❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
P❛r❡s ❞❡ ❈♦❞❛③③✐ ❡♠ ❙✉♣❡r❢í❝✐❡s ❞❡ ❱❛r✐❡❞❛❞❡s
❍♦♠♦❣ê♥❡❛s
♣♦r
❲❡❧✐♥t♦♥ ❞❡ ❖❧✐✈❡✐r❛ ●✐♠❛r❡③
❇r❛sí❧✐❛
Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)
Gp
Gimarez, Welinton de Oliveira
Pares de Codazzi em Superfícies de Variedades Homogêneas / Welinton de Oliveira Gimarez;
orientador João Paulo dos Santos. -- Brasília, 2016. 111 p.
Dissertação (Mestrado - Mestrado em Matemática) --Universidade de Brasília, 2016.
1. Pares de Codazzi. 2. Variedades homogêneas. 3. Conjectura de Milnor. 4. Curvatura Gaussiana
❆❣r❛❞❡❝✐♠❡♥t♦s
▼❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛ t♦❞♦s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ê①✐t♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❡ ❡♠ ❡s♣❡❝✐❛❧✿
✲ ➚ ❉❡✉s ♣♦r ♠❡ ♣❡r♠✐t✐r ❝❤❡❣❛r ❛té ❛q✉✐✳
✲ ❆♦s ♠❡✉s ♣❛✐s✱ ❆❞❤❡♠❛r ❡ ▲✉✐③❛✱ ♣❡❧❛s ♦r❛çõ❡s✱ ❛♣♦✐♦ ❡ ❡♥s✐♥❛♥❞♦✲♠❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❝♦♥str✉çã♦ ❡ ❝♦❡rê♥❝✐❛ ❞❡ ♠❡✉s ♣ró♣r✐♦s ✈❛❧♦r❡s✳
✲ ❆♦ ♠❡✉ ✐r♠ã♦✱ ❲❡❧t♦♥✱ ♠❡✉ ❡t❡r♥♦ ❛♠✐❣♦✱ q✉❡ s♦✉❜❡ ❡♥t❡♥❞❡r ♠✐♥❤❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ❛✉sê♥✲ ❝✐❛s✳
✲ ❆♦ ♦r✐❡♥t❛❞♦r✱ Pr♦❢❡ss♦r ❉r✳ ❏♦ã♦ P❛✉❧♦ ❞♦s ❙❛♥t♦s✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❝♦♥st❛♥t❡ ✐♥❝❡♥t✐✈♦✱ s❡♠♣r❡ ✐♥❞✐❝❛♥❞♦ ❛ ❞✐r❡çã♦ ❛ s❡r t♦♠❛❞❛✳ ❊ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ ❝♦♥✜❛♥ç❛ q✉❡ ❞❡♣♦s✐t♦✉ ❡♠ ♠✐♠✳ ▼✐♥❤❛ ❡t❡r♥❛ ❣r❛t✐❞ã♦✳
✲ ❆♦ t♦❞♦s ♠❡✉s ♣r♦❢❡ss♦r❡s✱ ♣❡❧♦s ✈❛❧✐♦s♦s ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ♠❡ ❢♦r♥❡❝❡r❛♠✳
✲ ❆♦s ❛♠✐❣♦s✱ ♣❡❧♦ ♣r❛③❡r ❞❡ s✉❛s ❛♠✐③❛❞❡s✱ ❝♦♥✈❡rs❛s✱ tr♦❝❛s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❛❥✉❞❛ ❡ ❝♦♥s❡❧❤♦s✳
✲ ❆♦ ❈◆Pq✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ à ❡st❡ tr❛❜❛❧❤♦✳ ❊♥✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s✳✳✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ ❞❡ ♣❛r❡s ❞❡ ❈♦❞❛③③✐ ❡♠ s✉♣❡r❢í❝✐❡s ❞❡ ✈❛r✐❡❞❛❞❡s ❤♦♠♦❣ê♥❡❛s tr✐❞✐♠❡♥s✐♦♥❛✐s✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ r❡s✉❧t❛❞♦ ❛❜str❛t♦ ♣❛r❛ ♣❛r❡s ❞❡ ❈♦❞❛③③✐ ❡♠ s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ♥ã♦✲♣♦s✐t✐✈❛ ❡ ♦ ❛♣❧✐❝❛♠♦s ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s ❞♦ t✐♣♦ ❊✜♠♦✈ ❡ ▼✐❧♥♦r ♣❛r❛ s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ♥❛s ❢♦r♠❛s ❡s♣❛❝✐❛✐s ♥ã♦✲ ❊✉❝❧✐❞✐❛♥❛s✳ P❛r❛ s✉♣❡r❢í❝✐❡s ❞❡ ❡s♣❛ç♦s ♣r♦❞✉t♦✱ ❛ té❝♥✐❝❛ ❞❡ ♣❛r❡s ❞❡ ❈♦❞❛③③✐ é ✉t✐❧✐③❛❞❛ ♥❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❞♦ t✐♣♦ ▲✐❡❜♠❛♥♥ ♣❛r❛ s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉s✲ s✐❛♥❛ ❝♦♥st❛♥t❡✳ ◆♦s ❡s♣❛ç♦s ❤♦♠♦❣ê♥❡♦s E(κ, τ),❝♦♠ τ 6= 0,❛♣r❡s❡♥t❛♠♦s ✉♠ ♣❛r ❞❡ ❈♦❞❛③③✐ ❞❡✜♥✐❞♦ s♦❜r❡ s✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡✱ ❝✉❥❛ s✉❛ (2,0)✲♣❛rt❡ é ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡
❆❜r❡s❝❤✲❘♦s❡♥❜❡r❣✳
P❛❧❛✈r❛s✲❈❤❛✈❡s✿ ♣❛r❡s ❞❡ ❈♦❞❛③③✐❀ ✈❛r✐❡❞❛❞❡s ❤♦♠♦❣ê♥❡❛s❀ ❝♦♥❥❡❝t✉r❛ ❞❡ ▼✐❧♥♦r❀ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❧✐♠✐t❛❞❛❀ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡❀ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣r❡s❡♥t ❛ st✉❞② ♦❢ ❈♦❞❛③③✐ ♣❛✐rs ♦♥ s✉r❢❛❝❡s ♦❢ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❤♦♠♦❣❡♥❡♦✉s ♠❛♥✐❢♦❧❞s✳ ■♥✐t✐❛❧❧②✱ ✇❡ ♣r❡s❡♥t ❛♥ ❛❜str❛❝t r❡s✉❧t ❛❜♦✉t ❈♦❞❛③③✐ ♣❛✐rs ♦♥ ❝♦♠♣❧❡t❡ s✉r❢❛❝❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ ●❛✉ss ❝✉r✈❛t✉r❡ ❛♥❞ ✇❡ ❛♣♣❧② ✐t t♦ ♦❜t❛✐♥ ❊✜♠♦✈ ❛♥❞ ▼✐❧♥♦r✬s t②♣❡ r❡s✉❧ts ❢♦r ❝♦♠♣❧❡t❡ s✉r❢❛❝❡s ✐♥ ♥♦♥✲❊✉❝❧✐❞✐❛♥ s♣❛❝❡ ❢♦r♠s✳ ❋♦r s✉r❢❛❝❡s ✐♥ ♣r♦❞✉❝t s♣❛❝❡s✱ t❤❡ t❡❝❤♥✐q✉❡ ♦❢ ❈♦❞❛③③✐ ♣❛✐rs ✐s ❛♣♣❧✐❡❞ ✐♥ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ▲✐❡❜♠❛♥♥✬s t②♣❡ r❡s✉❧t ❢♦r ❝♦♠♣❧❡t❡ s✉r❢❛❝❡s ✇✐t❤ ❝♦♥st❛♥t ●❛✉ss✐❛♥ ❝✉r✈❛t✉r❡✳ ■♥ t❤❡ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡s E(κ, τ),✇✐t❤τ 6= 0✱ ✇❡ ♣r❡s❡♥t ❛ ❈♦❞❛③③✐ ♣❛✐r ❞❡✜♥❡❞ ♦♥ s✉r❢❛❝❡s ✇✐t❤ ❝♦♥st❛♥t ♠❡❛♥ ❝✉r✈❛t✉r❡✱ ✇❤♦s❡(2,0)✲♣❛rt ✐s t❤❡ ❆❜r❡s❝❤✲
❘♦s❡♥❜❡r❣ ❞✐✛❡r❡♥t✐❛❧✳
❑❡②✇♦r❞s✿ ❈♦❞❛③③✐ ♣❛✐rs❀ ❤♦♠♦❣❡♥❡♦✉s ♠❛♥✐❢♦❧❞s❀ ▼✐❧♥♦r✬s ❝♦♥❥❡❝t✉r❡❀ ●❛✉ss✐❛♥ ❝✉r✈❛t✉r❡ ❧✐♠✐t❡❞❀ ❝♦♥st❛♥t ●❛✉ss✐❛♥ ❝✉r✈❛t✉r❡❀ ❝♦♥st❛♥t ♠❡❛♥ ❝✉r✈❛t✉r❡
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ Pr❡❧✐♠✐♥❛r❡s ✺
✶✳✶ ❈♦♥❥✉♥t♦ ❞❡ P❛r❡s ❋✉♥❞❛♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❊q✉❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ❚❡♥s♦r ❞❡ ❈♦❞❛③③✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✶ ❙❡❣✉♥❞❛ ❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✷ ❊q✉❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹✳✸ ❍✐♣❡rs✉♣❡r❢í❝✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✺ P❛r❡s ❞❡ ❈♦❞❛③③✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✻ ❙✉♣❡r❢í❝✐❡ ❞❡ ❘✐❡♠❛♥♥ ❡ ❋♦r♠❛ ◗✉❛❞rát✐❝❛ ❍♦❧♦♠♦r❢❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✻✳✶ ❆ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❍♦♣❢ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✷ ❙✉♣❡r❢í❝✐❡s ❈♦♠♣❧❡t❛s ❝♦♠ ❈✉r✈❛t✉r❛ ❊①trí♥s❡❝❛ ◆ã♦✲♣♦s✐t✐✈❛ ❡♠ H3 ❡ S3 ✸✹ ✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷ ❉❡✜♥✐çõ❡s ❡ ❘❡s✉❧t❛❞♦s ❇ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸ ❯♠❛ ❙♦❧✉çã♦ P❛r❝✐❛❧ ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ▼✐❧♥♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✹ P❛r❡s ❞❡ ❈♦❞❛③③✐ ❡♠ ❙✉♣❡r❢í❝✐❡s ❈♦♠♣❧❡t❛s ❝♦♠ ❈✉r✈❛t✉r❛ ◆ã♦✲♣♦s✐t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✹✳✶ ❙✉♣❡r❢í❝✐❡s ❈♦♠♣❧❡t❛s ❝♦♠ K≤0❡♠H3 ❡S3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✸ P❛r❡s ❞❡ ❈♦❞❛③③✐ ♥♦s ❊s♣❛ç♦s Pr♦❞✉t♦s ✹✼
