆s ❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ s❛t✐s❢❛③❡♠

dwi =

X

j

(20)

✶✾

❉❡♠♦♥str❛çã♦✳ P❛r❛ X, Y X(U)✱ s❡❣✉❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✷✳✶✼ ❞❡ ❬✶✷❪ q✉❡

dwi(X, Y) = X(wi(Y))−Y(wi(X))−wi([X, Y])

=XhY, eii −YhX, eii − h[X, Y], eii

=h∇XY, eii+hY,∇Xeii − h∇YX, eii − hX,∇Yeii − h[X, Y], eii

=X

k

(hY, ekihek,∇Xeii − hX, ekihek,∇Yeii)

=X

k

(wk(Y)wik(X)−wk(X)wik(Y))

=X

k

wik∧wk(X, Y).

Pr♦♣♦s✐çã♦ ✶✳✶✸✳ ◆❛s ♥♦t❛çõ❡s ❛❝✐♠❛✱ t❡♠♦s✱ ♣❛r❛ t♦❞♦ 1≤i, j n

dwij =

X

k

wik∧wkj+ Ωij. ✭✶✳✻✮

❉❡♠♦♥str❛çã♦✳ P❛r❛ X, Y X(M)✱ t❡♠♦s q✉❡

dwij(X, Y) = X(wij(Y))−Y(wij(X))−wij([X, Y])

= Xh∇Yei, eji −Yh∇Xei, eji − h∇[X,Y]ei, eji

= h∇X∇Yei− ∇Y∇Yei− ∇[X,Y]ei, eji

+h∇Yei,∇Xeji − h∇Xei,∇Yeji

= hR(X, Y)ei, eji+

X

k

h∇Yei, ekih∇Xej, eki

−X

k

h∇Xei, ekih∇Yej, eki

= Ωij(X, Y) +

X

k

(wik(Y)wjk(X)−wik(X)wjk(Y))

= Ωij(X, Y) +

X

k

wik∧wkj(X, Y).

❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❣❛r❛♥t❡ q✉❡ ❛s ✶−❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ✜❝❛♠ ✐♥t❡✐✲

(21)

✷✵

Pr♦♣♦s✐çã♦ ✶✳✶✹✳ ❙❡❥❛ {e1, . . . , en} ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ s♦❜r❡ ✉♠ ❛❜❡rt♦ U M ❡ s❡❥❛♠ wij ❡ θij ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡ ❢♦r♠❛s s✉❛✈❡s ❡♠ U✱ ❛♥t✐✲s✐♠étr✐❝♦s ✭wij = −wji✮ ❡ s❛t✐s❢❛③❡♥❞♦ ✭✶✳✺✮✳ ❊♥tã♦ wij = θij ♣❛r❛ t♦❞♦s 1≤i, j n✳

❉❡♠♦♥str❛çã♦✳ ❙❡❣✉❡ ❞❡ ✭✶✳✺✮ q✉❡✱ ♣❛r❛ 1≤in✱

X

j

(wij −θij)∧wj = 0.

P♦rt❛♥t♦✱ ♣❡❧♦ ❧❡♠❛ ❞❡ ❈❛rt❛♥ ✶✳✶✶✱ t❡♠✲s❡ ♣❛r❛ ❝❛❞❛ p❡♠ U

wij|p−θij|p =

X

k

hkij(p)wk|p.

❆ss✐♠✱ ✜❝❛♠ ❞❡✜♥✐❞❛s ❛s ❢✉♥çõ❡shk

ij :U →R✱ t❛✐s q✉❡hkij =h j

ik ❡wij−θij =

P

kh k

ijwk ♣❛r❛ t♦❞♦i, j✳ ❖❜s❡r✈❡ q✉❡

X

k

hkjiwk =wji−θji =−(wij −θij) = −

X

k

hkijwk.

▲♦❣♦✱ hk

ij =−hkji ❡

hkij =hkji =hijk =hkji =hjki =hjik =hkij.

➱ ♣♦ssí✈❡❧ ❡st❡♥❞❡r ❛♦s t❡♥s♦r❡s ❛ ♥♦çã♦ ❞❡ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡✳ ❙❡❥❛F

✉♠ ❦✲t❡♥s♦r✱ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❝♦✈❛r✐❛♥t❡ ∇F ❞❡F é ✉♠ (k+ 1)✲t❡♥s♦r ❞❛❞♦

♣♦r

∇F(X1, . . . , Xk, Y) = Y(F(X1, . . . , Xk))−F(∇YX1, . . . , Xk)

− · · · −F(X1, . . . ,∇YXk).

