Imersões isométricas de formas espaciais em Sn x R e Hn x R

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❡ ❚❡❝♥♦❧♦❣✐❛ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

■♠❡rsõ❡s ✐s♦♠étr✐❝❛s ❞❡ ❢♦r♠❛s ❡s♣❛❝✐❛✐s

❡♠

S

n

_×

R

❡

H

n

_×

R

.

❙❛♠✉❡❧ ❞❛ ❈r✉③ ❈❛♥❡✈❛r✐

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘✉② ❚♦❥❡✐r♦ ❞❡ ❋✐❣✉❡✐r❡❞♦ ❏✉♥✐♦r

■♠❡rsõ❡s ✐s♦♠étr✐❝❛s ❞❡ ❢♦r♠❛s ❡s♣❛❝✐❛✐s

❡♠

S

n

_×

R

❡

H

n

_×

R

_.

❙❛♠✉❡❧ ❞❛ ❈r✉③ ❈❛♥❡✈❛r✐

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❘✉② ❚♦❥❡✐r♦ ❞❡ ❋✐❣✉❡✐r❡❞♦ ❏✉♥✐♦r

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ❞♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳

❙ã♦ ❈❛r❧♦s ❏✉♥❤♦ ❞❡ ✷✵✶✺

❆✉t♦r ❖r✐❡♥t❛❞♦r

✷✵✵✵ ▼❛t❤❡♠❛t✐❝s ❙✉❜❥❡❝t ❈❧❛ss✐✜❝❛t✐♦♥✳ ✺✸❇✷✺✳

Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária/UFSCar

C221ii

Canevari, Samuel da Cruz.

Imersões isométricas de formas espaciais em Sn x R e Hn x R. / Samuel da Cruz Canevari. -- São Carlos : UFSCar, 2015.

97 f.

Tese (Doutorado) -- Universidade Federal de São Carlos, 2015.

1. Geometria diferencial. 2. Imersões isométricas. 3. Transformação de Ribaucour. 4. Hipersuperficies. 5. Variedades diferenciáveis. 6. Espaços produto. I. Título.

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t✉❞♦✳

●♦st❛r✐❛ ❞❡ ❡①♣r❡ss❛r ♠❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s à ♠✐♥❤❛ ❡s♣♦s❛ ❑❛t✐❛✱ ❛♦s ♠❡✉s ♣❛✐s ❋r❛♥❝✐s❝♦ ❡ ❙❛❧❡t❡✱ ❛♦s ♠❡✉s ✐r♠ã♦s ❚❤✐❛❣♦ ❡ ●❧❛✉❝♦✱ às ♠✐♥❤❛s ❝✉♥❤❛❞❛s ▼❛r✐❛♥❛ ❡ ▲❡✐♥❤❛✱ ❡ ❛♦ ♠❡✉ s♦❜r✐♥❤♦ ❡ ❛✜❧❤❛❞♦ ❆rt❤✉r✱ ♣❡❧♦ ❛♠♦r✱ ❝♦♠♣r❡❡♥sã♦✱ ❝❛r✐♥❤♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ✐♥❝❡♥t✐✈♦ ❝♦♥st❛♥t❡✳

❙♦✉ s✐♥❝❡r❛♠❡♥t❡ ❣r❛t♦ ❛♦ ❛♠✐❣♦ ❡ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❘✉② ❚♦❥❡✐r♦✱ ♣❡❧♦ ♣r♦✜ss✐♦♥❛❧✐s♠♦ ❡ ✐♥❝❡♥t✐✈♦ ❛ ♠✐♠ ❞❡❞✐❝❛❞♦✱ ❛tr❛✈és ❞❡ s✉❛ ❣r❛♥❞❡ ❡①♣❡r✐ê♥❝✐❛ ❡ ❛❞♠✐rá✈❡❧ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦✳

❆ t♦❞♦s ♠❡✉s ❛♠✐❣♦s✱ ♦♥❞❡ q✉❡r q✉❡ ❡st❡❥❛♠✱ ♣❡❧♦ ❝♦♥✈í✈✐♦ ❣♦st♦s♦ ❡ ❛♣♦✐♦ ❝♦♥st❛♥t❡❀ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❉❛♥✐❡❧ ❙✐❧✈❡✐r❛ ●✉✐♠❛rã❡s✱ ❋❡r♥❛♥❞♦ ▼❛♥✜♦✱ ●✉✐❧❤❡r♠❡ ❞❡ ❋r❡✐t❛s ❡ ❈❛r❧♦s ●♦♥ç❛❧✈❡s ❋✐❧❤♦ ♣❡❧❛s ❧♦♥❣❛s ❡ ✈❛❧✐♦s❛s ❞✐s❝✉ssõ❡s ♠❛t❡♠át✐❝❛s✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❈❛♠♣✉s Pr♦❢✳ ❆❧❜❡rt♦ ❈❛r✈❛❧❤♦ ❞❛ ❯❋❙✱ q✉❡ ♠❡ ❡①✐♠✐r❛♠ ❞❡ ♠✐♥❤❛s ❛tr✐❜✉✐çõ❡s ❛❝❛❞ê♠✐❝❛s t♦❞♦s ❡ss❡s ❛♥♦s✳

❘❡s✉♠♦

◆❡st❛ t❡s❡ ❝❧❛ss✐✜❝❛♠♦s ❛s ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f : Mm

c → Sm+p × R ❝♦♠

m ≥ 3✱ p ≤ m −3 ❡ c ≤ 1✱ ❡♠ q✉❡ Mm

c ❞❡♥♦t❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠

❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ c✳ ❖❜t❡♠♦s r❡s✉❧t❛❞♦s ♣❛r❝✐❛✐s s♦❜r❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❛s ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f : Mm

c → Hm+p × R ❝♦♠ m ≥ 3✱ p ≤ m − 3 ❡ c < 0✳

❈❛r❛❝t❡r✐③❛♠♦s ❛✐♥❞❛ ❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s f : M3 _{→ Q}4_{(c) ♣❛r❛ ❛s q✉❛✐s ❡①✐st❡ ♦✉tr❛}

✐♠❡rsã♦ ✐s♦♠étr✐❝❛ _f˜ _{: M}3 _{→ L}4_{✱ ❡♠ q✉❡ Q}4_{(c) ❡ L}4 _{❞❡♥♦t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✉♠❛}

❢♦r♠❛ ❡s♣❛❝✐❛❧ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ c ❡ ♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③ ❞❡ ❞✐♠❡♥sã♦ 4✳