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷ ❘❡s✉❧t❛❞♦s ❇ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷✳✶ ❊q✉❛çõ❡s ❞❡ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✸ ❙✉♣❡r❢í❝✐❡s ❞❡ ❘❡✈♦❧✉çã♦ ❈♦♠♣❧❡t❛s ❞❡ ❈✉r✈❛t✉r❛ ❈♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✸✳✹ P❛r ❞❡ ❈♦❞❛③③✐ ♣❛r❛ ❙✉♣❡r❢í❝✐❡s ❞❡ ❈✉r✈❛t✉r❛ ❈♦♥st❛♥t❡ ❡♠H2×R ❡S2×R ✳ ✳ ✺✹ ✸✳✺ ❚❡♦r❡♠❛ ❚✐♣♦ ▲✐❡❜♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✹ P❛r❡s ❞❡ ❈♦❞❛③③✐ ♥♦ ❊s♣❛ç♦ ❍♦♠♦❣ê♥❡♦ E(κ, τ) ✻✽
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✷ ❘❡s✉❧t❛❞♦s ❇ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✷✳✶ ❱❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ❍♦♠♦❣ê♥❡❛s E(κ, τ)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✷✳✷ ❙✉♣❡r❢í❝✐❡s ■♠❡rs❛s ❡♠E(κ, τ)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✹✳✷✳✸ ❍✲❙✉♣❡r❢í❝✐❡s ■♠❡rs❛s ❡♠ E(κ, τ) ❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❆❜r❡s❝❤✲❘♦s❡♥❜❡r❣ ✳ ✳ ✼✹ ✹✳✸ ❉✐❢❡r❡♥❝✐❛❧ ❆❜r❡s❝❤✲❘♦s❡♥❜❡r❣ ❡ P❛r❡s ❞❡ ❈♦❞❛③③✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✸✳✶ ❍✲❙✉♣❡r❢í❝✐❡s ■♠❡rs❛s ❡♠ E(κ, τ) ❝♦♠τ = 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✸✳✷ ❍✲❙✉♣❡r❢í❝✐❡s ■♠❡rs❛s ❡♠ E(κ, τ) ❝♦♠τ 6= 0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✹✳✹ ❈❧❛ss✐✜❝❛çã♦ ♣❛r❛ ❍✲❙✉♣❡r❢í❝✐❡s ■♠❡rs❛s ❡♠E(κ, τ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼
■♥tr♦❞✉çã♦
❯♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ M é ❞✐t❛ ❤♦♠♦❣ê♥❡❛ s❡ ❞❛❞♦s p, q ∈M ❡①✐st❡ ✉♠❛ ✐s♦♠❡tr✐❛
❞❡ M q✉❡ ❧❡✈❛ p ❡♠ q✳ ❖s ❡s♣❛ç♦s ❤♦♠♦❣ê♥❡♦s tr✐❞✐♠❡♥s✐♦♥❛✐s s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦s sã♦ ❝❧❛s✲
s✐✜❝❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞✐♠❡♥sã♦ ❞♦ ❣r✉♣♦ ❞❡ ✐s♦♠❡tr✐❛✱ q✉❡ ♣♦❞❡ s❡r ✸✱ ✹✱ ♦✉ ✻✳ ◆♦ ❝❛s♦ ❞❛ ❞✐♠❡♥sã♦ s❡r ✻✱ ❛ ✈❛r✐❡❞❛❞❡ é ✉♠❛ ❢♦r♠❛ ❡s♣❛❝✐❛❧✱ ♦✉ s❡❥❛✱ ♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ R3✱ ❛ ❡s❢❡r❛ tr✐✲ ❞✐♠❡♥s✐♦♥❛❧ S3 ♦✉ ♦ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦ tr✐❞✐♠❡♥s✐♦♥❛❧H3✳ ❙❡ ❛ ❞✐♠❡♥sã♦ ❞♦ ❣r✉♣♦ ❞❡ ✐s♦♠❡tr✐❛ ❢♦r ✸✱ ❛ ✈❛r✐❡❞❛❞❡ ♣♦ss✉✐ ❛ ❣❡♦♠❡tr✐❛ ❞♦ ❣r✉♣♦ ❞❡ ▲✐❡ ❙♦❧3✳ ◗✉❛♥❞♦ ❛ ❞✐♠❡♥sã♦ é ✹✱ ❛ ✈❛r✐❡❞❛❞❡ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❢❛♠í❧✐❛2−♣❛râ♠❡tr♦s✱κ, τ ∈R3, κ−4τ26= 0✱ ♥❛ q✉❛❧ ❞❡♥♦t❛♠♦s ♣♦rE(κ, τ). ❊st❛s ✈❛r✐❡❞❛❞❡s ❝♦rr❡s♣♦♥❞❡♠ ❛♦ ❡s♣❛ç♦ ♣r♦❞✉t♦M2(κ)×Rq✉❛♥❞♦ κ6= 0, τ = 0♦♥❞❡ M2 é ✉♠❛ ✈❛r✐❡❞❛❞❡ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛ ❞❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ κ✳ ◗✉❛♥❞♦τ 6= 0,t❡♠♦s ♦ ❡s♣❛ç♦
❍❡✐s❡♥❜❡r❣ ◆✐❧3 s❡κ= 0,♦ ❡s♣❛ç♦ ❞❡ r❡❝♦❜r✐♠❡♥t♦ ✉♥✐✈❡rs❛❧ ❞♦ ❣r✉♣♦ ❧✐♥❡❛r ❡s♣❡❝✐❛❧P SL^2(R) q✉❛♥❞♦ κ <0 ❡ ❛ ❡s❢❡r❛ ❞❡ ❇❡r❣❡r SB3(κ, τ)q✉❛♥❞♦ κ >0.
❯♠ ❢❛t♦ ✐♠♣♦rt❛♥t❡ ❞♦s ❡s♣❛ç♦s E(κ, τ) é q✉❡ ❡❧❡s sã♦ ✉♠❛ s✉❜♠❡rsã♦ ❘✐❡♠❛♥♥✐❛♥❛ s♦❜r❡ M2 ✭s✉♣❡r❢í❝✐❡ ❞❡ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡✮ ❝♦♠τ ❛ ❝✉r✈❛t✉r❛ ❞♦ ✜❜r❛❞♦✱ s❡♥❞♦ ❛s ✜❜r❛s tr❛❥❡tór✐❛s ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❞❡ ❑✐❧❧✐♥❣ ✉♥✐tár✐♦ ❞❡✜♥✐❞♦ ❡♠ E(κ, τ)✳
◆❡st❛ ❞✐ss❡rt❛çã♦✱ ❡st✉❞❛♠♦s ❛ ❡q✉❛çã♦ ❝❧áss✐❝❛ ❞❡ ❈♦❞❛③③✐ ❛ ♣❛rt✐r ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ❛❜str❛t♦✱ ❡ ❛ ✉t✐❧✐③❛♠♦s ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❛♥❛❧ít✐❝❛ ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s ❣❧♦❜❛✐s ♣❛r❛ s✉✲ ♣❡r❢í❝✐❡s ❡♠ ❞✐❢❡r❡♥t❡s ❛♠❜✐❡♥t❡s✳
❆ ❡q✉❛çã♦ ❈♦❞❛③③✐ ♣❛r❛ ✉♠❛ s✉♣❡r❢í❝✐❡ Σ✐♠❡rs❛ ♥♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ tr✐❞✐♠❡♥s✐♦♥❛❧ R3 é ❞❛❞❛ ♣♦r
∇XSY − ∇YSX−S[X, Y] = 0, X, Y ∈X(Σ).
❆q✉✐∇é ❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲ ❈✐✈✐t❛ ❞❛ ♣r✐♠❡✐r❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧I ❡Sé ♦ ♦♣❡r❛❞♦r ❢♦r♠❛✱
❞❡✜♥✐❞♦ ♣♦r II(X, Y) =I(SX, Y), ❡♠ q✉❡ II é ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ s✉♣❡r❢í❝✐❡✳ ❆
■♥tr♦❞✉çã♦ ✷
❈♦❞❛③③✐✱ ✐st♦ é✱ ♣❛r❡s ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s r❡❛✐s (I, II) ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ q✉❡ s❛t✐s❢❛③❡♠ ❛
❡q✉❛çã♦ ❞❡ ❈♦❞❛③③✐✱ ♦♥❞❡ I é ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✱ ❡ ❡①♣❧♦r❛r ❛s ♣♦ssí✈❡✐s ❝♦♥s❡q✉ê♥❝✐❛s
s✉♣❡r❢í❝✐❡✳
■♥s♣✐r❛❞♦s ♣❡❧♦s tr❛❜❛❧❤♦s ❞❡ ❏♦sé ●á❧✈❡③✱ ❆♥t♦♥✐♦ ▼❛rtí♥❡③ ❡ ❏♦sé ❚❡r✉❡❧ ❡♠ ❬✷✶❪ ❡ ❬✷✵❪✱ ❡st✉❞❛r❡♠♦s ♣❛r❡s ❞❡ ❈♦❞❛③③✐ ❡♠ s✉♣❡r❢í❝✐❡s ❡ s❡✉s ✐♥✈❛r✐❛♥t❡s ❛ss♦❝✐❛❞♦s✱ t❛✐s ❝♦♠♦ ❛s ❝✉r✈❛✲ t✉r❛s ♣r✐♥❝✐♣❛✐s✱ ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛✱ ❛ ❝✉r✈❛t✉r❛ ❡①trí♥s❡❝❛ ❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❍♦♣❢✳ ❱❡r❡♠♦s ❛ ♣❛rt✐r ❞❡st❡s tr❛❜❛❧❤♦s q✉❡ ❛ t❡♦r✐❛ ❞❡ ♣❛r❡s ❞❡ ❈♦❞❛③③✐ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ✈❛♥t❛❥♦s❛✱ ✉♠❛ ✈❡③ q✉❡✱ ❛ ♣❛rt✐r ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❛❜str❛t♦ ♣❛r❛ ♣❛r❡s ❞❡ ❈♦❞❛③③✐ ❡♠ s✉♣❡r❢í❝✐❡s✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s r❡s✉❧t❛❞♦s t✐♣♦s ▼✐❧♥♦r ❡ ❊✜♠♦✈ ♣❛r❛ s✉♣❡r❢í❝✐❡s ❡♠ H3 ❡ S3.