❆s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ❣❛r❛♥t❡♠ q✉❡ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❞❡✜♥❡ ✉♠ (k+ 1)✲t❡♥s♦r✳ ❆s ❝♦♠♣♦♥❡♥t❡s ❞❡ F ❡♠ ✉♠ r❡❢❡r❡♥❝✐❛❧

♦rt♦♥♦r♠❛❧ ❧♦❝❛❧ {e1, . . . , en}s❡rã♦ ❞❡♥♦t❛❞❛s ♣♦r

Fi1...ikj =∇F(ei1, . . . , eik, ej),

♦♥❞❡ i1, . . . , ik, j ∈ {1, . . . , n}✳

Pr♦♣♦s✐çã♦ ✶✳✶✺✳ ◆❛s ♥♦t❛çõ❡s ❛❝✐♠❛✱ t❡♠♦s

X

j

Fi1i2...ikjwj = dFi1...ik+

X

j

Fji2...ikwji1

+X

j

Fi1ji3...ikwji2 +· · ·+ X

j

Fi1...ik

(22)

✷✶

❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡✱ ♣♦r ❡①❡♠♣❧♦

X

j

Fi1i2ji4...ikwji3(ei) = X

j

F(ei1, ei2, ej, ei4, . . . , eik)h∇eiej, ei3i,

♦♥❞❡ i∈ {1, . . . , n}✱ ❝♦♠♦F é ❧✐♥❡❛r ❡ h∇eiej, ei3i=−hej,∇eiei3i✱ t❡♠♦s X

j

Fi1i2ji4...ikwji3(ei) =−F(ei1, ei2, X

j

h∇eiei3, ejiej, ei4, . . . , eik)

=F(ei1, ei2,∇eiei3, ei4, . . . , eik).

P♦rt❛♥t♦✱

X

j

Fi1i2...ikjwj(ei) =

X

j

Fi1i2...ikjδij =Fi1i2...iki =∇F(ei1, . . . , eik, ei)

= ei(F(ei1. . . . , eik))−F(∇eiei1, . . . , eik)− · · · −

F(ei1, . . . ,∇eieik) = dFi1i2...ik(ei) +

X

j

Fji2...ikwji1(ei)

+X

j

Fi1j...ikwji2(ei) +· · ·+ X

j

Fi1...ik−1wjik(ei).

❖❜s❡r✈❛çã♦ ✶✳✶✻✳ P❛r❛ ✉♠ r❡❢❡r❡♥❝✐❛❧ ❣❡♦❞és✐❝♦{e1, . . . , en}❡♠p✱ ♦✉ s❡❥❛✱

e1, . . . , en sã♦ ♦rt♦♥♦r♠❛✐s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡p ❡ ∇eiej|p = 0t❡♠♦s q✉❡ ∇Xej|p =

X

i

∇Xie

iej|p =

X

i

Xi(p)∇eiej|p✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ wij|p = 0 ❡ ❞❛í

❡♠ p✈❛❧❡

Fi1i2...ikj =∇F(ei1, ei2, . . . , eik, ej) =ej(Fi1i2...ik),

q✉❡r ❞✐③❡r✱ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ✉♠ t❡♥s♦r sã♦ ❛s ❞❡r✐✈❛❞❛s ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ss❡ t❡♥s♦r✳

❙❡❥❛ φ ✉♠ ✷✲t❡♥s♦r✱ ❞✐③❡♠♦s q✉❡ φ é s✐♠étr✐❝♦ q✉❛♥❞♦ φ(X, Y) =

φ(Y, X) ♣❛r❛ t♦❞♦ X, Y X(M)✱ ❡♠ t❡r♠♦s ❞❡ ✉♠ r❡❢❡r❡♥❝✐❛❧✱ ✐st♦ s✐❣✲

(23)

❈❛♣ít✉❧♦ ✷

❍✐r♣❡rs✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❡♠

❢♦r♠❛s ❡s♣❛❝✐❛✐s

✷✳✶ ■♠❡rsõ❡s ✐s♦♠étr✐❝❛s

◆❡st❛ s❡çã♦✱ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ t❡r♠✐♥♦❧♦❣✐❛s s♦❜r❡ s✉❜✈❛✲ r✐❡❞❛❞❡s✳

❙❡❥❛♠ (M ,f ge) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❞✐♠❡♥sã♦ m = n+k✱ M

✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦n ❡ ι:M Mf✉♠❛ ✐♠❡rsã♦✳ ❙❡ ❞❡✜♥✐♠♦s ✉♠❛

♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠M ♣♦rg =ι∗eg ✭✐st♦ é✱ ♣❛r❛X, Y T

pM, g(X, Y) =

e

g(ι∗X, ιY)✮ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ιé ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛✳ ❙❡✱ ❛❧é♠ ❞✐ss♦✱

ι é ✐♥❥❡t✐✈❛ t❡♠♦s q✉❡ M é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡Mf✳ ◆❡st❡