❆❜str❛❝t

■♥ t❤✐s t❤❡s✐s ✇❡ ❝❧❛ss✐❢② t❤❡ ✐s♦♠❡tr✐❝ ✐♠♠❡rs✐♦♥s f : Mm

c → Sm+p ×R ✇✐t❤

m ≥ 3✱ p ≤ m−3 ❛♥❞ c ≤1✱ ✇❤❡r❡ Mm

c ❞❡♥♦t❡s ❛ ❘✐❡♠❛♥♥✐❛♥ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❝♦♥st❛♥t

s❡❝t✐♦♥❛❧ ❝✉r✈❛t✉r❡ ❡q✉❛❧ t♦ c✳ ❲❡ ♦❜t❛✐♥ ♣❛rt✐❛❧ r❡s✉❧ts ♦♥ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❝ ✐♠♠❡rs✐♦♥s f :Mm

c →Hm+p×R✇✐t❤ m≥3✱p≤m−3❛♥❞ c≤0✳ ❲❡ ❛❧s♦ ❝❤❛r❛❝t❡r✐③❡

t❤❡ ❤②♣❡rs✉r❢❛❝❡s f : M3 _{→ Q}4_{(c) ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛♥♦t❤❡r ✐s♦♠❡tr✐❝ ✐♠♠❡rs✐♦♥}

˜

f :M3 _→L4_{✱ ✇❤❡r❡Q}4_{(c)❛♥❞L}4_{❞❡♥♦t❡ ❛4✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ ❢♦r♠ ♦❢ ❝♦♥st❛♥t s❡❝t✐♦♥❛❧}

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✺

✶✳✶ ■♠❡rsõ❡s ✐s♦♠étr✐❝❛s ❡♠ Qn(ǫ)_×R✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✶✳✶✳✶ ❈❧❛ss❡ A ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✶✳✶✳✷ ❙✉❜✈❛r✐❡❞❛❞❡s ❞❡ r♦t❛çã♦ ❡♠Qn(ǫ)_×R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✶✳✷ ❍✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ r♦t❛çã♦ ❞❡Q4

s(c)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷ ❯♠❛ ❝❧❛ss❡ ❞❡ s✉❜✈❛r✐❡❞❛❞❡s ❞❡ Q2m−3(ǫ)✳ ✸✵

✷✳✶ ❈❧❛ss❡ B ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✷✳✷ ❚r❛♥s❢♦r♠❛çã♦ ❞❡ ❘✐❜❛✉❝♦✉r ✲ Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✸ ❆ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❘✐❜❛✉❝♦✉r ♣❛r❛ ❛ ❝❧❛ss❡ B✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✷✳✹ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸ ■♠❡rsõ❡s ✐s♦♠étr✐❝❛s ❞❡ ❢♦r♠❛s ❡s♣❛❝✐❛✐s ❡♠ Qn(ǫ)×R✳ ✺✸

✸✳✶ ❘❡s✉❧t❛❞♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✷ ❉❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✸ ❉❡♠♦♥str❛çõ❡s ❞♦s ❚❡♦r❡♠❛s ✸✳✷ ❡ ✸✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✹ ❉❡♠♦♥str❛çã♦ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✸✳✺ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✸✳✻ ❊①❡♠♣❧♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹

✹ ❍✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ Q4(c) ❡ L4 ✼✽

✹✳✶ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✹✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✹✳✷ ❉❡♠♦str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✹✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✹✳✸ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✹✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾

❙❯▼➪❘■❖ ✈✐

✹✳✹ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵

❆ ❆❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ✉t✐❧✐③❛❞♦s ✾✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✾✻

■♥tr♦❞✉çã♦

❯♠ tó♣✐❝♦ ❝❡♥tr❛❧ ♥❛ t❡♦r✐❛ ❞❡ s✉❜✈❛r✐❡❞❛❞❡s é ♦ ❡st✉❞♦ ❞❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f :

Mm

c → Qm+p(˜c) ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ Mcm ❝♦♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡

c ❡ ❞✐♠❡♥sã♦m✳ ❆q✉✐✱ ❡ ❡♠ t♦❞♦ ♦ tr❛❜❛❧❤♦✱ QN(˜c) ❞❡♥♦t❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛

❝♦♠♣❧❡t❛ ❡ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛ ❞❡ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ ˜c❡ ❞✐♠❡♥sã♦N✳ ➱ ✉♠ ❢❛t♦ ❝♦♥❤❡❝✐❞♦ q✉❡ QN(˜c) é ✐s♦♠étr✐❝❛ à ❡s❢❡r❛ SN(˜c) _⊂RN+1 ❞❡ r❛✐♦ 1/√˜cs❡ c >˜ 0✱ ❛♦

❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ RN s❡ ˜c= 0 ❡ ❛♦ ❡s♣❛ç♦ ❤✐♣❡r❜ó❧✐❝♦

HN ={X = (x0, . . . , xN)∈LN+1 : hX, Xi=

1 ˜

c, x0 >0}

s❡ ˜c <0✱ ❡♠ q✉❡ LN+1 ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③ ❞❡ ❞✐♠❡♥sã♦ N + 1✳

❈♦♠ ❛✉①í❧✐♦ ❞❡ s✉❛ t❡♦r✐❛ ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s ❡①t❡r✐♦r♠❡♥t❡ ♦rt♦❣♦♥❛✐s✱ ❊✳ ❈❛rt❛♥ ♠♦str♦✉✱ ❞❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ q✉❡✱ s❡ f : Mm

c → Qm+p(˜c) é ✉♠❛ ✐♠❡rsã♦

✐s♦♠étr✐❝❛ ❝♦♠ c < ˜c✱ ❡♥tã♦ p≥m−1 ❡✱ s❡ p=m−1✱ ❡♥tã♦ f ♣♦ss✉✐✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ✜❜r❛❞♦ ♥♦r♠❛❧ ♣❧❛♥♦✳ P♦st❡r✐♦r♠❡♥t❡✱ ❏✳❉✳▼♦♦r❡ ✭❬✶✼❪✮ ❞❡s❡♥✈♦❧✈❡✉ ❛ t❡♦r✐❛ ❞❡ ❢♦r♠❛s ❜✐❧✐♥❡❛r❡s ❊✉❝❧✐❞✐❛♥❛s ❝♦♠ ✈❛❧♦r❡s ❡♠ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s ♠✉♥✐❞♦s ❞❡ ❢♦r♠❛s ❜✐❧✐♥❡❛r❡sh·,·i

♥ã♦✲❞❡❣❡♥❡r❛❞❛s✱ ❡st❡♥❞❡♥❞♦ ❛ t❡♦r✐❛ ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s ❡①t❡r✐♦r♠❡♥t❡ ♦rt♦❣♦♥❛✐s ❞❡ ❈❛rt❛♥✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❝❛s♦ ❡♠ q✉❡ h·,·i é ♣♦s✐t✐✈♦✲❞❡✜♥✐❞❛✳ ❈♦♠ ❛✉①í❧✐♦ ❞❡ s✉❛

t❡♦r✐❛✱ ▼♦♦r❡ ✐♥✐❝✐❛❧♠❡♥t❡ r❡♦❜t❡✈❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡✈✐❞♦ ❛ ❖✬◆❡✐❧❧ ❬✶✽❪✱ s❡❣✉♥❞♦ ♦ q✉❛❧ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ f : Mm

c → Qm+p(˜c)✱ ❝♦♠ c > ˜c ❡

p_≤m₋2✱ s❡ ❞❡❝♦♠♣õ❡ ♦rt♦❣♦♥❛❧♠❡♥t❡ ❝♦♠♦

αf =√ch·,·iη+γ, ✭✵✳✵✳✶✮

❡♠ q✉❡ η é ✉♠ ❝❛♠♣♦ ✉♥✐tár✐♦ ♥♦r♠❛❧ ❛ f✳ ❆❧é♠ ❞✐ss♦✱ s❡ p = m ₋1 ❡ αf_{(x) ♥ã♦ s❡}