❆♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠✱ ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✱ ❞♦✐s ❧❡♠❛s ❞❡ ❚✳ ❑❧♦t③ ▼✐❧♥♦r ❬✷✺❪✱ ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛✱ q✉❡ r❡❧❛❝✐♦♥❛♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣❛r❡s ❞❡ ❈♦❞❛③③✐ ❝♦♠ ❝✉r✈❛t✉r❛ ❡①trí♥s❡❝❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ♦✉ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❍♦♣❢ ❤♦❧♦♠♦r❢❛✳
❱❡r❡♠♦s q✉❡ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦Σé ✐♠❡rs❛ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ❡♠ ✉♠❛ ❢♦r♠❛
❡s♣❛❝✐❛❧ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ❡q✉❛çã♦ ❞❡ ❈♦❞❛③③✐ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❛r❛♥t✐r q✉❡ ❛ ✉s✉❛❧ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❍♦♣❢ é ❤♦❧♦♠♦r❢❛ s♦❜r❡ s✉♣❡r❢í❝✐❡ ♥♦ ❡s♣❛ç♦ ❤♦♠♦❣ê♥❡♦ E(κ, τ)✳ ◆♦ ❡♥t❛♥t♦✱ ❏✳❆✳❆❧❡❞♦✱ ❏✳▼✳ ❊s♣✐♥❛r ❡ ❏✳❆✳●á❧✈❡③✱ ✭✈❡r ❬✺❪✮✱ ♦❜t✐✈❡r❛♠ ✉♠ ♥♦✈♦ ♣❛r ❈♦❞❛③③✐ ❣❡♦♠étr✐❝♦ (A, II) s♦❜r❡
s✉♣❡r❢í❝✐❡ ❡♠ M2×R t❛❧ q✉❡ ❛(2,0)−♣❛rt❡ ❞❡II ❝♦♠ r❡s♣❡✐t♦ ❛ ❡str✉t✉r❛ ❝♦♥❢♦r♠❡ ❞❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ A é ❤♦❧♦♠♦r❢❛✳
P❛r❛ s✉♣❡r❢í❝✐❡s ♥♦s ❡s♣❛ç♦s ♣r♦❞✉t♦✱ ✉t✐❧✐③❛r❡♠♦s ❛ té❝♥✐❝❛ ❞❡ ♣❛r❡s ❞❡ ❈♦❞❛③③✐ ♣❛r❛ ❡st✉❞❛r s✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡✱ s❡❣✉✐♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞❡ ❆❧❡❞♦✱ ❊s♣✐♥❛r ❡ ●❛❧✈❡③ ❡♠ ❬✺❪ ❡ ❬✻❪✳ ❈❛r❛❝t❡r✐③❛r❡♠♦s ❛s ❝♦♥❞✐çõ❡s ❞❡ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ♣❛r❛ s✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ ❡✱ ✉♠❛ ✈❡③ ❡st❛❜❡❧❡❝✐❞❛s✱ ❞❡✜♥✐r❡♠♦s ✉♠ ♥♦✈♦ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ ♥❛ s✉♣❡r❢í❝✐❡ ❡♠ t❡r♠♦s ❞❛ ♣r✐♠❡✐r❛✱ s❡❣✉♥❞❛ ❢♦r♠❛s ❢✉♥❞❛♠❡♥t❛✐s ❡ ❞❛ ❢✉♥çã♦ ❛❧t✉r❛✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ t❛❧ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ é ✉♠ ♣❛r ❞❡ ❈♦❞❛③③✐ ❞❡ ❝✉r✈❛t✉r❛ ❡①trí♥s❡❝❛ ❝♦♥st❛♥t❡ q✉❛♥❞♦ ❛ s✉♣❡r❢í❝✐❡ t❡♠ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡✳
❱❡r❡♠♦s q✉❡ ❡ss❡ ♣❛r ❞❡ ❈♦❞❛③③✐ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦❜t❡r ✉♠ ❚❡♦r❡♠❛ ❞♦ t✐♣♦ ▲✐❡❜♠❛♥♥ ♣❛r❛ s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ❞❡ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ K >0 ❡♠ H2×R ♦✉ K > 1 ❡♠ S2×R✱ ❝❛r❛❝t❡r✐③❛♥❞♦✲❛s✱ ❝♦♠♦ ❛s ❡s❢❡r❛s r♦t❛❝✐♦♥❛❧♠❡♥t❡ s✐♠étr✐❝❛s ❞❡ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡✳
P❛r❛ ♦s ❡s♣❛ç♦sE(κ, τ),r❡ss❛❧t❛♠♦s q✉❡ ❛♣❡s❛r ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❞✐❢❡r❡♥❝✐❛❧ q✉❛❞rát✐❝❛ ❤♦❧♦♠♦r❢❛ s♦❜r❡ s✉♣❡r❢í❝✐❡ ❡♠ E(κ, τ)✱ τ 6= 0✱ ♥ã♦ ❡①✐st✐❛ ♣❛r ❞❡ ❈♦❞❛③③✐ ❡♠ s✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ✐♠❡rs❛s ♥❡ss❡s ❡s♣❛ç♦s✳ ■♥s♣✐r❛❞♦s ♣❡❧♦s ❡st✉❞♦s ❞❡ ❊s♣✐♥❛r ❡ ❚r❡❥♦s✱ ❡♠ ❬✶✺❪✱ ✈❡r❡♠♦s q✉❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❆❜r❡s❝❤✲❘♦s❡♥❜❡r❣ t❡♠ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❡♠ t❡r♠♦s ❞❡ ✉♠ ♣❛r ❞❡ ❈♦❞❛③③✐ ❞❡✜♥✐❞♦ ❡♠ H−s✉♣❡r❢í❝✐❡s ♥❡st❡ ❡s♣❛ç♦ q✉❛♥❞♦ τ 6= 0.
❖r❣❛♥✐③❛♠♦s ♦ tr❛❜❛❧❤♦ ❡♠ ❝✐♥❝♦ ❝❛♣ít✉❧♦s✱ ❝♦♠♦ s❡❣✉❡✳
◆♦ ❝❛♣ít✉❧♦ ✶✱ ✜①❛r❡♠♦s ❛s ♥♦t❛çõ❡s✱ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥♦ ❞❡❝♦rr❡r ❞♦ t❡①t♦✳ ❉❡✜♥✐♠♦s ♣❛r ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡✱ ❜❡♠ ❝♦♠♦ ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❡ ❡①trí♥s❡❝❛ ❡♠ r❡❧❛çã♦ ❛ ♣❛r❛♠❡tr✐③❛çõ❡s ❧♦❝❛✐s ❡ ✐s♦tér♠✐❝❛s✳ ❱❡r❡♠♦s ♦ t❡♥s♦r ❞❡ ❈♦❞❛③③✐ ❛ss♦❝✐❛❞♦ ❛ ✉♠ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ (I, II). ❊st❡ ♣♦r s✉❛ ✈❡③✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ t❡♦r✐❛
■♥tr♦❞✉çã♦ ✸
❈♦❞❛③③✐ ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡✳ ❱❡r❡♠♦s ❛ r❡❧❛çã♦ ❡♥tr❡ ♣♦♥t♦s ✉♠❜í❧✐❝♦s ❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❍♦♣❢✳ ❯♠❛ ♦✉tr❛ ❢❡rr❛♠❡♥t❛ ❜❛st❛♥t❡ ✉t✐❧✐③❛❞❛✱ s❡rá ❛♥á❧✐s❡ ❝♦♠♣❧❡①❛✱ ♥❛ q✉❛❧ ♥♦s ❛✉①✐❧✐❛rá ♥❛ t❡♦r✐❛ ❞❡ s✉♣❡r❢í❝✐❡ ❞❡ ❘✐❡♠❛♥♥ ❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s ❤♦❧♦♠♦r❢❛s✳
❖ ❝❛♣✐t✉❧♦ ✷ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞♦s tr❛❜❛❧❤♦s ❬✷✶❪ ❡ ❬✷✵❪✳ ■♥st✐❣❛❞♦ ♣❡❧♦s tr❛❜❛❧❤♦s ❞❡ ❍✐❧✲ ❜❡rt✱ ❛♣r❡s❡♥t❛r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❊✜♠♦✈✱ ♥♦ q✉❛❧ ♥♦s ❞✐③ q✉❡ ♥❡♥❤✉♠❛ s✉♣❡r❢í❝✐❡ ♣♦❞❡ s❡r ✐♠❡rs❛ ♥♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ t❛❧ q✉❡ ♥❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛✱ s❡❥❛ ❝♦♠♣❧❡t❛ ❡ t❡♥❤❛ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛K ≤const <0.❱❡r❡♠♦s t❛♠❜é♠ q✉❡ ❛♣❡s❛r ❞♦s ♣r♦❣r❡ss♦s s✐❣♥✐✜✲
❝❛t✐✈♦s ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠ ❝✉r✈❛t✉r❛ ♥❡❣❛t✐✈❛✱ q✉❡stõ❡s ✐♠♣♦rt❛♥t❡s s✉❣❡r✐❞❛s ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❍✐❧❜❡rt ♣❡r♠❛♥❡❝❡♠ s❡♠ r❡s♣♦st❛ ❛té ❤♦❥❡✳ ❊♥tr❡ ♦s ♣r♦❜❧❡♠❛s ❛❜❡rt♦s ♠❡♥✲ ❝✐♦♥❛r❡♠♦s ❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❏♦♥❤ ▼✐❧♥♦r✳ ❊st✉❞❛r❡♠♦s ❛ s❡❣✉✐♥t❡ s♦❧✉çã♦ ♣❛r❝✐❛❧ ❞❛ ❝♦♥❥❡❝t✉r❛ ❞❡ ▼✐❧♥♦r✱ ♦❜t✐❞❛ ♣♦r ❙♠②t❤ ❡ ❳❛✈✐❡r ✭✈❡r ❬✷✵❪✱[✷✾]✮✿
ψ : Σ −→ R3 ✉♠❛ s✉♣❡r❢í❝✐❡ ✐♠❡rs❛ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ❞❡ ❝✉r✈❛t✉r❛ ♥ã♦ ♣♦s✐t✐✈❛✳ ❙❡ ✉♠❛ ❞❛s s✉❛s ❢✉♥çõ❡s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s k2
i s❛t✐s❢❛③ ki2 ≥const >0, ❡♥tã♦ ψ(Σ) é
✉♠ ❝✐❧✐♥❞r♦ ❣❡♥❡r❛❧✐③❛❞♦ ❡♠R3✳
❱❡r❡♠♦s ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♥t❡r❡ss❛♥t❡ ❞❡st❡ r❡s✉❧t❛❞♦ ♣❛r❛ s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ❝♦♠ ❝✉r✲ ✈❛t✉r❛ ❞❡ ●❛✉ss ♥ã♦✲♣♦s✐t✐✈❛ ❡♠ R3,❝♦♠ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❧✐♠✐t❛❞❛ ❛ ♣❛rt✐r ❞❡ ③❡r♦✳ ❖ t❡♦r❡♠❛ ❛❝✐♠❛ ♣r♦♣õ❡ ✉♠❛ s♦❧✉çã♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ❝♦♥❥❡❝t✉r❛ ❞❡ ▼✐❧♥♦r✱ ♥♦ ❝❛s♦ K ≤ 0✳ ❆♣r❡s❡♥t❛✲