❝❛s♦✱ ❝❤❛♠❛♠♦s Mf❞❡ ✈❛r✐❡❞❛❞❡ ❛♠❜✐❡♥t❡✳

❚♦❞♦s ♦s r❡s✉❧t❛❞♦s ❞❡ss❡ ❝❛♣ít✉❧♦ ✈❛❧❡♠ ♣❛r❛ q✉❛❧q✉❡r ✐♠❡rsã♦ ✐s♦♠é✲ tr✐❝❛✱ ✉♠❛ ✈❡③ q✉❡ ♦s r❡s✉❧t❛❞♦s sã♦ ❧♦❝❛✐s ❡ q✉❛❧q✉❡r ✐♠❡rsã♦ é ❧♦❝❛❧♠❡♥t❡ ✉♠ ♠❡r❣✉❧❤♦✳ P♦rt❛♥t♦✱ s❡♠♣r❡ q✉❡ ♥ã♦ ❤♦✉✈❡r ♣❡r✐❣♦ ❞❡ ❝♦♥❢✉sã♦✱ ✐❞❡♥✲ t✐✜❝❛r❡♠♦s p M ❝♦♠ ι(p) ∈ Mf✱ ❝❛❞❛ V TpM ❝♦♠ ι∗(V) ∈ Tι(q)Mf ❡ ❞❡♥♦t❛r❡♠♦s ❛s ♠étr✐❝❛s ❞❡ M ❡Mf❛♣❡♥❛s ♣♦r h , i

❊♠ ❝❛❞❛ ♣♦♥t♦pM✱ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡TpMfs❡♣❛r❛ ♦ ♥❛ s♦♠❛ ❞✐r❡t❛ ♦rt♦❣♦♥❛❧

TpMf=TpM ⊕TpM⊥. ❖ ❝♦♥❥✉♥t♦

T M⊥ = a

p∈M

TpM⊥

é ❝❤❛♠❛❞♦ ❞❡ ✜❜r❛❞♦ ♥♦r♠❛❧✳ ❉❡♥♦t❛r❡♠♦s ♣♦rX(M)⊥♦ ❡s♣❛ç♦ ❞❛s s❡çõ❡s

❞❡ T M⊥✳ ❯♠ r❡❢❡r❡♥❝✐❛❧(E

1, . . . , Em) ♣❛r❛ Mf❡♠ ✉♠ ❛❜❡rt♦ Ue ⊂Mfé ❞✐t♦

(24)

✷✸

❛❞❛♣t❛❞♦ à ✐♠❡rsã♦ s❡ ♦s ♣r✐♠❡✐r♦s n ✈❡t♦r❡s (E1|p, . . . , En|p) ❣❡r❛♠ TpM ❡♠ ❝❛❞❛ pUeM✳ ❙❡❣✉❡ q✉❡ (En+1|p, . . . , Em|p) ❣❡r❛ TpM⊥✳

✷✳✷ ▼❛✐s s♦❜r❡ r❡❢❡r❡♥❝✐❛✐s ♠ó✈❡✐s

❆❣♦r❛✱ r❡❧❛❝✐♦♥❛♠♦s ❛s ❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ❡ ❝✉r✈❛t✉r❛ ❞❛ ✈❛r✐❡❞❛❞❡ ❛♠❜✐❡♥t❡ ❡ ❞❛ s✉❜✈❛r✐❡❞❛❞❡✳ ❆❧é♠ ❞✐ss♦✱ ❞❡✜♥✐♠♦s ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❞❡♠♦♥str❛♠♦s ❛ ❡q✉❛çã♦ ❞❡ ●❛✉ss ✭✷✳✹✮✳

❙❡❥❛♠ Mfn+k ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❡ ι : Mn Mfn+k ✉♠❛ ✐♠❡r✲ sã♦ ✐s♦♠étr✐❝❛✳ P❛r❛ p ∈ M✱ ❝♦♥s✐❞❡r❡ ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❧♦❝❛❧ {e1, . . . , en, en+1, . . . , en+k}❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛Ue ❞❡ p❡♠ Mf❛❞❛♣t❛❞❛ à ✐♠❡r✲ sã♦ ❡ s❡❥❛ U =M Ue✳

❆ss♦❝✐❛❞♦ ❛ ❡st❡ r❡❢❡r❡♥❝✐❛❧✱ t❡♠♦s ♦ ❝♦rr❡❢❡r❡♥❝✐❛❧ ❞✉❛❧ {w1, . . . , wn+k}✳ ❈♦♥✈❡♥❝✐♦♥❛♠♦s ♦s í♥❞✐❝❡s✿

1i, j, k, . . . n, 1A, B, C, . . . n+k, n+ 1 α, β, γ, . . .n+k.