❞❡❝♦♠♣õ❡ ❝♦♠♦ ❡♠(0.0.1)✱ ▼♦♦r❡ ♣r♦✈♦✉ q✉❡f ❞❡✈❡ t❡r ✜❜r❛❞♦ ♥♦r♠❛❧ ♣❧❛♥♦ ❡♠x✳ ❯♠ ♣♦♥t♦x_∈M ♥♦ q✉❛❧αf _{s❡ ❞❡❝♦♠♣õ❡ ❝♦♠♦ ❡♠(0.0.1)é ❞❡♥♦♠✐♥❛❞♦ ✉♠ ♣♦♥t♦ ❢r❛❝❛♠❡♥t❡}

■♥tr♦❞✉çã♦ ✷

✉♠❜í❧✐❝♦ ♣❛r❛ f✳ ❙❡ t♦❞♦s ♦s ♣♦♥t♦s x∈M ❢♦r❡♠ ❢r❛❝❛♠❡♥t❡ ✉♠❜í❧✐❝♦s ♣❛r❛ f✱ ❞✐r❡♠♦s q✉❡ f é ❢r❛❝❛♠❡♥t❡ ✉♠❜í❧✐❝❛✳

P♦st❡r✐♦r♠❡♥t❡✱ ❉❛❥❝③❡r ❡ ❚♦❥❡✐r♦ ♠♦str❛r❛♠ ❡♠([✼])q✉❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛

f: Mm

c → Qm+p(˜c)✱ ❝♦♠ c > ˜c ❡ p ≤ m−2 é✱ ❧♦❝❛❧♠❡♥t❡ ❡♠ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦

❡ ❞❡♥s♦ ❞❡ Mm_{✱ ✉♠❛ ❝♦♠♣♦s✐çã♦ f = i◦h✱ ❡♠ q✉❡ i: M}m

c → Qm+1(˜c) é ✉♠❛ ✐♥❝❧✉sã♦

✉♠❜í❧✐❝❛ ❡ h: U _→Q_˜cm+p é ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠ ❛❜❡rt♦U _⊂Qm+1(˜c) ❝♦♥t❡♥❞♦

i(Mm

c )✳ ❖ ♠❡s♠♦ r❡s✉❧t❛❞♦ é ❛✐♥❞❛ ✈á❧✐❞♦ s❡ p = m −1 ❡ f é ❢r❛❝❛♠❡♥t❡ ✉♠❜í❧✐❝❛✳

❊①❡♠♣❧♦s ❡①♣❧í❝✐t♦s ❞❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f: Mm

c → Q2m−1(˜c)✱ c > ˜c✱ s❡♠ ♣♦♥t♦s

❢r❛❝❛♠❡♥t❡ ✉♠❜í❧✐❝♦s✱ ❢♦r❛♠ ❝♦♥str✉í❞♦s ❡♠ ([✾]) ✉s❛♥❞♦ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❘✐❜❛✉❝♦✉r✳

◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ❛s ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f: Mm

c → Qn(ǫ)× R✱ ǫ ∈

{−1,1}✳ ❖❜s❡r✈❡ q✉❡ Qn(ǫ)× R ❛❞♠✐t❡ ✉♠ ♠❡r❣✉❧❤♦ ✐s♦♠étr✐❝♦ ❝❛♥ô♥✐❝♦ ❡♠ En+2✱

❡♠ q✉❡ En+2 ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ Rn+2 ♦✉ ♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③ Ln+2✱ ❝♦♥❢♦r♠❡

s❡❥❛ ǫ = 1 ♦✉ −1✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ǫ = 1✱ t❛❧ ❡st✉❞♦ s❡ ✐♥s❡r❡ ♥♦

♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ ❞❡ ❡st✉❞❛r ❛s ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s ❞❡ ✈❛r✐❡❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ❝♦♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ ♥♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦✳

❙✉♣❡r❢í❝✐❡s ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ ❞❡Q2(ǫ)_×R❢♦r❛♠ ❡st✉❞❛❞❛s ❡♠

❬✶❪ ❡ ❬✷❪✱ ❝♦♠ ê♥❢❛s❡ ❡♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❣❧♦❜❛✐s✳ ❖s ❛✉t♦r❡s ♠♦str❛r❛♠ q✉❡ ♥ã♦ ❡①✐st❡♠ s✉♣❡r❢í❝✐❡s ❝♦♠♣❧❡t❛s ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ c❡♠ S2_×R✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ H2 _×R✮ q✉❛♥❞♦ c < −1 ❡ 0 < c < 1 ✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ c < −1✮✳ ▼♦str❛r❛♠ t❛♠❜é♠

q✉❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛ ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ c >1✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱

c >0✮ ❞❡ S2_×R ✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ H2_×R✮ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞❡ r♦t❛çã♦✱ ❡ s✉❛s ❝✉r✈❛s

❣❡r❛tr✐③❡s ❢♦r❛♠ ❞❡t❡r♠✐♥❛❞❛s ❡①♣❧✐❝✐t❛♠❡♥t❡✳

❆s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ Qm(ǫ)×R❝♦♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡c❡ ❞✐♠❡♥sã♦ m _≥ 3 ❢♦r❛♠ ❝❧❛ss✐✜❝❛❞❛s ♣♦r ❋✳▼❛♥✜♦ ❡ ❘✳ ❚♦❥❡✐r♦ ❡♠ ❬✶✹❪✳ ❙❡ m _≥ 4 ❡ ǫ = 1

✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ǫ = −1✮✱ ♠♦str♦✉✲s❡ q✉❡ t❛✐s ❤✐♣❡rs✉♣❡r❢í❝✐❡ s♦♠❡♥t❡ ❡①✐st❡♠✱ ♠❡s♠♦

❧♦❝❛❧♠❡♥t❡✱ ♣❛r❛ c _≥ 1 ✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ c _{≥ −}1✮✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ t❛✐s ✈❛❧♦r❡s ❞❡ c✱ ✉♠❛ t❛❧ ❤✐♣❡rs✉♣❡r❢í❝✐❡ s❡♠♣r❡ s❡ ❡st❡♥❞❡ ❛ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛ ❞❡ r♦t❛çã♦✳ ◆♦ ❝❛s♦ m= 3✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❝❧❛ss❡ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♥ã♦✲r♦t❛❝✐♦♥❛✐s ❞❡Qm(ǫ)_×R❝♦♠

❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ c✱ ❝♦♠ ǫc ∈ (0,1)✳ ◗✉❛❧q✉❡r ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❞❡ss❛ ❝❧❛ss❡

é ❞❛❞❛ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❡♠ t❡r♠♦s ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉♣❡r❢í❝✐❡s ♣❛r❛❧❡❧❛s ❞❡ Q3(ǫ) ❝♦♠