r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ✉s❛♥❞♦ r❡s✉❧t❛❞♦s ❞♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦✱ ❝♦♠♦ ♦ t❡♦r❡♠❛ ❙❛❝❦st❡❞❡r✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛r❡♠♦s s✉❛s ❡①t❡♥sã♦ ♣❛r❛ ❛s ❢♦r♠❛s ❡s♣❛❝✐❛✐s ♥ã♦✲❊✉❝❧✐❞✐❛♥❛s✱ ✉t✐❧✐③❛♥❞♦ ♣❛r❛ ✐ss♦ ♦ t❡♦r❡♠❛ ❞❡ ❍✉❜❡r ❡ ❛ t❡♦r✐❛ ❞❡ ♣❛r❡s ❞❡ ❈♦❞❛③③✐✳ ◆❛ ✈❡r❞❛❞❡✱ ♠♦str❛r❡♠♦s q✉❡✿
◆❡♥❤✉♠❛ s✉♣❡r❢í❝✐❡ ♣♦❞❡ s❡r ✐♠❡rs❛ ❡♠ H3 ✭r❡s♣✳ S3✮ s❡ é ❝♦♠♣❧❡t❛ ♥❛ ♠étr✐❝❛ ❘✐✲ ❡♠❛♥♥✐❛♥❛ ✐♥❞✉③✐❞❛✱ ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ K ≤ −1 ✭r❡s♣✳ K ≤ const < 0✮ ❡
✉♠❛ ❞❡ s✉❛s ❝✉r✈❛t✉r❛ ♣r✐♥❝✐♣❛✐s ki s❛t✐s❢❛③❡♥❞♦ ki2≥ε >0, ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ε
♣♦s✐t✐✈❛✳
❙♦❜r❡ ❛ ❣❡♦♠❡tr✐❛ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠ ✜♥s ❞❡ ❝✉r✈❛t✉r❛ ♥ã♦✲♣♦s✐t✐✈❛ ❡♠ ❢♦r♠❛s ❡s♣❛❝✐❛✐s ♥ã♦✲❊✉❝❧✐❞✐❛♥❛ ♣r♦✈❛r❡♠♦s✿
❈♦♥s✐❞❡r❡ ✉♠❛ ✐♠❡rsã♦ ❝♦♠♣❧❡t❛ ❡♠ H3 ✭r❡s♣✳ S3✮ q✉❡ ❢♦r❛ ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♠✲ ♣❛❝t♦ s❛t✐s❢❛ç❛✿
• ❛ ❝✉r✈❛t✉r❛ ●❛✉ss K ≤ −1 ✭r❡s♣✳ K ≤const <0✮ ❡
• k2
i ≥ε >0, ε∈R ♦♥❞❡ ki é ✉♠❛ ❞❡ s✉❛s ❝✉r✈❛t✉r❛ ♣r✐♥❝✐♣❛✐s✳
❊♥tã♦✱ ❛ ✐♠❡rsã♦ t❡♠ ❝✉r✈❛t✉r❛ t♦t❛❧ ✜♥✐t❛ ❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ t❡♠ t♦♣♦❧♦❣✐❛ ✜♥✐t❛ ❡ ár❡❛ ✜♥✐t❛✳
■♥tr♦❞✉çã♦ ✹
♣❛r❛ s✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ ❡①✐st❡ ✉♠ ♣❛r ❞❡ ❈♦❞❛③③✐ r❡❧❛❝✐♦♥❛❞♦ à s✉❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛✱ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❢✉♥çã♦ ❛❧t✉r❛✳ ❆ ♣❛rt✐r ❞❡st❡ ♥♦✈♦ ♣❛r✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ❚❡♦r❡♠❛ ❞♦ t✐♣♦ ▲✐❡❜♠❛♥♥ ♣❛r❛ s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ❞❡ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ ❡♠ H2×R❡S2×R❝✉❥❛ ❞❡♠♦♥str❛çã♦ é ❜❛s❡❛❞❛ ♥❛ ✈❡rsã♦ ❛❜str❛t❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ▲✐❡❜♠❛♥♥ ❡ ♥♦s r❡s✉❧t❛❞♦s ❞❛❞♦s ♣♦r ❆❧❡❞♦✱ ❊s♣✐♥❛r ❡ ●á❧✈❡③ ❡♠ ❬✻❪ ❡ ❊s♣✐♥❛r ❬✶✹❪✳ ❆❧é♠ ❞✐ss♦✱ ❡st❡ ♣❛r t❡♠ ❝✉r✈❛t✉r❛ ❡①trí♥s❡❝❛ ❝♦♥st❛♥t❡✱ ♦ q✉❡ ♥♦s ❞á ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❢♦r♠❛ q✉❛❞rát✐❝❛ ❤♦❧♦♠♦r❢❛ ♣❛r❛ s✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ❡♠H2×R❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ ♠❛✐♦r q✉❡ ✉♠ ❡♠S2×R.❊ss❡ ❢❛t♦✱ ❝♦♠♦ ♦❝♦rr❡ ♣❛r❛ s✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ❬✶❪✱ é ❛ ❝❤❛✈❡ ♣❛r❛ ♦❜t❡r ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ❞❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡✳ ❆ss✐♠✱ ♦❜t❡r❡♠♦s ✉♠ ❚❡♦r❡♠❛ ❞♦ t✐♣♦ ❞❡ ▲✐❡❜♠❛♥♥✱ ♦✉ s❡❥❛✱ ♣r♦✈❛r❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛ ❞❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ ❡♠ H2×R❡ ✉♠❛ ú♥✐❝❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛ ❞❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ♠❛✐♦r q✉❡ ✶ ❡♠ S2×R✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠❡tr✐❛s ❞♦ ❡s♣❛ç♦ ❛♠❜✐❡♥t❡✳ ❊st❛s s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s sã♦ ♣r❡❝✐s❛♠❡♥t❡ ❛s s✉♣❡r❢í❝✐❡s ❞❡ r❡✈♦❧✉çã♦✳
❖ ❝❛♣ít✉❧♦ ✹✱ ❞❡❞✐❝❛r❡♠♦s ❛♦ ❡st✉❞♦ ❞♦ ❛rt✐❣♦ ❬✶✺❪✳ ■♥t❡r♣r❡t❛r❡♠♦s ♦ ♣❛r ❞❡ ❈♦❞❛③③✐ ❞❛ ❞✐❢❡r❡♥❝✐❛❧ ❆❜r❡s❝❤✲❘♦s❡♥❜❡r❣ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡♦♠étr✐❝❛s✳ ❉✐s❝✉t✐r❡♠♦s ♦ ❝❛s♦ ❝♦♥❤❡❝✐❞♦ ❞❛ H−s✉♣❡r❢í❝✐❡ ❡♠ ✉♠ ❡s♣❛ç♦ ♣r♦❞✉t♦ M2×R✳ ❉❡♣♦✐s✱ ♦❜t❡r❡♠♦s ✉♠ ♣❛r ❈♦❞❛③③✐ ❣❡♦♠é✲ tr✐❝♦ ❛ss♦❝✐❛❞♦ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❆❜r❡s❝❤✲❘♦s❡♥❜❡r❣ s♦❜r❡ q✉❛❧q✉❡rH−s✉♣❡r❢í❝✐❡ ✐♠❡rs❛ ❡♠E(κ, τ) q✉❛♥❞♦ τ 6= 0✳ ❋✐♥❛❧✐③❛r❡♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦
♣❛r❛ H−s✉♣❡r❢í❝✐❡s ❡♠E(κ, τ).
❈❛♣ít✉❧♦
1
Pr❡❧✐♠✐♥❛r❡s
◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐r❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ✉t✐❧✐③❛❞♦s ♥♦ r❡st❛♥t❡ ❞♦ tr❛❜❛❧❤♦✳ ❉❡✜♥✐✲ r❡♠♦s ✉♠ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡✱ ✐♥s♣✐r❛❞♦s ♣❡❧❛s ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ❢♦r♠❛s q✉❛❞rát✐❝❛s ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡♠ R3✳ ❉✐s❝✉t✐r❡♠♦s ❛s r❡♣r❡s❡♥t❛çõ❡s ❞❡ ✉♠ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ ❝♦♠ r❡s♣❡✐t♦ ❛ s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥❛ s✉♣❡r❢í❝✐❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♣❛râ♠❡✲ tr♦s ✐s♦tér♠✐❝♦s✱ ❛ss✐♠ ❝♦♠♦ ❛s r❡♣r❡s❡♥t❛çõ❡s ❞❛ ❝♦♠♣❧❡①✐✜❝❛çã♦ ❞❡ t❛✐s ♣❛r❡s ❡♠ r❡❧❛çã♦ ❛♦s ♣❛râ♠❡tr♦s ❝♦♠♣❧❡①♦s ❛ss♦❝✐❛❞♦s✳ ❉❡✜♥✐r❡♠♦s ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛ ❡ ●❛✉ss✐❛♥❛ ❞❡ ✉♠ ♣❛r ❢✉♥✲ ❞❛♠❡♥t❛❧ ❡ ♦❜t❡♠♦s s✉❛s ❡①♣r❡ssõ❡s ❡♠ t❡r♠♦s ❞❡ ♣❛râ♠❡tr♦s ❧♦❝❛✐s✳ ❉✐s❝✉t✐r❡♠♦s ❛ ♥♦çã♦ ❞❡ ♣♦♥t♦ ✉♠❜í❧✐❝♦ ❞❡ ✉♠ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ ❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❍♦♣❢ ❞♦ ♣❛r✳
❇❛s❡❛❞♦ ❡♠ ❬✹❪ ❡ ❬✶✹❪✱ ❝♦♠❡ç❛r❡♠♦s ❧✐st❛♥❞♦ ✈ár✐♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❡ r❡s✉❧t❛❞♦s q✉❡ ✉s❛r❡♠♦s ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✳ ■r❡♠♦s s✉♣♦r q✉❡ ❛ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ é s❡♠♣r❡ C∞ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r
Σ✉♠❛ s✉♣❡r❢í❝✐❡ ♦r✐❡♥t❛❞❛✳ ❉❡t❛❧❤❡s ❛❞✐❝✐♦♥❛✐s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✾❪✱ ❬✽❪✱ ❬✶✾❪✱ ❬✷✷❪✱ ❬✷✹❪
❡ ❬✷✼❪✳
✶✳✶ ❈♦♥❥✉♥t♦ ❞❡ P❛r❡s ❋✉♥❞❛♠❡♥t❛✐s
❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❙❡❥❛♠ Q(Σ) ♦ ❝♦♥❥✉♥t♦ ❞❡ ❢♦r♠❛s ❜✐❧✐♥❡❛r❡s s✐♠étr✐❝❛s s♦❜r❡ Σ✱ q✉❡ ✐❞❡♥✲
t✐✜❝❛r❡♠♦s ❝♦♠♦ s✉❛ ❢♦r♠❛ q✉❛❞rát✐❝❛ ❛ss♦❝✐❛❞❛ ❡ R(Σ) ♦ ❝♦♥❥✉♥t♦ q✉❡ ❞❡✜♥❡ ✉♠❛ ♠étr✐❝❛
❘✐❡♠❛♥♥✐❛♥❛✳ ❆ ✉♠ ♣❛r (I, II) t❛❧ q✉❡ (I, II)∈ R(Σ) × Q(Σ)≡ P(Σ) ❝❤❛♠❛r❡♠♦s ♣❛r ❢✉♥✲
❞❛♠❡♥t❛❧ s♦❜r❡ Σ✳ ❚♦❞♦ ❝♦♥❥✉♥t♦ P(Σ)❝❤❛♠❛r❡♠♦s ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s ❢✉♥❞❛♠❡♥t❛✐s✳
❊①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐❥❡t✐✈❛ ❡♥tr❡ Q(Σ) ❡ ♦ ❝♦♥❥✉♥t♦ S(Σ,h,i) ❞❡ ❡♥❞♦♠♦r✜s♠♦
❛✉t♦❛❞❥✉♥t♦ ❞❡X(Σ)✱ ❝♦♠ r❡s♣❡✐t♦ à ♠étr✐❝❛ h,i ≡I ✜①❛❞❛✱ ✐st♦ é✱
S(Σ,h,i) ={S:X(Σ)−→X(Σ);hSX, Yi=hX, SYi, ∀ X, Y ∈X(Σ)}.
