❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
❈❯❘❙❖ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❘❡♥❛t♦ ❖❧✐✈❡✐r❛ ❚❛r❣✐♥♦
❆ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s
♠í♥✐♠❛s ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s ✹✲❞✐♠❡♥s✐♦♥❛✐s
❘❡♥❛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♦✳
❚❛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✳✮
❆❣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✳
❘❡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✳
❆❜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), c≤0, ✇❤♦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❡c≤0✱ ✇❡ ♣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❡❞✳
❈♦♥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 ✸✻
■♥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❡✳
✾
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)✭c≤0✮✳ ❆❧é♠ ❞✐ss♦✱ ❛♣❧✐❝❛♠♦s ♦ ♠ét♦❞♦ ❞♦ r❡❢❡r❡♥❝✐❛❧ ♠ó✈❡❧
❈❛♣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, g∈C∞(M)✳
❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s k✲t❡♥s♦r❡s ❡♠ M✱ ❞❡♥♦t❛❞♦ ♣♦r Tk(M)✱ é ✉♠
✶✶
❡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),
✶✷
✭❛✮ ∇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
✶✸
❊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
k−Y 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 Z−f∇[X,Y]Z
= f∇X∇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
✶✹
❚❛♠❜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
✶✺
◆❛ 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
✶✻
◆♦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♦ ✶✳✽✳ ❙❡❥❛ σ⊂TpM ✉♠ 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❣✐♥❛ ✶✵✻ ❞❡ ❬✺❪✳
✶✼
✶✳✸ ❙♦❜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.
✶✽
❙❡❥❛ 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
✶✾
❉❡♠♦♥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❡✐✲
✷✵
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≤i≤n✱
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
✷✶
❉❡♠♦♥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✐❣✲
❈❛♣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, ι∗Y)✮ ❡♥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♦p∈M✱ ♦ ♣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♦
✷✸
❛❞❛♣t❛❞♦ à ✐♠❡rsã♦ s❡ ♦s ♣r✐♠❡✐r♦s n ✈❡t♦r❡s (E1|p, . . . , En|p) ❣❡r❛♠ TpM ❡♠ ❝❛❞❛ p∈Ue∩M✳ ❙❡❣✉❡ 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✿
1≤i, j, k, . . . ≤n, 1≤A, 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
✷✹
❆❧é♠ ❞✐ss♦✱ 0 =dwα =Piwαi∧wi ❡ ♣❡❧♦ ❧❡♠❛ ❞❡ ❈❛rt❛♥ ✶✳✶✶ t❡♠✲s❡
wiα =
X
j
hαijwj. ✭✷✳✸✮
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
˜
hαijw˜iw˜j =
X
ij
akialjhαklariwrasjws =δkrδlshαklwrws
=X
kl
hαklwkwl = 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
hαii
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
✷✺
❋✐♥❛❧♠❡♥t❡✱
˜
H =X
α 1 n X i ˜
hαii
˜
eα =
X α,γ 1 n X i,k,l,β
aβαakialihβkl
aγαeγ
= X α,β,γ,k,l 1 na β αa γ αδ
klhβ kleγ =
X
β,γ,k
1
nδ
βγhβ kkeγ
=X β 1 n X k
hβkk
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:Mn→Mfn+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
hαikwk
∧X
l
hαjlwl
=−1
2
X
kl
X
α
(hαikhαjl−hαilhαjk)
wk∧wl.
❆ss✐♠✱ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦ ❞❡ ●❛✉ss
Rijkl =Reijkl−
X
α
(hαikhαjl−hαilhαjk). ✭✷✳✹✮
❆❣♦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
n−m 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
n−m n
✷✻
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 = −pn−nmx2 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
n−mϕj, 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
n−m n f0+
r
m nfn+1, ei =fi, i= 1, . . . , n,
en+1 =
r
n−m 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
n−m n ϕ0+
r
m nϕn+1, wi =ϕi, i= 1, . . . , n,
wn+1 =
r
n−m n ϕ0−
r
m nϕn+1.
✷✼
r❡♥❝✐❛❧ (wA)sã♦ ❡♥tã♦ ❞❛❞❛s ♣♦r
w0,j =−wj,0 =
r
n−m 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
n−m 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+ (n−m)µ=m
r
n−m
m −(n−m)
r
m
n−m = 0.
❆❧é♠ ❞✐ss♦✱ ❛ ♥♦r♠❛ ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ é ❞❛❞❛ ♣♦r
S =X
ij
✷✽
✷✳✸ ❖ ▲❛♣❧❛❝✐❛♥♦ ❞❛ ♥♦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✐❝♦ ❡♠ p∈U✱ ❡♥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
✷✾
✭❜✮ 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) =g∇f +f∇g❀
✭❝✮ 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) =f∆f +
|∇f|2.
❉❡♠♦♥str❛çã♦✳ P❛r❛ ✭❛✮ ❡ ✭❜✮✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ ♣❛r❛ t♦❞♦ Z ∈ X(M)✱
t❡♠♦s
✸✵
❖ ✐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(g∇f+f∇g) = div(g∇f) + div(f∇g)
=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.