❝✉r✈❛t✉r❛ ♥✉❧❛✳

■♥tr♦❞✉çã♦ ✸

Sm+p _×R ❝♦♠ m ≥ 3✱ p ≤ m−3 ❡ c ≤ 1✳ ◆❡ss❛s ❤✐♣ót❡s❡s✱ ♠♦str❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡

q✉❡ ✉♠❛ t❛❧ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ♥ã♦ ❡①✐st❡✱ ♠❡s♠♦ ❧♦❝❛❧♠❡♥t❡✱ s❡ c < 0✳ ❆❧é♠ ❞✐ss♦✱

♠♦str❛♠♦s q✉❡ ♦ ❝❛s♦ c= 0 ♣♦❞❡ ♦❝♦rr❡r ❛♣❡♥❛s s❡ p=m−3 ❡✱ ❛❞❡♠❛✐s✱f ❞❡✈❡ s❡r ✉♠ ❝✐❧✐♥❞r♦ ✈❡rt✐❝❛❧ s♦❜r❡ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦m₋1❡ ❝✉r✈❛t✉r❛ ♥✉❧❛ ❞❡S2m−3✳ ◆♦

❝❛s♦ ❡♠ q✉❡c= 1✱ ♣r♦✈❛♠♦s q✉❡f(Mm

c )é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ ✉♠❛ ❢❛t✐❛

Sm+p_×{t_}❞❡Sm+p_×R✳ ❖ ❝❛s♦ ♠❛✐s ✐♥t❡r❡ss❛♥t❡ é ❛q✉❡❧❡ ❡♠ q✉❡c_∈(0,1)✱ ♦ q✉❛❧ ♦❝♦rr❡

t❛♠❜é♠ s♦♠❡♥t❡ q✉❛♥❞♦p=m−3✳ ❖❜t✐✈❡♠♦s ✉♠❛ ❝♦♥str✉çã♦ ❡①♣❧í❝✐t❛ ❞❡ t❛✐s ✐♠❡rsõ❡s

❡♠ t❡r♠♦s ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s ♣❛r❛❧❡❧❛s ♣❡rt❡♥❝❡♥t❡s ❛ ✉♠❛ ❝❡rt❛ ❝❧❛ss❡ ❞❡ s✉❜✈❛r✐❡❞❛❞❡s ❞❡ ❞✐♠❡♥sã♦ m−1❞❡Q2m−3(ǫ)✱ ❛ q✉❛❧ ❞❡♥♦♠✐♥❛♠♦s ❞❡ ❝❧❛ss❡B✳

▼♦str❛♠♦s q✉❡ q✉❛❧q✉❡r ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ g: Mm−1 _{→ Q}2m−3_{(ǫ) ♥❛ ❝❧❛ss❡ B ♣♦❞❡ s❡r}

❝♦♥str✉í❞❛✱ ✈✐❛ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛s s✉❜✈❛r✐❡❞❛❞❡s✱ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❞❡

Qm(c)q✉❡ ❛❞♠✐t❡ ✉♠❛ ♦✉tr❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❡♠R_m2m₋−₃4✱ ❡♠ q✉❡R2_mm₋−₃4 ❞❡♥♦t❛ ♦ ❡s♣❛ç♦

♣s❡✉❞♦✲❊✉❝❧✐❞✐❛♥♦ ❞❡ ❞✐♠❡♥sã♦ 2m−4 ❝✉❥❛ ♠étr✐❝❛ ♣♦ss✉✐ í♥❞✐❝❡ m−3. P❛r❛ m = 4✱

♦❜t✐✈❡♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦✱ ❝♦♠ ✐♥t❡r❡ss❡ ♣ró♣r✐♦✱ ❞❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❡ tê♠ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡✳ ❈♦♥str✉í♠♦s ❡①❡♠♣❧♦s ❡①♣❧í❝✐t♦s ❞❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s ♥❛ ❝❧❛ss❡B✱ ❛ ♣❛rt✐r

❞♦s q✉❛✐s ♣r♦❞✉③✐♠♦s ❡①❡♠♣❧♦s ❡①♣❧í❝✐t♦s ❞❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛sf :Mm

c →Q2m−3(ǫ)×R✱

❝♦♠ m ≥ 3 ❡ ǫc ∈ (0,1)✳ P❛r❛ ǫ = 1✱ ❛s ❝♦♠♣♦s✐çõ❡s f˜= i◦f ❞❡ t❛✐s ✐♠❡rsõ❡s ❝♦♠ ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛ i: S2m−3 _×R _→ R2m−1 ❢♦r♥❡❝❡♠✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♥♦✈♦s ❡①❡♠♣❧♦s ❞❡

✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s _f˜_:Mm

c →R2m−1 s❡♠ ♣♦♥t♦s ❢r❛❝❛♠❡♥t❡ ✉♠❜í❧✐❝♦s✳

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ f : Mm

c → Sm+p ×R ❝♦♠ m ≥ 3✱ p ≤ m− 2 ❡ c ≤ 1✱ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡

❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ Mm

c ❡①✐st❡ ❛♣❡♥❛s s❡c= 1❡f é ✉♠❛ ✐♥❝❧✉sã♦ t♦t❛❧♠❡♥t❡ ❣❡♦❞és✐❝❛

❞❡Sm ❡♠ ✉♠❛ ❢❛t✐❛Sm+p_{× {}t}❞❡Sm+p_×R✳ ❙❡p=m−3✱ ❛❧é♠ ❞❡ t❛✐s ❡①❡♠♣❧♦s tr✐✈✐❛✐s

❡①✐st❡♠ ❛♣❡♥❛s ❛q✉❡❧❡s ❡♠ q✉❡ c = 0 ❡ f é ✉♠ ❝✐❧✐♥❞r♦ ✈❡rt✐❝❛❧ s♦❜r❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ g: N₀m−1 →S2m−3✱ ❝♦♠ N₀m−1 ❝♦♠♣❧❡t❛✳

❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ✉♠❛ ❞❡s❝r✐çã♦ ❞♦ ❝♦♥t❡ú❞♦ ❞♦s ✈ár✐♦s ❝❛♣ít✉❧♦s ❞❡st❛ t❡s❡✳ ❈♦♠❡ç❛♠♦s ♥♦ss♦ t❡①t♦ ✐♥tr♦❞✉③✐♥❞♦✱ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❛❧❣✉♠❛s ❢❡rr❛♠❡♥t❛s út❡✐s ❛♦ ❡st✉❞♦ ❞❛s s✉❜✈❛r✐❡❞❛❞❡s ❞❡Qm+p(ǫ)_×R✱ ❝♦♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡✳ ❙ã♦

❞❛❞❛s ❛s ❡q✉❛çõ❡s ❞❡ ●❛✉ss✱ ❈♦❞❛③③✐ ❡ ❘✐❝❝✐✱ ❛❧é♠ ❞❡ ❡q✉❛çõ❡s ❛❞✐❝✐♦♥❛✐s ❡♥✈♦❧✈❡♥❞♦ ❛s ♣❛rt❡s t❛♥❣❡♥t❡ ❡ ♥♦r♠❛❧ ❞♦ ❝❛♠♣♦ ∂/∂tt❛♥❣❡♥t❡ ❛♦ s❡❣✉♥❞♦ ❢❛t♦r ❞❡Qm+p(ǫ)_×R✳ ❚❛✐s