❆ss✐♠✱ ❛ss♦❝✐❛❞❛ ❛ S∈ S(Σ,h,i) ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❢♦r♠❛ q✉❛❞rát✐❝❛✱ IIS✱ ♣♦r
✶✳✷ ❊q✉❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s ✻
❡ ❞❛❞❛ II∈ Q(Σ)✱ ♣♦❞❡♠♦s ❞❡✜♥✐rS ∈ S(Σ,h,i)♣♦r
II(X, Y) =hSX, Yi, X, Y ∈X(Σ),
S ❛ q✉❛❧ ❝❤❛♠❛♠♦s ♦♣❡r❛❞♦r ❢♦r♠❛✳
P♦rt❛♥t♦✱ s❡ ✜①❛r♠♦s h,i ∈ R(Σ)✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❡♥❞♦♠♦r✜s♠♦ ❛✉t♦❛❞❥✉♥t♦ ♦✉ ❢♦r♠❛
q✉❛❞rát✐❝❛ ♣❛r❛ ❞❡✜♥✐r ✉♠ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ ✉s❛♥❞♦ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❡♥tr❡Q(Σ)❡S(Σ,h,i)✳ ❊♥tã♦✱
✉♠ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ s♦❜r❡ Σ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❝♦♥s✐❞❡r❛r ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ h,i ❡♠ Σ ❡
✉♠ ❡♥❞♦♠♦r✜s♠♦ ❛✉t♦❛❞❥✉♥t♦ (❝♦♠ r❡s♣❡✐t♦ ❛h,i) ❞❡ X(Σ)✳
❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ ❞✐st✐♥t❛ ❡♠ ❝❛❞❛ ❝❛s♦✿
• P❛r❛ ❢♦r♠❛s q✉❛❞rát✐❝❛s ✉s❛r❡♠♦s (I, II);
• P❛r❛ ❡♥❞♦♠♦r✜s♠♦ ❛✉t♦❛❞❥✉♥t♦✱ ❛ ♥♦t❛çã♦ s❡rá (h,i, S).
❉❛❞♦ ✉♠ ♣❛r ❢✉♥❞❛♠❡♥t❛❧(I, II)∈ P(Σ)✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛H✱ ❛ ❝✉r✈❛t✉r❛
❡①trí♥s❡❝❛K(I, II)≡K❡ ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐ski✱i= 1,2✱ ❞❡(I, II)❝♦♠♦ ❛ ♠❡t❛❞❡ ❞♦ tr❛ç♦✱
♦ ❞❡t❡r♠✐♥❛♥t❡ ❡ ♦s ✈❛❧♦r❡s ♣ró♣r✐♦s ❞♦ ❡♥❞♦♠♦r✜s♠♦ S✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✜♥✐♠♦s t❛♠❜é♠ ❛
❝✉r✈❛t✉r❛ ❛ss✐♠étr✐❝❛ ❞❡ (I, II) ♣♦r
q=q(I, II) =H2−K(I, II) = (k1−k2)
2
4 .
❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❉✐r❡♠♦s q✉❡ ♦ ♣❛r ❢✉♥❞❛♠❡♥t❛❧ (I, II) ≡(h,i, S) é ✉♠❜í❧✐❝♦ ❡♠ p∈ Σ ✭♦✉
p ∈ Σ é ♣♦♥t♦ ✉♠❜í❧✐❝♦ ❞♦ ♣❛r (I, II)✱ s❡ II é ♣r♦♣♦r❝✐♦♥❛❧ à I ❡♠ p✱ ✐st♦ é✱ II =αI ❡♠ p✱
♣❛r❛ ❛❧❣✉♠ α r❡❛❧✳
✶✳✷ ❊q✉❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s
➱ ❝♦♥✈❡♥✐❡♥t❡ ❝♦♥t✐♥✉❛r ❡s❝r❡✈❡♥❞♦ ♣❛rt❡ ❞♦ q✉❡ ✈✐♠♦s ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✳ ❙❡❥❛♠ (x, y) ♣❛râ♠❡tr♦s ❧♦❝❛✐s ❡♠ t♦r♥♦ ❞❡U ❞❡ Σ✱(I, II)∈ P(Σ)❡s❝r❡✈❡♠♦s
I = Edx2+ 2F dxdy+Gdy2 ✭✶✳✶✮ II = edx2+ 2f dxdy+gdy2,
✶✳✷ ❊q✉❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s ✼
Pr♦♣♦s✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛♠ (U, ϕ= (x, y)) ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❧♦❝❛❧ ❡♠ Σ ❡ (I, II) ∈ P(Σ) t❛❧
q✉❡
I = Edx2+ 2F dxdy+Gdy2 II = edx2+ 2f dxdy+gdy2.
❊♥tã♦✱ ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❡ ❛ ❝✉r✈❛t✉r❛ ❡①trí♥s❡❝❛ sã♦ ❞❛❞❛s ♣♦r
H =H(I, II) = Eg+Ge−2F f 2(EG−F2)
K =K(I, II) = eg−f
2
EG−F2. ❆❧é♠ ❞✐ss♦✱ ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❞❡ (I, II) sã♦
k1 = H+
p
H2−K
k2 = H−
p
H2−K ❉❡♠♦♥str❛çã♦✳ ❱❡r ❬✽❪✳
❈♦♠♦(Σ, g) é ✉♠❛ s✉♣❡r❢í❝✐❡ ♠✉♥✐❞❛ ❞❡ ✉♠❛ ♠étr✐❝❛ g,❡♥tã♦ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❝♦♥❡①ã♦ ∇ ❛✜♠ s♦❜r❡ Σ✱ ❛ s❛❜❡r✱ ❛ ❝♦♥❡①ã♦ ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ❞❡I✳
❉❡♥♦t❛r❡♠♦s ♣♦r {∂x, ∂y} ♦s ❝❛♠♣♦s ❝♦♦r❞❡♥❛❞♦s ❛ss♦❝✐❛❞♦s à ♣❛r❛♠❡tr✐③❛çã♦✳ ❊♥tã♦ ❛s
❢✉♥çõ❡sΓk
ij ❞❡✜♥✐❞❛s ❡♠U ♣♦r
∇∂x∂x = Γ111∂x+ Γ211∂y
∇∂x∂y = Γ112∂x+ Γ212∂y
∇∂y∂y = Γ122∂x+ Γ222∂y
sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❝♦♥❡①ã♦ ∇ ❡♠ U ❛ss♦❝✐❛❞❛ ❛♦s ♣❛râ♠❡tr♦s (x, y)✱ ❝❤❛♠❛❞♦s sí♠❜♦❧♦s ❞❡
❈❤r✐st♦✛❡❧ ❛ss♦❝✐❛❞♦s à ♠étr✐❝❛I ❡ sã♦ ❞❛❞♦s ♣♦r
Γ111 = 1
2(EG−F2)(GEx−2F Fx+F Ey)
Γ211 = − 1
2(EG−F2)(EEy−2EFx+F Ex)
Γ112 = 1
2(EG−F2)(GEy−F Gx) ✭✶✳✷✮
Γ212 = 1
2(EG−F2)(EGx−F Ey)
Γ122 = − 1
2(EG−F2)(GGx−2GFy+F Gy)
Γ222 = 1
✶✳✸ ❚❡♥s♦r ❞❡ ❈♦❞❛③③✐ ✽
P♦r ♦✉tr♦ ❧❛❞♦✱ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❛ss♦❝✐❛❞♦ à ❝♦♥❡①ã♦ ∇ é ♦ t❡♥s♦r ❞❡ ❝✉r✈❛t✉r❛ R ❞❛❞♦ ♣♦r✿
R(X, Y)Z =∇Y∇XZ− ∇X∇YZ+∇[X,Y]Z, X, Y, Z∈X(Σ), q✉❡ ♠❡❞❡ ♦ q✉❛♥t♦ ❞❡✐①❛ ❛ ♠étr✐❝❛ ❞❡ s❡r ♣❧❛♥❛✱ ✐st♦ é✱ t❡♠ ❝✉r✈❛t✉r❛ ③❡r♦✳
❆ss♦❝✐❛❞❛ ❛♦ t❡♥s♦r ❞❡ ❝✉r✈❛t✉r❛R ❡stá ❛ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❞❡I✱ q✉❡ é ❞❛❞❛ ♣♦r K(I)(p) = hR(Xp, Yp)Xp, Ypi
||Xp×Yp||2 ∀
p∈S,
♦♥❞❡ Xp✱Yp ∈❚pS sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡
||Xp×Yp||=
q
||Xp||2||Yp||2− hXp, Ypi2.
◆♦t❡♠♦s q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡K(I) ♥ã♦ ❞❡♣❡♥❞❡ ❞♦s ✈❡t♦r❡s ❡s❝♦❧❤✐❞♦s✱ ❡ s✐♠ ❞♦s ♣♦♥t♦s ♦♥❞❡
❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦✳ ❘❡❢❡r✐♠♦s ❛ K(I) ❝♦♠♦ ❛ ❝✉r✈❛t✉r❛ ✐♥trí♥s❡❝❛ ♦✉ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✱
q✉❛♥❞♦ ❡st✐✈❡r♠♦s tr❛❜❛❧❤❛♥❞♦ ❝♦♠ s✉♣❡r❢í❝✐❡s✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✷ ✭❱❡r ❬✾❪✮✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠ ❝✉r✈❛t✉r❛ s❡❝✲
❝✐♦♥❛❧ ❝♦♥st❛♥t❡ K (1,0,−1). ❊♥tã♦ M é ✐s♦♠étr✐❝❛ ❛ M /f Γ, ♦♥❞❡ Mf é Sn ✭s❡ K = 1✮✱ Rn
✭s❡ K = 0✮ ♦✉ Hn ✭s❡ K = −1✮✱ Γ é ✉♠ s✉❜❣r✉♣♦ ❞♦ ❣r✉♣♦ ❞❛s ✐s♦♠❡tr✐❛s ❞❡ Mf q✉❡ ♦♣❡r❛
❞❡ ♠♦❞♦ ♣r♦♣r✐❛♠❡♥t❡ ❞❡s❝♦♥tí♥✉♦ ❡♠ M ,f ❡ ❛ ♠étr✐❝❛ ❞❡ M /f Γ é ❛ ✐♥❞✉③✐❞❛ ♣❡❧♦ r❡❝♦❜r✐♠❡♥t♦
π :Mf−→M /f Γ.
✶✳✸ ❚❡♥s♦r ❞❡ ❈♦❞❛③③✐
❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❉❛❞♦ (h,i, S) ∈ P(Σ)✱ ♦ t❡♥s♦r ❞❡ ❈♦❞❛③③✐ ❛ss♦❝✐❛❞♦ ❛ S é ❛ ❛♣❧✐❝❛çã♦ TS :X(Σ)×X(Σ)−→X(Σ)❞❛❞❛ ♣♦r
TS(X, Y) =∇XSY − ∇YSX−S[X, Y] ; X, Y ∈X(Σ).