❊♠Ue t❡♠♦s q✉❡ ❛s ❢♦r♠❛s wA, wAB s❛t✐s❢❛③❡♠ ❛s ❡q✉❛çõ❡s ❞❡ ❡str✉t✉r❛

dwA =

X

B

wAB∧wB, wAB+wBA= 0,

dwAB =

X

C

wAC∧wCB +ΩeAB, ΩeAB =

1 2

X

C,D

e

RABCDwC ∧wD. ✭✷✳✶✮

❙❡ X X(U)✱ ❡♥tã♦ wα(X) = hX, eαi = 0✱ ♣♦✐s✱ eα X(U)✳ ▲♦❣♦✱

❝♦♥❝❧✉í♠♦s q✉❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦ ❡♠ U ❡ ❝❛♠♣♦s ❡♠ X(U)

dwi =

X

C

wiC ∧wC =

X

j

wij ∧wj+

X

α

wiα ∧wα =

X

j

wij ∧wj,

♦✉ s❡❥❛✱ ❛s r❡str✐çõ❡s ❞❡st❛s ❢♦r♠❛s ❛ U s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ ✭✶✳✺✮ ❡ ♣♦rt❛♥t♦

❛ ♣r♦♣♦s✐çã♦ ✶✳✶✹ ❣❛r❛♥t❡ q✉❡ ❡ss❛s r❡str✐çõ❡s sã♦✱ ❞❡ ❢❛t♦✱ ❛s ❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ❞❡ M✳ ❆ss✐♠✱ ❛s ❡q✉❛çõ❡s ❞❡ ❡str✉t✉r❛s ❞❡ M sã♦ ❞❛❞❛s ♣♦r

dwi =

X

j

wij ∧wj, wij +wji = 0,

dwij =

X

k

wik∧wkj + Ωij, Ωij =

1 2

X

k,l

(25)

✷✹

❆❧é♠ ❞✐ss♦✱ 0 =dwα =Piwαi∧wi ❡ ♣❡❧♦ ❧❡♠❛ ❞❡ ❈❛rt❛♥ ✶✳✶✶ t❡♠✲s❡

wiα =

X

j

ijwj. ✭✷✳✸✮

P♦rt❛♥t♦✱ ✜❝❛♠ ❞❡✜♥✐❞❛s ❛s ❢✉♥çõ❡s hα

ij : U → R✳ ❱✐st♦ q✉❡ ❛s ❢♦r♠❛s

wiα sã♦ s✉❛✈❡s t❡♠✲s❡ q✉❡ ❛s ❢✉♥çõ❡s hij t❛♠❜é♠ sã♦ s✉❛✈❡s✳ ❆ ♣❛rt✐r ❞❛í✱

❞❡✜♥✐♠♦s ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ✐♠❡rsã♦ ♥❛ ❞✐r❡çã♦ eα ♣♦r

IIα =X

ij

hαijwiwj.

❙❡{e˜1, . . . ,e˜n+k}❢♦r ♦✉tr♦ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❡♠ Ue ❛❞❛♣t❛❞♦ à ✐♠❡r✲ sã♦✱ ❝♦♠˜eα =eα❡e˜j =akjek ✭❛q✉✐ ❡st❛♠♦s ✉s❛♥❞♦ ❛ ❝♦♥✈❡♥çã♦ ❞❡ ❊✐♥st❡✐♥✮✱ ❡♥tã♦ w˜j =akjwk✱

˜

wαi(X) = h∇eX˜eα,˜eii=h∇eXeα, akieki

=akih∇eXeα, eki=akiwαk(X)

❡ ˜hα

ij = ˜wαi(˜ej) =akialjwαk(el) = akialjhαkl✳ P♦rt❛♥t♦✱

˜

IIα =X

ij

˜

ijw˜iw˜j =

X

ij

akialjklariwrasjws =δkrδlshαklwrws

=X

kl

klwkwl = IIα,

♦✉ s❡❥❛✱ IIα ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❡ ❡stá ❣❧♦❜❛❧✲

♠❡♥t❡ ❞❡✜♥✐❞❛✳ ❯♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ι : Mn → Mfn+k é t♦t❛❧♠❡♥t❡

❣❡♦❞és✐❝❛ s❡ IIα≡0 ♣❛r❛ t♦❞♦α✳

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ♦ ❝❛♠♣♦

H =X

α

1

n

X

i

ii

eα.