❡q✉❛çõ❡s ❞❡t❡r♠✐♥❛♠✱ ❛ ♠❡♥♦s ❞❡ ♠♦✈✐♠❡♥t♦s rí❣✐❞♦s✱ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡Qm+p(ǫ)×R✳

■♥tr♦❞✉çã♦ ✹

❝❧❛ss❡ A, q✉❡ ♣♦ss✉❡♠ ❛ ♣r♦♣✐❡❞❛❞❡ ❞❡ q✉❡ ❛ ❝♦♠♣♦♥❡♥t❡ t❛♥❣❡♥t❡ ❞♦ ❝❛♠♣♦ ∂/∂t é ✉♠ ❛✉t♦✈❡t♦r ❞❡ t♦❞♦s ♦s ♦♣❡r❛❞♦r❡s ❞❡ ❢♦r♠❛ ❞❡ f✳ ❙ã♦ ❞❛❞❛s t❛♠❜é♠ ❛ ❞❡✜♥✐çã♦ ❡ ❛s ♣❛r❛♠❡tr✐③❛çõ❡s ❞❛s s✉❜✈❛r✐❡❞❛❞❡s ✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❤✐♣❡rs✉♣❡r❢í❝✐❡s✮ ❞❡ r♦t❛çã♦ ❡♠

Qm+p(ǫ)_×R ✭r❡s♣❡❝t✐✈❛♠❡♥t❡✱ Qn_s(c)✮ q✉❡ tê♠ ✉♠❛ ❝✉r✈❛ ❝♦♠♦ ❣❡r❛tr✐③✱ ❡♠ q✉❡ Qn_s(c)

❞❡♥♦t❛ ✉♠❛ ❢♦r♠❛ ❡s♣❛❝✐❛❧ ♣s❡✉❞♦✲❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❞✐♠❡♥sã♦ n✱ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ c ❡ í♥❞✐❝❡ s_{∈ {}0,1_}✳

❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❛s ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s g: Mm−1 _→

Q2m−3(ǫ) ♣❡rt❡♥❝❡♥t❡s à ❝❧❛ss❡ _B✳ ❖❜t❡♠♦s ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❘✐❜❛✉❝♦✉r ♣❛r❛ t❛❧

❝❧❛ss❡✱ ❛ q✉❛❧ ♣❡r♠✐t❡ ♦❜t❡r ♥♦✈♦s ❡①❡♠♣❧♦s ❞❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s ♥❡ss❛ ❝❧❛ss❡ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❞❛❞❛ ❡ ❞❡ ✉♠❛ s♦❧✉çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❞❡ ❊❉P✬s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣r♦❞✉③✐♠♦s ❡①❡♠♣❧♦s ❡①♣❧í❝✐t♦s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ss❛ ❝❧❛ss❡ ❛ ♣❛rt✐r ❞❡ ✉♠❛ s♦❧✉çã♦ tr✐✈✐❛❧✳ ❚❛✐s ❡①❡♠♣❧♦s sã♦ ✉s❛❞♦s ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡ ♣❛r❛ ❝♦♥str✉✐r ❡①❡♠♣❧♦s ❡①♣❧í❝✐t♦s ❞❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f :Mm

c →Q2m−3(ǫ)×R❝♦♠ ǫc∈(0,1) ✭❱❡r ❙❡çã♦ 3.6✮✳

◆♦ ❈❛♣ít✉❧♦ ✸ ♠♦str❛♠♦s ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦✱ ❞❡s❝r✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✱ s♦❜r❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f : Mm

c → Sm+p ×R ❝♦♠ m ≥ 3✱ p ≤ m−3✳

❖❜t❡♠♦s ❛✐♥❞❛ r❡s✉❧t❛❞♦s ♣❛r❝✐❛✐s ♣❛r❛ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f : Mm

c → Hm+p ×R ❝♦♠

m ≥3❡ p≤m−3✳

◆♦ q✉❛rt♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❡st✉❞❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡r♠✐♥❛r ❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡sf: M3 _→Q4_{(c)♣❛r❛ ❛s q✉❛✐sM}3_{❛❞♠✐t❡ t❛♠❜é♠ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛}

♥♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③ L4✳ ❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡ss❡ ❝❛♣ít✉❧♦ ❞á ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s

s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ q✉❡ ♣♦ss✉❡♠ três ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❞✐st✐♥t❛s✳ ◆♦ ✜♥❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✱ ❡①✐❜✐♠♦s ❡①❡♠♣❧♦s ❡①♣❧í❝✐t♦s ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❡ sã♦ s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦✳ ❚❛❧ ♣r♦❜❧❡♠❛ ❢♦✐ ❡st✉❞❛❞♦✱ ❡♠ ♠❛✐♦r ❣❡♥❡r❛❧✐❞❛❞❡✱ ❡♠ ❬✸❪✳

❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ ❡♥✉♥❝✐❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ sã♦ ✉t✐❧✐③❛❞♦s ♥♦ ❞❡❝♦rr❡r ❞❡st❛ t❡s❡✳ ❖s r❡s✉❧t❛❞♦s ♥ã♦ ❞❡♠♦♥str❛❞♦s sã♦ ❛❝♦♠♣❛♥❤❛❞♦s ❞❡ r❡❢❡rê♥❝✐❛s ♥❛s q✉❛✐s s✉❛s ❞❡♠♦♥str❛çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s✳

✶✳✶ ■♠❡rsõ❡s ✐s♦♠étr✐❝❛s ❡♠

Q

(ǫ)

_×

R

✳

❖ ❝♦♥t❡ú❞♦ ❞❡st❛ s❡çã♦ ❡stá ❜❛s❡❛❞♦ ❡♠ ❬✶✻❪ ❡ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❢♦r♥❡❝❡r ❛❧❣✉♠❛s ❢❡rr❛♠❡♥t❛s út❡✐s ❛♦ ❡st✉❞♦ ❞❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s✱ ❡♠ Qn(ǫ) _× R✱ ❞❡ ✈❛r✐❡❞❛❞❡s

❘✐❡♠❛♥♥✐❛♥❛s ❞❡ ❞✐♠❡♥sã♦ m ≥3❡ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡✳

❙❡❥❛♠ f : Mm _→Qn_{(ǫ)×R ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❡} ∂

∂t ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s

✉♥✐tár✐♦s t❛♥❣❡♥t❡s ❛♦ s❡❣✉♥❞♦ ❢❛t♦r ❞❡ Qn(ǫ)×R✳ ❊♥tã♦✱ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡

T ❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s η ♥♦r♠❛✐s ❛ f ✜❝❛♠ ❞❡✜♥✐❞♦s ♣♦r ∂

∂t =f∗T +η. ✭✶✳✶✳✶✮

❉❡r✐✈❛♥❞♦ (1.1.1)❡ ✉s❛♥❞♦ q✉❡ _∂t∂ é ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ♣❛r❛❧❡❧♦ ❡♠Qn(ǫ)_×R✱