❱❡❥❛♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❚❡♥s♦r ❞❡ ❈♦❞❛③③✐ q✉❡ s❡❣✉❡♠ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦✳
Pr♦♣♦s✐çã♦ ✶✳✸✳✶✳ ❙❡❥❛ h,i ∈ R(Σ). ❊♥tã♦✿
✶✳ TS é ❛♥t✐ss✐♠étr✐❝♦❀
✷✳ TS é ♠✉❧t✐❧✐♥❡❛r ❡♠ C∞(Σ)✱ ✐st♦ é✱
TS(f1X1+f2X2) = f1TS(X1, Y) +f2TS(X2, Y)
TS(X, f1Y1+f2Y2) = f1TS(X, Y1) +f2TS(X, Y2)
✶✳✸ ❚❡♥s♦r ❞❡ ❈♦❞❛③③✐ ✾
✸✳ ❉❛❞♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s X, Y ∈X(Σ)❡ f ∈ C∞(Σ),t❡♠♦s
Tf S(X, Y) =f TS(X, Y) +X(f)SY −Y(f)SX.
❉❡♠♦♥str❛çã♦✳ ✶✳ ❙❡❥❛♠X, Y ∈X(Σ).❈♦♠♦[X, Y] =−[Y, X]✱ ❡♥tã♦
TS(X, Y) = ∇XSY − ∇YSX−S[X, Y] = −(∇YSX− ∇XSY +S[X, Y]) = −(∇YSX− ∇XSY −S[Y, X]) = TS(Y, X).
✷✳ ❙❡❥❛♠X1, X2, Y ∈X(Σ)❡ f1, f2 ∈ C∞✳ ❚❡♠♦s q✉❡
TS(f1X1+f2X2, Y) =∇f1X1+f2X2SY − ∇YS(f1X1+f2X2)−S[f1X1+f2X2, Y].
▼❛s
∇f1X1+f2X2 =f1∇X1SY +f2∇X2SY,
∇YS(f1X1+f2X2) = ∇Y(f1SX1+f2SX2)
= f1∇YSX1+Y(f1)SX1+f2∇YSX2+Y(f2)SX2, ❡
S[f1X1+f2X2, Y] = S[f1X1, Y] +S[f2X2, Y]
= S(f1[X1, Y]−Y(f1)X1) +S(f2[X2, Y]−Y(f2)X2)
= f1S[X1, Y]−Y(f1)SX1+f2S[X2, Y]−Y(f2)SX2. ▲♦❣♦✱
TS(f1X1+f2X2, Y) =f1Ts(X1, Y) +f2Ts(X2, Y). ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♦❜t❡♠♦s
✶✳✸ ❚❡♥s♦r ❞❡ ❈♦❞❛③③✐ ✶✵
✸✳ ❙❡❥❛♠ ❳✱❨∈X(Σ)❡ f ∈ C∞(Σ),❡♥tã♦
Tf S(X, Y) = ∇Xf SY − ∇Yf SX−f S[X, Y]
= f∇XSY +X(f)SY −f∇YSX−Y(f)SX−f S[X, Y] = f(∇XSY − ∇YSX−S[X, Y]) +X(f)SY −Y(f)SX = f T.
❙❡rá út✐❧ ❡s❝r❡✈❡r♠♦s ❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❝♦♦r❞❡♥❛❞❛✳ ❉❛❞♦s (x, y) ♣❛râ✲
♠❡tr♦s ❧♦❝❛✐s s♦❜r❡ U ∈ Σ ❡ ✉s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ✈✐st❛s ❞♦ ❚❡♥s♦r ❞❡ ❈♦❞❛③③✐✱ é s✉✜❝✐❡♥t❡
❝♦♥❤❡❝❡r ❝♦♠♦ TS ❛t✉❛ s♦❜r❡ ♦s ❝❛♠♣♦s ❢✉♥❞❛♠❡♥t❛✐s{∂x, ∂y}✳
Pr♦♣♦s✐çã♦ ✶✳✸✳✷✳ ❉❛❞♦ (I, II) ≡ (h,i, S) ∈ P(Σ)✱ s❡❥❛ (x, y) ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❧♦❝❛❧ ❡♠
U ⊂Σ t❛❧ q✉❡ I ❡ II sã♦ ❡s❝r✐t❛s ❝♦♠♦ (1.1).❊♥tã♦
hTS(∂x, ∂y), ∂xi = fx−ey+eΓ112−gΓ211+f(Γ212−Γ111) hTS(∂x, ∂y), ∂yi = gx−fy +eΓ122−gΓ212+f(Γ222−Γ112) ❉❡♠♦♥str❛çã♦✳ ❙❡ (h,i, S)∈ P(Σ)✱ ❡♥tã♦ ❞❡ (1.1)
hTS(∂x, ∂y), ∂xi = h∇∂xS∂y− ∇∂yS∂x, ∂xi = ∂xhS∂y, ∂xi − hS∂y,∇∂x∂xi
−∂yhS∂x, ∂xi+hS∂x,∇∂y∂xi = fx− hS∂y,Γ111∂x+ Γ211∂yi −ey
+hS∂x,Γ112∂x+ Γ212∂yi
= fx−ey+eΓ112−gΓ211+f(Γ212−Γ111). ❆♥❛❧♦❣❛♠❡♥t❡✱
hTS(∂x, ∂y), ∂yi=gx−fy+eΓ122−gΓ212+f(Γ222−Γ112).
❆ ♠✉❧t✐❧✐♥❡❛r✐❞❛❞❡ ❞❡TS(X, Y) ♥♦s ♣❡r♠✐t❡ ❡st❡♥❞❡r ♦ ❚❡♥s♦r ❞❡ ❈♦❞❛③③✐ à ❢✉♥çõ❡s ❞✐❢❡r❡♥✲
❝✐á✈❡✐s ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ t♦♠❡♠♦s f ∈ C∞(Σ) ❡ ❝♦♥s✐❞❡r❡♠♦s S = f Id ∈ S(Σ,h,i)✳ ❊♥tã♦✱
❝♦♠♦ ∇é ❧✐✈r❡ ❞❡ t♦rçã♦ ❡TId= 0✱ t❡♠♦s q✉❡Tf(X, Y) =X(f)Y −Y(f)X= [X, Y] (f).
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✶✶
❡q✉❛çã♦ ❞❡ ❈♦❞❛③③✐✱ ✐st♦ é
∇XSY − ∇YSX−S[X, Y] = 0, X, Y ∈X(Σ),
♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r✱
TS(X, Y) = 0, X, Y ∈X(Σ),
♦ q✉❡ ✐♠♣❧✐❝❛✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✸✳✷✱ ❛s ❡q✉❛çõ❡s ✉s✉❛✐s ❞❡ ❈♦❞❛③③✐✲▼❛✐♥❛r❞✐✳
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s
❊st❛ s❡çã♦ ❢♦✐ ❜❛s❡❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ❱■ ❞❡ ❬✾❪✳
❉❡✜♥✐çã♦ ✶✳✹✳✶✳ ❉❛❞❛ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛f :Mn−→Mn+m✱ ♣❛r❛ ❝❛❞❛p∈M,♦ ♣r♦❞✉t♦
✐♥t❡r♥♦ ❡♠TpM ❞❡❝♦♠♣õ❡ ❡st❡ ❡s♣❛ç♦ ♥❛ s♦♠❛ ❞✐r❡t❛TpM =TpM⊕(TpM)⊥,♦♥❞❡(TpM)⊥,é ♦
❝♦♠♣❧❡♠❡♥t♦ ♦rt♦❣♦♥❛❧ ❞❡ TpM ❡♠TpM .▲♦❣♦✱ s❡ v∈TpM✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡rv=vT +v⊥,❡♠
q✉❡ vT ∈TpM é ❝❤❛♠❛❞❛ ❝♦♠♣♦♥❡♥t❡ t❛♥❣❡♥❝✐❛❧ ❞❡ v ❡ v⊥ ∈TpM⊥ é ❝❤❛♠❛❞❛ ❛ ❝♦♠♣♦♥❡♥t❡
♥♦r♠❛❧ ❞❡ v✳
Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳ ❙❡❥❛ ∇ ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ M✳ ❙❡ X, Y ∈ X(M), ❡♥tã♦ ∇XY = (∇XY)⊥✱ ♦♥❞❡X✱Y sã♦ q✉❛✐sq✉❡r ❡①t❡♥sõ❡s ❧♦❝❛✐s ❞❡X✱Y ❛ M✱ ❞❡✜♥❡ ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛
❛ss♦❝✐❛❞❛ à ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ❞❡ M.
✶✳✹✳✶ ❙❡❣✉♥❞❛ ❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧
◆❡st❛ s❡çã♦ ❝♦♠♣❛r❛r❡♠♦s ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡M ❝♦♠ ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡M
❛tr❛✈és ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧✳ ❙❡ X, Y ∈X✱ ❡♥tã♦
∇XY = (∇XY) T + (
∇XY)⊥=∇XY + (∇XY)⊥,
❞❡ ♠♦❞♦ q✉❡
(∇XY)
⊥ =∇
XY − ∇XY
é ✉♠ ❝❛♠♣♦ ❧♦❝❛❧ ❡♠ M ♥♦r♠❛❧ à M.
❉❡✜♥✐çã♦ ✶✳✹✳✷✳ ❆ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ M é ❛ ❛♣❧✐❝❛çã♦ B : X(M)×X(M) −→
N(M) ❞❡✜♥✐❞❛ ♣♦rB(X, Y) = (∇XY)⊥,♦♥❞❡ X✱ Y sã♦ ❡①t❡♥sõ❡s ❧♦❝❛✐s ❞❡X, Y a M .
Pr♦♣♦s✐çã♦ ✶✳✹✳✷✳ ❆ ❙❡❣✉♥❞❛ ❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ é ❜✐❧✐♥❡❛r ❡ s✐♠étr✐❝❛✳
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✶✷
❆❣♦r❛ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❙❡❣✉♥❞❛ ❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧ s❡❣✉♥❞♦ ♦ ✈❡t♦r ♥♦r♠❛❧ η∈ (TpM)⊥✳
❙❡❥❛ p∈M.❉❡✜♥✐♠♦s ❛ ❛♣❧✐❝❛çã♦ Hη :TpM×TpM −→R❞❛❞❛ ♣♦r
Hη(x, y) =hB(x, y), ηi,
x, y∈TpM.
❉❡✜♥✐çã♦ ✶✳✹✳✸✳ ❆ ❢♦r♠❛ q✉❛❞rát✐❝❛ IIη ❞❡✜♥✐❞❛ ❡♠ TpM ♣♦r
IIη(x) =Hη(x, x)
é ❝❤❛♠❛❞❛ ❙❡❣✉♥❞❛ ❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ f ❡♠p, s❡❣✉♥❞♦ ♦ ✈❡t♦r ♥♦r♠❛❧ η.