❉♦ ♠❡s♠♦ ♠♦❞♦✱ s❡ {e˜1, . . . ,e˜n+k} ❢♦r ♦✉tr♦ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❡♠ Ue ❛❞❛♣t❛❞♦ à ✐♠❡rsã♦✱ ❝♦♠ e˜j =akjek ❡ e˜α =aβαeβ ❡♥tã♦

˜

wαi(X) =h∇eX˜eα,e˜ii=h∇eXaβαeβ,e˜ii

=hX(aβα)eβ,˜eii+haβα∇eXeβ, aikeki=aβαa k

iwβk(X),

˜

hαii = ˜wαi(˜ei) = aαβakialiwβk(el) =

X

β,k,l

(26)

✷✺

❋✐♥❛❧♠❡♥t❡✱

˜

H =X

α 1 n X i ˜

ii

˜

eα =

X α,γ 1 n X i,k,l,β

αakialikl

α

= X α,β,γ,k,l 1 na β αa γ αδ

klhβ kleγ =

X

β,γ,k

1

βγhβ kkeγ

=X β 1 n X k

kk

eβ =H.

■st♦ é✱ H ✐♥❞❡♣❡♥❞❡ ❞♦ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❡ ❡stá ❣❧♦❜❛❧♠❡♥t❡ ❞❡✜♥✐❞♦✳

❋✐❝❛ ❛ss✐♠ ❞❡✜♥✐❞♦ ✉♠ ❝❛♠♣♦ H X(M)⊥✱ s❡♥❞♦ ♦ s❡✉ ✈❛❧♦r ❡♠ p

M ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ✈❡t♦r ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❞❡ ι ❡♠ p✳ ❯♠❛ ✐♠❡rsã♦

✐s♦♠étr✐❝❛ ι:MnMfn+k é ♠í♥✐♠❛ s❡H 0✱ ✐✳❡✳✱ s❡P ih

α

ii= 0 ♣❛r❛ t♦❞♦

α✳

❉❛s ❡q✉❛çõ❡s ✭✷✳✶✮ ❡ ✭✷✳✷✮✱ ♦❜t❡♠♦s

1 2

X

kl

Rijklwk∧wl =

X

α

wiα∧wαj+

1 2

X

kl

e

Rijklwk∧wl.

▼❛s

X

α

wiα∧wαj =−

X

α

X

k

ikwk

∧X

l

jlwl

=−1

2

X

kl

X

α

(hαikjliljk)

wk∧wl.

❆ss✐♠✱ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦ ❞❡ ●❛✉ss

Rijkl =Reijkl−

X

α

(hαikjliljk). ✭✷✳✹✮

❆❣♦r❛ ❛♣❧✐❝❛r❡♠♦s ♦ ♠ét♦❞♦ ❞♦ r❡❢❡r❡♥❝✐❛❧ ♠ó✈❡❧ ♣❛r❛ ♠♦str❛r q✉❡ ♦ t♦r♦ ❞❡ ❈❧✐✛♦r❞ é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ♠í♥✐♠❛ ❞❛ ❡s❢❡r❛✳

❊①❡♠♣❧♦ ✷✳✶✳ ❙❡❥❛♠ m ❡ n ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s t❛❧ q✉❡ m < n✱

❞✐③❡♠♦s q✉❡

Mm,n−m =Sm

r

m n

×Sn−m

r

nm n

é ♦ t♦r♦ ❞❡ ❈❧✐✛♦r❞✳ ❙❡ x1 : Sm pmn → Rm+1 ❡ x2 : Sn−m

q

n−m n

→ Rn−m+1 ❞❡♥♦t❛♠ ❛s ✐♠❡rsõ❡s ❝❛♥ô♥✐❝❛s ❡♥tã♦

x= (x1, x2) :Sm

r

m n

×Sn−m

r

nm n

(27)

✷✻

é ✉♠❛ ✐♠❡rsã♦ ❞♦ t♦r♦ ❞❡ ❈❧✐✛♦r❞ ♥❛ ❡s❢❡r❛Sn+1✳ ❙❡rá ♠♦str❛❞♦ q✉❡M m,n−m é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ♠í♥✐♠❛ ❞❛ ❡s❢❡r❛ Sn+1 ❝♦♠ ❛ ♥♦r♠❛ ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛

❢✉♥❞❛♠❡♥t❛❧ ✐❣✉❛❧ ❛ n✳ ❙❡❥❛ f0, f1, . . . , fm ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ♣❛r❛

Rm+1 t❛❧ q✉❡ f

0 = −pmnx1 é ♥♦r♠❛❧ ❛ Sm pmn

❡ s❡❥❛ ϕ0, ϕ1, . . . , ϕm s❡✉ ❝♦rr❡❢❡r❡♥❝✐❛❧ ❞✉❛❧✳ ❙✐♠✐❧❛r♠❡♥t❡✱ ♣❛r❛ Sn−mqn−nm ❡♠ Rn−m+1✱ ❡s❝♦❧❤❡✲