♦❜t❡♠♦s ❞❛s ❡q✉❛çõ❡s ❞❡ ●❛✉ss ❡ ❈♦❞❛③③✐ q✉❡

0 = ∇¯X

∂

∂t = ¯∇Xf∗T + ¯∇Xη

= f∗∇XT +αf(X, T)−f∗AfηX+∇⊥Xη

= f∗(∇XT −AfηX) +αf(X, T) +∇⊥Xη,

❡♠ q✉❡∇ ❡∇¯ ❞❡♥♦t❛♠ ❛s ❝♦♥❡①õ❡s ❞❡ ▲❡✈✐✲❈✐✈✐t❛ ❞❡Mm _{❡ Q}n_{(ǫ)×R✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱}

Pr❡❧✐♠✐♥❛r❡s ✻

❡♥q✉❛♥t♦ ∇⊥ _{❡ α}f_{(·,·) ❞❡♥♦t❛♠ ❛ ❝♦♥❡①ã♦ ♥♦r♠❛❧ ❡ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ f✱}

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❛í✱

∇XT =AfηX ✭✶✳✶✳✷✮

❡

αf(X, T) = _−∇⊥_Xη ✭✶✳✶✳✸✮

♣❛r❛ t♦❞♦ X ∈X_(M)✳ ❆q✉✐✱ ❡ ❡♠ t♦❞❛ ❡st❛ t❡s❡✱_Af

η ❞❡♥♦t❛ ♦ ♦♣❡r❛❞♦r ❞❡ ❢♦r♠❛ ❞❡f ♥❛

❞✐r❡çã♦ ❞❡ η✱ ❞❛❞♦ ♣♦r✿

hAf_ηX, Yi=hαf(X, Y), ηi

♣❛r❛ q✉❛✐sq✉❡r X, Y _∈X_(M)✳ ◆♦t❡ q✉❡_T é ♦ ❣r❛❞✐❡♥t❡ ❞❛ ❢✉♥çã♦ ❛❧t✉r❛

h=hf , i˜ ∗ ∂ ∂ti,

❡♠ q✉❡ i:Qn(ǫ)×R_→En+2 ❞❡♥♦t❛ ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✱ s❡♥❞♦ En+2 ♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ Rn+2 q✉❛♥❞♦ ǫ= 1 ♦✉ ♦ ❡s♣❛ç♦ ▲♦r❡♥t③✐❛♥♦ Ln+2 q✉❛♥❞♦ǫ =₋1✱ ❡f˜:=i_◦f✳ ❉❡ ❢❛t♦✱

h∇h, Xi=X(h) =Xhf , i˜ ∗ ∂

∂ti=hf˜∗X, i∗ ∂

∂ti=hX, Ti

♣❛r❛ t♦❞♦ X _∈X_(M)✳

❆s ❡q✉❛çõ❡s ❞❡ ●❛✉ss✱ ❈♦❞❛③③✐ ❡ ❘✐❝❝✐ ♣❛r❛ f sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ✭✈❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✶✸❪✮

R(X, Y)Z =ǫ(X∧Y − hY, TiX∧T +hX, TiY ∧T)Z+Af_α(Y,Z)X−Af_α(X,Z)Y, ✭✶✳✶✳✹✮

(_∇⊥

Xα)(Y, Z)−(∇⊥Yα)(X, Z) = ǫ(hX, ZihY, Ti − hY, ZihX, Ti)η ✭✶✳✶✳✺✮

❡

Pr❡❧✐♠✐♥❛r❡s ✼

❡♠ q✉❡ X, Y, Z ∈ X_(M)✱ _{ξ ∈ Γ(N}f_M) ❡ _{(X ∧Y)Z = hY, ZiX − hX, ZiY.} ❆ ❡q✉❛çã♦

✭✶✳✶✳✺✮ é ❡q✉✐✈❛❧❡♥t❡ ❛

(∇XAf)(Y, ζ)−(∇YAf)(X, ζ) = ǫhη, ζi(X∧Y)T. ✭✶✳✶✳✼✮

❊♠❜♦r❛ ✐ss♦ ♥ã♦ s❡❥❛ ✉t✐❧✐③❛❞♦ ♥♦ q✉❡ s❡❣✉❡✱ ✈❛❧❡ ❛ ♣❡♥❛ ♠❡♥❝✐♦♥❛r q✉❡ ❛s ❡q✉❛çõ❡s ✭✶✳✶✳✷✮ ✲ ✭✶✳✶✳✻✮ ❞❡t❡r♠✐♥❛♠ ❝♦♠♣❧❡t❛♠❡♥t❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ f : Mm _{→ Q}n_{(ǫ)×R ❛ ♠❡♥♦s ❞❡ ✉♠❛ ✐s♦♠❡tr✐❛ ❞❡ Q}n_{(ǫ)× R ✭✈❡r ❈♦r♦❧ár✐♦ ✸ ❞❡}

❬✶✸❪✮✳

❱❛♠♦s ❛❣♦r❛ r❡❧❛❝✐♦♥❛r ❛s s❡❣✉♥❞❛s ❢♦r♠❛s ❢✉♥❞❛♠❡♥t❛✐s ❡ ❛s ❝♦♥❡①õ❡s ♥♦r♠❛✐s ❞❡f ❡ f˜✳ Pr✐♠❡✐r♦ ♥♦t❡ q✉❡ νˆ=π◦ié ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ♥♦r♠❛❧ ✉♥✐tár✐♦ ❞❛ ✐♥❝❧✉sã♦ i: Qn(ǫ)_×R_→En+2, ǫ_{∈ {−}1,1_}✱ ❡♠ q✉❡π : En+1_×R_→En+1 é ❛ ♣r♦❥❡çã♦✱ ❡

˜

∇Zνˆ = π∗i∗Z =i∗Z − hi∗Z, i∗ ∂ ∂tii∗

∂ ∂t

= i∗(Z− hZ, ∂ ∂ti

∂ ∂t)

♣❛r❛ t♦❞♦ Z ∈X_(Qn_(ǫ)×R)✱ ❡♠ q✉❡_∇˜ é ❛ ❝♦♥❡①ã♦ ❞❡ _En+2_. ▲♦❣♦

Ai_ˆνZ =₋Z +_hZ, ∂ ∂ti

∂

∂t. ✭✶✳✶✳✽✮

❖s ❡s♣❛ç♦s ♥♦r♠❛✐s✱ Nf_{M ❡ N}f˜_{M, ❞❡ f ❡ f˜✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦ r❡❧❛❝✐♦♥❛❞♦s}

♣♦r

Nf˜M =i∗NfM ⊕s♣❛♥{ν},

❡♠ q✉❡ ν = ˆν◦f =π◦f˜✳ ❉❛❞♦ ξ∈Γ(Nf_{M)✱ ♦❜t❡♠♦s ❞❡ ✭✶✳✶✳✽✮ q✉❡}

˜

∇Xi∗ξ = i∗∇¯Xξ+αi(f∗X, ξ)