◆♦t❡ q✉❡ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ IIη é ❜✐❧✐♥❡❛r ❡ s✐♠étr✐❝❛✳ ❆ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r IIη ❡stá
❛ss♦❝✐❛❞❛ à ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r ❛✉t♦❛❞❥✉♥t❛ Sη :TpM −→TpM ❞❛❞❛ ♣♦r
hSη(x), yiHη(x, y) =hB(x, y), ηi.
Pr♦♣♦s✐çã♦ ✶✳✹✳✸✳ ❙❡❥❛♠ p ∈ M✱ x ∈ TpM ❡ η ∈ (Tp)⊥✳ ❙❡❥❛ N ✉♠❛ ❡①t❡♥sã♦ ❧♦❝❛❧ ❞❡ η
♥♦r♠❛❧ ❛ M✳ ❊♥tã♦Sη(x) =−(∇xN)T. ✶✳✹✳✷ ❊q✉❛çõ❡s ❋✉♥❞❛♠❡♥t❛✐s
❉❛❞❛ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ f : Mn −→ Mn+m, t❡♠♦s✱ ❡♠ ❝❛❞❛ p ∈ M, ❛ ❞❡❝♦♠♣♦s✐çã♦ TpM = TpM ⊕(TpM)⊥✱ q✉❡ ✈❛r✐❛ ❞✐❢❡r❡♥❝✐❛✈❡❧♠❡♥t❡ ❝♦♠ p. ■st♦ s✐❣♥✐✜❝❛ q✉❡✱ ❧♦❝❛❧♠❡♥t❡✱ ❛
♣❛rt❡ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ T M , q✉❡ s❡ ♣r♦❥❡t❛ s♦❜r❡ M, s❡ ❞❡❝♦♠♣õ❡ ❡♠ ✉♠ ✜❜r❛❞♦ t❛♥❣❡♥t❡ T M ❡ ❡♠ ✉♠ ✜❜r❛❞♦ ♥♦r♠❛❧ T M⊥.
❉❛❞♦sX ❡ η,s❛❜❡♠♦s q✉❡ (∇Xη)T =−Sη(X)✳ ❆❣♦r❛ ❡st✉❞❛r❡♠♦s ♦ ❝♦♠♣♦♥❡♥t❡ ♥♦r♠❛❧ ❞❡
∇Xη✱ q✉❡ s❡rá ❝❤❛♠❛❞❛ ❛ ❝♦♥❡①ã♦ ♥♦r♠❛❧ ∇⊥ ❞❛ ✐♠❡rsã♦✳ ◆♦t❡♠♦s q✉❡
∇⊥Xη= (∇Xη)N =∇Xη−(∇Xη)T =∇Xη+Sη(X).
❱❡r✐✜❝❛♠♦s ❢❛❝✐❧♠❡♥t❡ q✉❡ ❛ ❝♦♥❡①ã♦ ♥♦r♠❛❧∇⊥ ♣♦ss✉✐ ❛s ♣r♦♣r✐❡❞❛❞❡s ✉s✉❛✐s ❞❡ ✉♠❛ ❝♦♥❡✲ ①ã♦✱ ✐st♦ é✱ é ❧✐♥❡❛r ❡♠ X✱ ❛❞✐t✐✈❛ ❡♠η✱ ❡
∇⊥X(f η) =f∇⊥Xη+X(f)η,
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✶✸
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ ❝❛s♦ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡✱ ✐♥tr♦❞✉③✐♠♦s ❛ ♣❛rt✐r ❞❡∇⊥ ✉♠❛ ♥♦çã♦ ❞❡ ❝✉r✈❛t✉r❛ ♥♦ ✜❜r❛❞♦ ♥♦r♠❛❧ q✉❡ é ❝❤❛♠❛❞❛ ❝✉r✈❛t✉r❛ ♥♦r♠❛❧ R⊥ ❞❛ ✐♠❡rsã♦ ❡ ❞❡✜♥✐❞❛ ♣♦r
R⊥(X, Y)η=∇⊥Y∇⊥Xη− ∇⊥X∇⊥Yη+∇⊥[X,Y]η.
Pr♦♣♦s✐çã♦ ✶✳✹✳✹ ✭❊q✉❛çã♦ ❞❡ ❲❡✐♥❣❛rt❡♥✮✳ ❙❡❥❛♠ X, Y ∈X(M) ❡ N ∈(TpM)⊥✳ ❊♥tã♦✱ ❡♠
M ✈❛❧❡ h∇XN✱ Yi=−hN, B(X, Y)i,♦♥❞❡ X, Y , N sã♦ q✉❛✐sq✉❡r ❡①t❡♥sõ❡s ❧♦❝❛✐s ❞❡ X, Y, N ❛
M .
❉❡♠♦♥str❛çã♦✳ ❈♦♠♦hN , Yi= 0 ❡♠M ❡ X é t❛♥❣❡♥t❡ ❛ M✱ t❡♠♦s XhN , Yi= 0.
▼❛s ❡♠ M,
XhN , Yi = h∇XN , Yi+hN ,∇XYi
= h∇XN , Yi+hN , B(X, Y) +∇XYi = h∇XN , Yi+hN , B(X, Y)i+hN ,∇XYi = h∇XN , Yi+hN , B(X, Y)i.
Pr♦♣♦s✐çã♦ ✶✳✹✳✺ ✭❊q✉❛çã♦ ❞❡ ●❛✉ss✮✳ P❛r❛ t♦❞♦ X, Y, Z, W ∈TpM ✈❛❧❡
R(X, Y, Z, T) =R(X, Y, Z, T) +hB(X, T), B(Y, Z)i − hB(Y, T), B(X, Z)i.
❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡
∇XY =∇XY +B(X, Y)
❡
∇Xη=∇⊥Xη−Sη(X).
❊♥tã♦
R(X, Y)Z = ∇Y∇XZ− ∇X∇YZ+∇[X,Y]Z
= ∇Y(∇XZ+B(X, Z))− ∇X(∇YZ+B(Y, Z)) +∇[X,Y]Z+B([X, Y], Z)
= ∇Y∇XZ+B(Y,∇XZ) +∇⊥YB(X, Y)−SB(X,Y)Y −∇X∇YZ+B(X,∇YZ)− ∇⊥XB(Y, Z) +SB(Y,Z)X
+∇[X,Z]Z+B([X, Y], Z)
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✶✹
❋❛③❡♥❞♦ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❛ ú❧t✐♠❛ ❡①♣r❡ssã♦ ❝♦♠ T,♦❜t❡♠♦s✿
hR(X, Y)Z, Ti=hR(X, Y)Z, Ti − hSB(X,Z)Y, Ti+hSB(Y,Z)X, Ti. ❈♦♠♦hSη(x), yi=hB(x, y), ηi,t❡♠♦s
hR(X, Y)Z, Ti=hR(X, Y)Z, Ti − hB(Y, T), B(X, Z)i+hB(X, T), B(Y, Z)i.
❈♦r♦❧ár✐♦ ✶✳✹✳✶✳ ❙❡ p∈ ▼ ❡ X, Y ∈TpM sã♦ ✈❡t♦r❡s ♦rt♦♥♦r♠❛✐s✱ ✈❛❧❡
K(X, Y)−K(X, Y) =hB(X, Y), B(X, Y)i − |B(X, Y)|2.
Pr♦♣♦s✐çã♦ ✶✳✹✳✻ ✭❊q✉❛çã♦ ❞❡ ❘✐❝❝✐✮✳
hR(X, Y)η, ξi=hR⊥(X, Y)η, ξi+hSη(SξX), Yi − hSξ(SηX), Yi.
❉❡♠♦♥str❛çã♦✳
R(X, Y)η=∇Y∇Xη− ∇X∇Yη+∇[X,Y]η
=∇Y(∇⊥Xη−Sη(X))− ∇X(∇Y⊥η−Sη(Y)) +∇[⊥X,Y]η−Sη([X, Y]) =∇⊥Y∇⊥Xη−S∇⊥
XηY − ∇YSη(X)−B(Y, Sη(X))− ∇
⊥
X∇⊥Yη+S∇⊥
YηX +∇XSη(Y) +B(X, Sη(Y)) +∇⊥[X,Y]η−Sη([X, Y])
=R⊥(X, Y)η−B(X, Sη(Y))−B(Y, Sη(X))−S∇⊥
XηY − ∇YSη(X) +S∇⊥
YηX+∇XSη(Y)−Sη[X, Y].
❋❛③❡♥❞♦ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝♦♠ξ
hR(X, Y)η, ξi = hR⊥(X, Y)η, ξi+hB(X, Sη(Y), ξi − hB(Y, Sη(X), ξi = hR⊥(X, Y)η, ξi+hSη(X), Sη(Y)i − hSη(Y), Sη(X)i.
❆❣♦r❛ ✈❛♠♦s ♦❜t❡r ❛ ❡q✉❛çã♦ ❞❡ ❈♦❞❛③③✐✳ ❙❡❥❛♠ X(M)⊥ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❛♠♣♦s ♥♦r♠❛✐s ❡
B :X(M)×X(M)×X(M)⊥−→D(M) t❛❧ q✉❡
B(X, Y, η) =hB(X, Y), ηi.
❉❡✜♥❛♠♦s
(∇XB)(Y, Z) =∇X(B(Y, Z))−B(∇XY, Z)−B(Y,∇XZ),
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✶✺
Pr♦♣♦s✐çã♦ ✶✳✹✳✼ ✭❊q✉❛çã♦ ❞❡ ❈♦❞❛③③✐✮✳ ❈♦♠ ❛ ♥♦t❛çã♦ ❛❝✐♠❛
hR(X, Y)Z, ηi= (∇YB)(X, Z, η)−(∇XB)(Y, Z, η)
❉❡♠♦♥str❛çã♦✳
hR(X, Y)Z, ηi=h∇Y∇XZ− ∇X∇YZ+∇[X,Y]Z, ηi
=h∇Y(∇XZ+B(X, Z))− ∇X(∇YZ+B(Y, Z)) +∇[X,Y]Z+B([X, Y], Z), ηi
=hB(Y,∇XZ)−B(X,∇YZ) +∇YB(X, Y)− ∇XB(Y, Z) +B(∇XY − ∇YX, Z), ηi
=h∇YB(X, Z)−B(∇YX, Z)−B(X,∇YZ)− (∇XB(Y, Z)−B(∇XY, Z)−B(Y,∇XZ)), ηi =h(∇YB)(X, Y), ηi − h(∇XB)(Y, Z), ηi = (∇YB)(X, Z, η)−(∇XB)(Y, Z, η).