♠♦s ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ fm+1, . . . , fn+1 t❛❧ q✉❡ fn+1 = −pnnmx2 é ♥♦r♠❛ ❛ Sn−mqn−m

n

❡ s❡❥❛ ϕm+1, . . . , ϕn+1 s❡✉ ❝♦rr❡❢❡r❡♥❝✐❛❧ ❞✉❛❧✳ ❙❡❥❛

(ϕAB)A,B=0,1,...,n+1 ❛s ❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ❞♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦Rn+2 ❝♦♠ r❡s✲ ♣❡✐t♦ ❛♦ r❡❢❡r❡♥❝✐❛❧ (ϕA)A=0,1,...,n+1✳ ❊ss❛s ❢♦r♠❛s✱ r❡str✐t❛s ❛ Mm,n−m s❛t✐s✲ ❢❛③❡♠

ϕ0 =ϕn+1= 0,

ϕ0,i =−ϕi,0 =−

r

n

mϕi, i= 1, . . . , m,

ϕj,n+1 =−ϕn+1,j =

r

n

nmϕj, j =m+ 1, . . . , n,

ϕAB =−ϕBA= 0 ♣❛r❛ A = 0,1, . . . , m❡ B =m+ 1, . . . , n+ 1.

✭✷✳✺✮

❚♦♠❛♠♦s ✉♠ ♥♦✈♦ r❡❢❡r❡♥❝✐❛❧ ♦rt♦❣♦♥❛❧ e0, . . . , en+1 ♣❛r❛ Rn+2 ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

e0 =

r

nm n f0+

r

m nfn+1, ei =fi, i= 1, . . . , n,

en+1 =

r

nm n f0−

r

m nfn+1.

❊♥tã♦ e0 é ♥♦r♠❛❧ ❛ Sn+1 ❡ en+1 é ♥♦r♠❛❧ ❛ Mm,n−m✳ ❙❡❥❛ w0, . . . , wn+1 ♦ s❡✉ r❡❢❡r❡♥❝✐❛❧ ❞✉❛❧✳ ❚❡♠♦s

w0 =

r

nm n ϕ0+

r

m nϕn+1, wi =ϕi, i= 1, . . . , n,

wn+1 =

r

nm n ϕ0−

r

m nϕn+1.

(28)

✷✼

r❡♥❝✐❛❧ (wA)sã♦ ❡♥tã♦ ❞❛❞❛s ♣♦r

w0,j =−wj,0 =

r

nm n ϕ0,j+

r

m

nϕn+1,j ♣❛r❛ j = 1, . . . , n, w0,n+1 =−wn+1,0 =−ϕ0,n+1,

wij =ϕij ♣❛r❛ i, j = 1, . . . , n,

wi,n+1 =−wn+1,i =

r

nm n ϕi,0−

r

m

nϕi,n+1 ♣❛r❛ i= 1, . . . , n.

❘❡str✐♥❣✐♥❞♦ ❡ss❛s ❢♦r♠❛s ❛Mm,n−m ❝♦♠ ❛✉①í❧✐♦ ❞❡ ✭✷✳✺✮✱ ♦❜t❡♠♦s

(wAB) =

          

w11 . . . w1m λw1 ✳✳✳ . . . ✳✳✳ 0 ✳✳✳

wm+1 . . . wmm λwm

wm+1m+1 . . . wm+1n µwm+1

0 ✳✳✳ . . . ✳✳✳ ✳✳✳ wnm+1 . . . wnn µwn

−λw1 . . . −λwm −µwm+1 . . . −µwn 0

           ,

♦♥❞❡ µ = −p m

n−m ❡ λ =

q

n−m

m ✳ ❉❛í ♦❜t❡♠♦s q✉❡ ❛s ❝♦♠♣♦♥❡♥t❡s hij ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ sã♦ ❞❛❞❛s ♣♦r

(hij) =

         λ ✳✳✳ λ µ ✳✳✳ µ          .

❖s ❡❧❡♠❡♥t♦s ♥ã♦ ✐♥❞✐❝❛❞♦s sã♦ ♥✉❧♦s✳ ❈❛❧❝✉❧❛♥❞♦ ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ♦❜té♠✲ s❡

H =mλ+ (nm)µ=m

r

nm

m −(n−m)

r

m

nm = 0.