= ₋f˜∗Af_ξX+i∗∇⊥Xξ+hX, Tihξ, ηiν

❞♦♥❞❡ s❡❣✉❡ q✉❡

Pr❡❧✐♠✐♥❛r❡s ✽

❡

˜

∇⊥Xi∗ξ=i∗∇⊥Xξ+hX, Tihξ, ηiν, ✭✶✳✶✳✾✮

❡♠ q✉❡ _∇˜⊥ _{é ❛ ❝♦♥❡①ã♦ ♥♦r♠❛❧ ❞❡ f .˜ P♦r ♦✉tr♦ ❧❛❞♦✱}

˜

∇Xν= ˜∇Xνˆ◦f = ˜∇f∗Xνˆ= ˜f∗(X− hX, TiT)− hX, Tii∗η,

❧♦❣♦

Af_ν˜X =−X+hX, TiT

♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

Afν˜T =−kηk2T ❡ A

˜

f

νX =−X, s❡ X ∈ {T}⊥, ✭✶✳✶✳✶✵✮

❡

˜

∇⊥Xν =−hX, Tii∗η. ✭✶✳✶✳✶✶✮

✶✳✶✳✶ ❈❧❛ss❡

A

❉❡♥♦t❛r❡♠♦s ♣♦r A ❛ ❝❧❛ss❡ ❞❛s ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s f : Mm _{→ Q}n_{(ǫ) × R ❝♦♠}

❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ q✉❡ ♦ ❝❛♠♣♦ T✱ ❞❡✜♥✐❞♦ ❡♠ (1.1.1)✱ é ✉♠ ❛✉t♦✈❡t♦r ❞❡ t♦❞♦s ♦s

♦♣❡r❛❞♦r❡s ❞❡ ❢♦r♠❛ ❞❡ f✳ ❊①❡♠♣❧♦s tr✐✈✐❛✐s ❞❡ t❛✐s ✐♠❡rsõ❡s sã♦ ♦s ❝✐❧✐♥❞r♦s s♦❜r❡ ✐♠❡rsõ❡s ✐s♦♠étr✐❝❛s g: Nm−1 _{→ Q}n_{(ǫ)✱ ♣❛r❛ ♦s q✉❛✐s M}m _{é ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣r♦❞✉t♦}

Nm−1_{×R ❡f =g×id✱ ❡♠ q✉❡ id: R→Ré ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡✳ ❊♠ t❛✐s ❡①❡♠♣❧♦s✱ ♦}

❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ♥♦r♠❛✐sη❡♠ ✭✶✳✶✳✶✮ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✳ ❊①❡♠♣❧♦s ♠❛✐s ✐♥t❡r❡ss❛♥t❡s sã♦ ❝♦♥str✉í❞♦s ❝♦♠♦ s❡❣✉❡✳

❙❡❥❛ g : Nm−1 _{→ Q}n_{(ǫ) ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠}

❝♦♥❥✉♥t♦ ♦rt♦♥♦r♠❛❧ ④ξ1, ξ2, ..., ξk⑥ ❞❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ♥♦r♠❛✐s ♣❛r❛❧❡❧♦s ❛♦ ❧♦♥❣♦

❞❡ g✳ ❊st❛ ❤✐♣ót❡s❡ é s❛t✐s❢❡✐t❛ ❧♦❝❛❧♠❡♥t❡✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ g ♣♦ss✉✐ ✜❜r❛❞♦ ♥♦r♠❛❧ ♣❧❛♥♦✳ ❆ss✐♠✱ ♦ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ E ❞❡ ♣♦st♦ k ❞♦ ✜❜r❛❞♦ ♥♦r♠❛❧ Ng_{N ❞❡ g✱ ❣❡r❛❞♦}

♣♦rξ1, ξ2, ..., ξk,é ♣❛r❛❧❡❧♦ ❡ ♣❧❛♥♦✳ ❙❡❥❛♠j : Qn(ǫ)→Qn(ǫ)×R❡i: Qn(ǫ)×R→En+2

Pr❡❧✐♠✐♥❛r❡s ✾

❊♥tã♦✱ ♦ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ _E˜ _{❞❡ N}˜g_{N ❝✉❥❛ ✜❜r❛ E}˜_{(x)✱ ❡♠ x ∈ N}m−1_{, é ❣❡r❛❞❛ ♣♦r}

˜

ξ0, ξ˜1, ..., ξ˜k+1✱ é t❛♠❜é♠ ♣❛r❛❧❡❧♦ ❡ ♣❧❛♥♦✳ ❉❡✜♥❛ ✉♠❛ ✐s♦♠❡tr✐❛ ❞❡ ✜❜r❛❞♦s ✈❡t♦r✐❛✐s

φ : Nm−1 _×Ek+2 _→E˜ _♣♦r

φx(y) :=φ(x, y) = k+1

X

i=0

yiξ˜i, ✭✶✳✶✳✶✷✮

♣❛r❛ y= (y0, y1, ..., yk+1)∈Ek+2. ❙❡❥❛

f : Mm :=Nm−1_×I _→Qn(ǫ)_×R

❞❛❞❛ ♣♦r

˜

f(x, s) := (i_◦f)(x, s) = φx(γ(s)) = k+1

X

i=0

γi(s) ˜ξi(x) ✭✶✳✶✳✶✸✮

❡♠ q✉❡ γ : I →Qk(ǫ)×R_⊂ Ek+2✱ γ = (γ0, ..., γk, γk+1), é ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r s✉❛✈❡ t❛❧

q✉❡ ǫγ2

0 +γ12+...+γk2 =ǫ ❡γk+1 ♣♦ss✉✐ ❞❡r✐✈❛❞❛ ♥ã♦ ♥✉❧❛ ❡♠ t♦❞♦ ♣♦♥t♦✳

❚❡♦r❡♠❛ ✶✳✶✳ ❬✶✻❪ ❆ ❛♣❧✐❝❛çã♦ f ❞❡✜♥❡✱ ❡♠ ♣♦♥t♦s r❡❣✉❧❛r❡s✱ ✉♠❛ ✐♠❡rsã♦ ♥❛ ❝❧❛ss❡A. ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ q✉❛❧q✉❡r ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ f :Mm _→Qn_{(ǫ)×R✱ m ≥ 2✱ ♥❛ ❝❧❛ss❡ A}

é✱ ❧♦❝❛❧♠❡♥t❡✱ ❞❛❞❛ ❞❡st❛ ❢♦r♠❛✳

❯♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ✉♠ ♣♦♥t♦ (x, s) _∈ Mm _{= N}m−1 _×R

s❡r r❡❣✉❧❛r ♣❛r❛ f é ❞❛❞❛ ♥❛ ♣❛rt❡(ii) ❞❛ Pr♦♣♦s✐çã♦ ✶✳✹ ❛❜❛✐①♦✳

❆ ❛♣❧✐❝❛çã♦ _f˜_{é ✉♠ t✉❜♦ ♣❛r❝✐❛❧ s♦❜r❡ ˜g ❝♦♠ ✜❜r❛ γ✱ ♥♦ s❡♥t✐❞♦ ❞❛❞♦ ❡♠ ❬✹❪✳}

●❡♦♠❡tr✐❝❛♠❡♥t❡✱ _f˜_{(M) é ♦❜t✐❞❛ tr❛♥s♣♦rt❛♥❞♦ ♣❛r❛❧❡❧❛♠❡♥t❡ ❛ ❝✉r✈❛φx(γ(I))✱ ❝♦♥t✐❞❛}