❖❜s❡r✈❛çã♦ ✶✳✹✳✶✳ ❙❡ ♦ ❡s♣❛ç♦ ❛♠❜✐❡♥t❡ M t❡♠ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡✱ ❛ ❡q✉❛çã♦ ❞❡ ❈♦❞❛③③✐ s❡
r❡❞✉③ ❛✿
(∇YB)(X, Z, η) = (∇XB)(Y, Z, η). ✶✳✹✳✸ ❍✐♣❡rs✉♣❡r❢í❝✐❡s
❉❡✜♥✐çã♦ ✶✳✹✳✹✳ ❙❡ ❛ ❝♦❞✐♠❡♥sã♦ ❞❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ i:Mn−→Mn+k é ✶✱ ✐st♦ é✱ K = 1,
❞✐③❡♠♦s q✉❡ M é ✉♠❛ ❍✐♣❡rs✉♣❡r❢í❝✐❡✳
❚♦♠❛♥❞♦ ❞♦✐s ✈❡t♦r❡s x, y♦rt♦♥♦r♠❛✐s ✈✐♠♦s✱ ❡♠ ❞❡❝♦rrê♥❝✐❛ ❞❛ ❡q✉❛çã♦ ❞❡ ●❛✉ss✱ q✉❡ K(x, y)−K(x, y) =hB(x, x), B(y, y)i − |B(x, y)|2. ✭✶✳✸✮
P❛r❛ ♦ ❝❛s♦ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡✱ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ❛❞♠✐t❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s✳
◗✉❛♥❞♦M é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡✱ ❡①✐st❡♠ ❛♣❡♥❛s ❞✉❛s ❡s❝♦❧❤❛s ♣❛r❛ ♦ ✈❡t♦r ✉♥✐tár✐♦ ♥♦r♠❛❧✳
❙❡ M ❡M sã♦ ❛♠❜❛s ♦r✐❡♥tá✈❡✐s ❡ ❡s❝♦❧❤❡♠♦s ♦r✐❡♥t❛çã♦ ♣❛r❛M ❡M ,❡♥tã♦ t❡♠♦s ✉♠❛ ❡s❝♦❧❤❛
ú♥✐❝❛ ♣❛r❛ ♦ ✈❡t♦r ✉♥✐tár✐♦ ♥♦r♠❛❧✿ s❡{e1, ..., en} é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ♦r✐❡♥t❛❞❛ ❞❡TpM✱ ❡s✲
❝♦❧❤❡♠♦sη❞❡ t❛❧ ❢♦r♠❛ q✉❡{e1, ..., en, η}é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ♦r✐❡♥t❛❞❛ ❞❡TpM✳ ■ss♦ ♣r♦❞✉③
✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❧ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠M✳ ❊st❛ ❡s❝♦❧❤❛ ✜①❛ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❡
♣♦❞❡♠♦s ♥♦s r❡❢❡r✐r✱ s✐♠♣❧❡s♠❡♥t❡✱ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧Hη ❞❡ M ❡ ❛♦ ♦♣❡r❛❞♦r ❢♦r♠❛
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✶✻
❢♦r♠❛❞❛ ♣♦r ❛✉t♦✈❡t♦r❡s✱ ✐st♦ é✱ Sη(ei) = λiei, ♦♥❞❡ ei sã♦ ❛s ❞✐r❡çõ❡s ♣r✐♥❝✐♣❛✐s ❡ ♦s λi = ki
❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s✳
Pr♦♣♦s✐çã♦ ✶✳✹✳✽✳ ❙❡ M é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡✱
K(ei, ej)−K(ei, ej) =λiλj.
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ p ∈ M ❡ η ∈(TpM)⊥✱|η|= 1. ❙❡❥❛ {e1, ..., en} ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡
TpM, ♣❛r❛ ❛ q✉❛❧Sη é ❞✐❛❣♦♥❛❧✱ ✐st♦ é✱ Sη(ei) =λiei,♦♥❞❡ λi sã♦ ♦s ✈❛❧♦r❡s ♣ró♣r✐♦s ❞❡Sη.
◆♦t❡ q✉❡
B(ei, ej) =Hη(ei, ej)η =hSη(ei), ejiη=λiδijη.
P♦rt❛♥t♦ ✭✶✳✸✮ s❡ ❡s❝r❡✈❡
K(ei, ej)−K(ei, ej) =λiλj.
❉❡✜♥✐çã♦ ✶✳✹✳✺✳ ❙❡ M é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡✱ ❞❡✜♥✐♠♦s ❛ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss✲❑r♦♥❡❝❦❡r
❞❡M ♣♦rK = detSη =k1·. . .·kn❡ ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❞❡M ♣♦rH= 1
ntrSη =
k1+. . .+kn
n .
❚❡♦r❡♠❛ ✶✳✹✳✶ ✭❊❣r❡❣✐✉♠ ❞❡ ●❛✉ss✮✳ ❙❡ M2 é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❡♠R3✱ ❡♥tã♦ ♣❛r❛ q✉❛❧q✉❡r p ∈ M ❡ ♣❛r❛ q✉❛✐sq✉❡r ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s X, Y ❞❡ TpM, K(p) = K(X, Y)✳
P♦rt❛♥t♦✱ ❛ ❝✉r✈❛t✉r❛ ❞❡ ●❛✉ss é ✉♠ ✐♥✈❛r✐❛♥t❡ ✐s♦♠étr✐❝♦ ❞❡ (M, g).
Pr♦♣♦s✐çã♦ ✶✳✹✳✾✳ ❙❡❥❛ f :M −→M ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ❝♦❞✐♠❡♥sã♦ ✐❣✉❛❧ ❛ ✶✳ ❙❡❥❛♠ X, Y, Z ❝❛♠♣♦s t❛♥❣❡♥t❡s ❡ η ✉♠ ❝❛♠♣♦ ♥♦r♠❛❧ ✉♥✐tár✐♦ ❛ M. ❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s s❡
✈❡r✐✜❝❛♠✿
✶✮ ❊q✉❛çã♦ ❞❡ ●❛✉ss
R(X, Y)Z−R(X, Y)Z =hSX, ZiSY − hSY, ZiSX
✷✮ ❊q✉❛çã♦ ❞❡ ❈♦❞❛③③✐
R(X, Y)N =∇XSY − ∇YSX−S([X, Y]).
◗✉❛♥❞♦ M t❡♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡✱ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ s❡ r❡❞✉③ ❛
∇X(SY)− ∇Y(SX) =S([X, Y]),
❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ♠♦str❛r❡♠♦s ❛ ❡q✉❛çã♦ ❞❡ ❈♦❞❛③③✐✳ ❖❜s❡r✈❡ q✉❡∇Xη= (∇Xη)T+ (∇Xη)N =−Sη(X) +∇⊥Xη,✐st♦ é✱∇⊥Xη=∇Xη+Sη(X).❙❡❥❛ η(q)∈(T M)⊥ t❛❧ q✉❡hη, ηi= 1✳
❊♥tã♦ Xhη, ηi = 0. ❙❡ X∈T M✱ s❡❣✉❡ q✉❡h∇Xη, ηi= 0✱ ❞❛í t❡♠♦s ∇Xη∈T M. ❈♦♥s❡q✉❡♥t❡✲
♠❡♥t❡ ∇⊥
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✶✼
❙❛❜❡♠♦s q✉❡ s❡ ♦ ❡s♣❛ç♦ ❛♠❜✐❡♥t❡M t❡♠ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡✱ ❛ ❡q✉❛çã♦ ❞❡ ❈♦❞❛③③✐ s❡ r❡❞✉③
❛
(∇YB)(X, Z, η) = (∇XB)(Y, Z, η).
▼❛s ♥♦t❡ q✉❡
(∇XB)(Y, Z, η) = XB(Y, Z, η)−B(∇XY, Z, η)−B(Y,∇XZ, η) = XhB(Y, Z), ηi − hB(∇XY, Z), ηi − hB(Y,∇XZ), ηi = XhSηY, Zi − hSη(∇XY), Zi − hSηY,∇XZi
= h∇X(SηY), Zi − hSη(∇XY), Zi.
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱
(∇YB)(X, Z, η) =h∇Y(SηX), Zi − hSη(∇YX), Zi.
▲♦❣♦✱∇X(SηY)−Sη(∇XY)− ∇Y(SηY) +Sη(∇YX) = 0,✐st♦ é✱
∇X(SηY)− ∇Y(SηX)−Sη([X, Y]) = 0.
▼♦str❛r❡♠♦sR(X, Y)N =∇XS(Y)− ∇YS(X)−S([X, Y]).
❚❡♠♦s q✉❡
R(X, Y)η = ∇Y∇Xη− ∇X∇Yη+∇[X,Y]η
= ∇Y(∇⊥Xη−SηX)− ∇X(∇Y⊥η−SηY) +∇[⊥X,Y]η−Sη([X, Y])
= ∇X(SηY)− ∇Y(SηX)−Sη([X, Y]). ✭✶✳✹✮
◆♦t❡♠♦s q✉❡
h∇XSY − ∇YSX, Ni = h∇XSY, Ni − h∇YSX, Ni
= XhSY, Ni − hSY,∇XNi −YhSX, Ni+hSX,∇YNi.
❈♦♠♦
hSY, Ni = hSX, Ni= 0
hSY,∇XNi = hSY,−SηXi
✶✳✹ ■♠❡rsõ❡s ■s♦♠étr✐❝❛s ✶✽
t❡♠♦s
h∇XSY − ∇YSX, Ni = −hSY,−SXi+hSX,−SYi = hSY, SXi − hSX, SYi
= 0.
■st♦ é✱ ♦ ❝❛♠♣♦∇XSY − ∇YSX é ✉♠ ❝❛♠♣♦ t❛♥❣❡♥t❡ ❛M.▲♦❣♦✱ ♣♦❞❡♠♦s tr♦❝❛r∇XSY −
∇YSX ♣♦r ∇XSY − ∇YSX ♥❛ ❡q✉❛çã♦ (1.4)✱ ♦❜t❡♥❞♦
R(X, Y)η=∇XSη(Y)− ∇YSη(X)−Sη([X, Y])
❱❡❥❛♠♦s ❛❣♦r❛ ❝♦♠♦ ✜❝❛ ❛ ❡q✉❛çã♦ ❞❡ ●❛✉ss✳
❙❡❥❛♠X, Y, Z, T ∈X(M)❡ ◆∈X(M)⊥ ✉♠ ❝❛♠♣♦ ♥♦r♠❛❧ ✉♥✐tár✐♦✳ ❙❡ ❛ ❝♦❞✐♠❡♥sã♦ ❢♦r ✐❣✉❛❧
❛ ✶✱ ❡♥tã♦
B(X, T) =λN,
❞♦♥❞❡
λ=hB(X, T), Ni=hSX, Ti.
▲♦❣♦
B(X, T) =hSX, TiN.
❙❡❣✉❡ ❞❛í q✉❡ ❛ ❡q✉❛çã♦ ❞❡ ●❛✉ss ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❝♦♠♦
hR(X, Y)Z−R(X, Y)Z, Ti=hhSY, TiN,hSX, ZiNi − hhSX, TiN,hSY, ZiNi,
✐st♦ é✱
hR(X, Y)Z−R(X, Y)Z, Ti=hhSX, ZiSY − hSY, ZiSX, Ti.
◆♦t❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r é ✈á❧✐❞❛ ♣❛r❛ q✉❛❧q✉❡r T ∈X(M) ❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ s✉❜st✐t✉✐♥❞♦
T ♣♦rN,♦❜t❡♠♦s
hR(X, Y)Z−R(X, Y)Z, Ni= 0,
♦✉ s❡❥❛ R(X, Y)Z−R(X, Y)Z ∈X(M)✳ P♦rt❛♥t♦✱