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

S =X

ij

(29)

✷✽

✷✳✸ ❖ ▲❛♣❧❛❝✐❛♥♦ ❞❛ ♥♦r♠❛ ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛

❛♦ q✉❛❞r❛❞♦

❊♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛M ❡①✐t❡♠ ❣❡♥❡r❛❧✐③❛çõ❡s ♥❛t✉r❛✐s ❞♦s ❢❛♠♦✲

s♦s ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s ❡♠R3✿ ❣r❛❞✐❡♥t❡✱ ❞✐✈❡r❣❡♥t❡ ❡ ▲❛♣❧❛❝✐❛♥♦✳ ◆❡st❛ s❡çã♦ ✉s❛♠♦s ♦ ♠ét♦❞♦ ❞♦ r❡❢❡r❡♥❝✐❛❧ ♠ó✈❡❧ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❛ ♥♦r♠❛ ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❛♦ q✉❛❞r❛❞♦✳

❙❡❥❛ f : Mn R ✉♠❛ ❢✉♥çã♦ s✉❛✈❡✱ ♦ ❣r❛❞✐❡♥t❡ ❞❡ f✱ ❞❡♥♦t❛❞♦ ♣♦r

∇f✱ é ♦ ú♥✐❝♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ q✉❡ s❛t✐s❢❛③

h∇f, Xi=X(f), ♣❛r❛ t♦❞♦ X ∈X(M).

❙❡❥❛ X ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ s✉❛✈❡ ❡♠ Mn✳ ❆ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ X é ❛ ❢✉♥çã♦

divX :Mn R✱ ❞❛❞❛ ♣♦r

(divX)(p) = tr{Yp →(∇YX)p}, ♣❛r❛ p∈M

❋✐♥❛❧♠❡♥t❡✱ ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡f✱ ❞❡♥♦t❛❞♦ ♣♦r∆f✱ é ❛ ❢✉♥çã♦∆f = div(f)✳

❱❡❥❛♠♦s ♣r✐♠❡✐r♦ q✉❛✐s ❛s ❡①♣r❡ssõ❡s ❞♦ ❣r❛❞✐❡♥t❡✱ ❞✐✈❡r❣❡♥t❡ ❡ ▲❛♣❧❛❝✐✲ ❛♥♦ ❡♠ t❡r♠♦s ❞❡ ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❧♦❝❛❧✳

Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡❥❛♠ f : Mn R ✉♠❛ ❢✉♥çã♦ s✉❛✈❡✱ X ✉♠ ❝❛♠♣♦

✈❡t♦r✐❛❧ s✉❛✈❡ ❡♠ Mn {e

1, . . . , en} ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❡♠ ✉♠❛ ✈✐✲ ③✐♥❤❛♥ç❛ ❛❜❡rt❛ U M✳ ❊♥tã♦ ❡♠ U t❡♠♦s

✭❛✮ ∇f =X

i

ei(f)ei❀

✭❜✮ divX =X

i

{ei(ai)− h∇eiei, Xi}, ♦♥❞❡ X =

X

i

aiei❀

✭❜✮ ∆f =X

i

{ei(ei(f))−(∇eiei)f}✳

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ ♦ r❡❢❡r❡♥❝✐❛❧ ❢♦r ❣❡♦❞és✐❝♦ ❡♠ pU✱ ❡♥tã♦ ❡♠p t❡♠♦s

✭❜✬✮ divX =ei(ai) ❀

✭❝✬✮ ∆f =X

i

ei(ei(f))✳

❉❡♠♦♥str❛çã♦✳

✭❛✮ P♦❞❡♠♦s ❡s❝r❡✈❡r ∇f = Piaiei ❡♠ U✳ ❙❡♥❞♦ {e1, . . . , en} ✉♠ r❡✲ ❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ t❡♠♦s ai = h∇f, eii = ei(f)✱ ♣♦rt❛♥t♦✱ ∇f =

P

(30)

✷✾

✭❜✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈❡r❣ê♥❝✐❛ ❞❡ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧✱ t❡♠♦s

divX = tr{Y → ∇YX}=

X

i

h∇eiX, eii

=X

i

{eihX, eii − hX,∇eieii}=

X

i

{ei(ai)− hX,∇eieii}.

✭❝✮ P❡❧♦ ✐t❡♠ ✭❛✮ t❡♠♦s q✉❡ ∆f = Piei(f)ei ❡♠ U✳ ❉♦ ✐t❡♠ ✭❝✮ t✐r❛♠♦s q✉❡

∆f =X

i

{ei(ei(f)))− h∇f,∇eieii}=

X

i

{ei(ei(f))−(∇eiei)f}.

❖s ✐t❡♠ ✭❜✬✮ ❡ ✭❝✬✮ s❡❣✉❡♠ ❞❛ ❞❡✜♥✐çã♦ ❞❡ r❡❢❡r❡♥❝✐❛❧ ❣❡♦❞és✐❝♦✳

❖❜s❡r✈❛çã♦ ✷✳✸✳ ❆s ❡q✉❛çõ❡s ❛❝✐♠❛ ❞❡✐①❛♠ ❝❧❛r♦ q✉❡ ∇f✱ divX ❡ ∆f sã♦

s✉❛✈❡s✳

❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❛r✐t♠ét✐❝❛s ❞♦ ❣r❛❞✐❡♥t❡✱ ❞✐✲ ✈❡r❣❡♥t❡ ❡ ▲❛♣❧❛❝✐❛♥♦✳

Pr♦♣♦s✐çã♦ ✷✳✹✳ ❙❡ f, g : Mn R sã♦ ❢✉♥çõ❡s s✉❛✈❡s ❡ X, Y sã♦ ❝❛♠♣♦s ✈❡t♦r✐❛✐s s✉❛✈❡s ❡♠ Mn✱ ❡♥tã♦

✭❛✮ ∇(f +g) = f+g❀

✭❜✮ ∇(f g) =gf +fg❀

✭❝✮ div(X+Y) = divX+ divY❀

✭❞✮ div(f X) =fdivX+h∇f, Xi

✭❡✮ ∆(f g) = g∆f +f∆g+ 2h∇f,∇gi✳

✭❡✬✮ ❊♠ ♣❛rt✐❝✉❧❛r✱

1 2∆(f

2) =ff +

|∇f|2.

❉❡♠♦♥str❛çã♦✳ P❛r❛ ✭❛✮ ❡ ✭❜✮✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ ♣❛r❛ t♦❞♦ Z X(M)

t❡♠♦s

(31)

✸✵

❖ ✐t❡♠ ✭❝✮ é ✐♠❡❞✐❛t♦✳ ◗✉❛♥t♦ ❛ ✭❜✮✱ s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡

div(f X) =X

i

h∇ei(f X), eii=

X

i

hei(f)X+f∇eiX, eii

=X

i

{hX, ei(f)eii+fh∇eiX, eii}

=fdivX+h∇f, Xi,

♦♥❞❡ {e1, . . . , en} é ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛

U M✳ ❋✐♥❛❧♠❡♥t❡✱ ♣❛r❛ ♦ ✐t❡♠ ✭❡✮ ✉s❛♠♦s ♦s ✐t❡♠ ✭❜✮✱ ✭❝✮ ❡ ✭❞✮✱

∆(f g) = div((f g)) = div(gf+fg) = div(gf) + div(fg)

=gdiv(f) +fdiv(g) + 2h∇f,gi=g∆f+f∆g+ 2h∇f,gi.

❆ ♣❛rt✐r ❞❛q✉✐✱ ✈♦❧t❛♠♦s ♥♦ss❛ ❛t❡♥çã♦ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s✳ ❙❡❥❛ι:Mn

f

Mn+1✉♠❛ s✉❜✈❛r✐❡❞❛❞❡✳ ❊♠ ❝❛❞❛ ♣♦♥t♦ ❞❡M ❡①✐st❡♠ ❡①❛t❛♠❡♥t❡ ✷ ✈❡t♦r❡s ✉♥✐tár✐♦s ♥♦r♠❛✐s ❛ M✳ ❙❡M é ♦r✐❡♥tá✈❡❧ ✭♦ q✉❡ ♣♦❞❡♠♦s ❛ss✉♠✐r ♣❛ss❛♥❞♦

❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡M✮ ❡♥tã♦ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠ ú♥✐❝♦ ♥♦r♠❛❧N✳ ❙❡❥❛♠ p✉♠ ♣♦♥t♦ ❞❡M ❡{e1, . . . , en, en+1 =N}✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛Ue ❞❡p❡♠Mf❛❞❛♣t❛❞♦ à ✐♠❡rsã♦✳ ❊♥tã♦✱ ♣❡❧❛s ❡q✉❛çõ❡s ❞❛ s❡çã♦

✷✳✷ t❡r❡♠♦s

wi,n+1 =

X

j

hijwj, hij =hn+1ij =hji,

dwi =

X

j

wij ∧wj,

dwij =

X

k

wik∧wkj+wi,n+1∧wn+1,j+Ωeij. ✭✷✳✻✮

◆❡st❡ ❝❛s♦✱ só ❡①✐t❡✱ ❛ ♠❡♥♦s ❞❡ ♦r✐❡♥t❛çã♦✱ ✉♠❛ ú♥✐❝❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥✲ ❞❛♠❡♥t❛❧ ❡♠ U =Ue M✱ ❛ s❛❜❡r IIn+1 =Pijhijwiwj✱ ❞❡♥♦t❛r❡♠♦s IIn+1✱

s✐♠♣❧❡s♠❡♥t❡✱ ♣♦r h✳ ❆ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❡ ❛ ❝✉r✈❛t✉r❛

♠é❞✐❛ ❞❡ Mn sã♦ ❞❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r✿

K = det(hij)❡ H =

1

n

X

i

hii.

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