♥♦ ❡s♣❛ç♦ ♥♦r♠❛❧ ❞❡ g˜❡♠ x_∈Nn−1_{✱ ❝♦♠ r❡s♣❡✐t♦ à ❝♦♥❡①ã♦ ♥♦r♠❛❧ ❞❡ g˜✳}

❙❡❥❛♠ f : Mm _{→ Q}n_{(ǫ) × R ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❡ f}˜ _{= i ◦ f✱ ❡♠ q✉❡}

i:Qn(ǫ)_×R_→En+2 é ❛ ✐♥❝❧✉sã♦ ❝❛♥ô♥✐❝❛✳

❈♦r♦❧ár✐♦ ✶✳✷✳ ❬✶✻❪ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

(i) ❖ ❝❛♠♣♦ T ❡♠ ✭✶✳✶✳✶✮ é ♥ã♦ ♥✉❧♦ ❡ f˜♣♦ss✉✐ ✜❜r❛❞♦ ♥♦r♠❛❧ ♣❧❛♥♦❀

(ii) f ♣♦ss✉✐ ✜❜r❛❞♦ ♥♦r♠❛❧ ♣❧❛♥♦ ❡ ♣❡rt❡♥❝❡ à ❝❧❛ss❡ _A❀

(iii) ˜f é ❧♦❝❛❧♠❡♥t❡ ❞❛❞❛ ❝♦♠♦ ❡♠ ✭✶✳✶✳✶✸✮ ❡♠ t❡r♠♦s ❞❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ g :

Nm−1 _→Qn_{(ǫ)❝♦♠ ✜❜r❛❞♦ ♥♦r♠❛❧ ♣❧❛♥♦ ❡ ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r s✉❛✈❡γ : I →Q}k_(ǫ)×R⊂

Pr❡❧✐♠✐♥❛r❡s ✶✵

❆ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡ ❞❡s❝r❡✈❡ ❛ ❞✐❢❡r❡♥❝✐❛❧✱ ♦ ❡s♣❛ç♦ ♥♦r♠❛❧ ❡ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ _f˜_{❞❡✜♥✐❞❛ ❡♠ ✭✶✳✶✳✶✸✮✳ ❉❛❞♦s x ∈ N}m−1_{✱ X ∈T}

xN

❡ s ∈ I✱ ❞❡♥♦t❡ ♣♦r XH _{♦ ú♥✐❝♦ ✈❡t♦r ❡♠ T}

(x,s)M t❛❧ q✉❡ π1∗XH =X ❡ π2∗XH = 0✱ ❡♠ q✉❡ π1 :Mm →Nm−1 ❡π2 :Mm →I sã♦ ❛s ♣r♦❥❡çõ❡s ❝❛♥ô♥✐❝❛s✳

Pr♦♣♦s✐çã♦ ✶✳✸✳ ❙ã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

(i) ❆ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ f˜é ❞❛❞❛ ♣♦r

˜

f∗(x, s)XH = ˜g∗(x)(γ0(s)I −

k

X

i=1

γi(s)Agξi(x))X ✭✶✳✶✳✶✹✮

♣❛r❛ t♦❞♦ X ∈TxN✱ ❡♠ q✉❡ I é ♦ ❡♥❞♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ TxN✱ ❡

˜

f∗(x, s) ∂

∂s =φx(γ

′_{(s)). ✭✶✳✶✳✶✺✮}

(ii) ❆ ❛♣❧✐❝❛çã♦ f˜✭❡✱ ♣♦rt❛♥t♦✱ f✮ é ✉♠❛ ✐♠❡rsã♦ ❡♠ (x, s) s❡✱ ❡ s♦♠❡♥t❡ s❡✱

Ps(x) :=γ0(s)I−

k

X

i=1

γi(s)Agξi(x) = −A

˜

g

φx(¯γ(s)) =−A

˜

g

φx(γ(s)), ✭✶✳✶✳✶✻✮

❡♠ q✉❡ ¯γ(s) = (γ0(s), ..., γk(s),0), é ✉♠ ❡♥❞♦♠♦r✜s♠♦ ✐♥✈❡rsí✈❡❧ ❞❡ TxN.

(iii) ❙❡ f˜é ✉♠❛ ✐♠❡rsã♦ ❡♠ (x, s), ❡♥tã♦

N₍f˜_x,s)M = ˜j∗E(x)⊥⊕φx(γ′(s)⊥)⊂Nxg˜N, ✭✶✳✶✳✶✼✮

❡♠ q✉❡ E(x)⊥ _❡γ′_(s)⊥ _{❞❡♥♦t❛♠ ♦s ❝♦♠♣❧❡♠❡♥t♦s ♦rt♦❣♦♥❛✐s ❞❡ E(x) ❡♠ N}g

xN ❡ ❞❡γ′(s)

❡♠ Ek+2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡

N₍f˜_x,s)M =i∗N₍f_x,s)M ⊕s♣❛♥{(π◦f˜)(x, s)}=i∗N₍f_x,s)M ⊕φx(γ(s)), ✭✶✳✶✳✶✽✮

❡♠ q✉❡ π:En+2 _=En+1_×R→En+1 _{é ❛ ♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛✳}

(iv) ❙❡ f˜é ✉♠❛ ✐♠❡rsã♦ ❡♠ (x, s), ❡♥tã♦

Af_ξ˜(x, s)XH _{= (P}

Pr❡❧✐♠✐♥❛r❡s ✶✶

♣❛r❛ q✉❛✐sq✉❡r ξ _∈N₍f˜_x,s)M ❡ X _∈TxN,

Af_ξ˜(x, s)∂

∂s = 0, s❡ ξ∈˜j∗E(x)

⊥ _{✭✶✳✶✳✷✵✮}

❡

Af_φ˜_x(ζ)(x, s) ∂

∂s =

hγ′′_{(s), ζi}

hγ′_{(s), γ}′_(s)i ∂

∂s, s❡ ζ ∈E

k+2_,

hζ, γ′(s)_i= 0. ✭✶✳✶✳✷✶✮ ❆❧é♠ ❞✐ss♦✱

Af_ζ(x, s) =Af_i˜∗ζ(x, s) ✭✶✳✶✳✷✷✮

♣❛r❛ t♦❞♦ ζ ∈Ek+2_.

❉❡♠♦♥str❛çã♦✿ ❉❛❞❛ ✉♠❛ ❝✉r✈❛ s✉❛✈❡ β :J _→Nm−1_{✱ ❝♦♠ 0∈J✱β(0) =x ❡ β}′_{(0) =X✱} ♣❛r❛ ❝❛❞❛ s ∈ I s❡❥❛ βs : J → Mm ❞❛❞❛ ♣♦r βs(t) = (β(t), s). ❊♥tã♦ βs(0) = (x, s) ❡

β′

s(0) =XH✳ ❆ss✐♠

˜

f∗(x, s)XH = d

dt|t=0f˜(βs(t)) = d dt|t=0

k+1

X

i=0

γi(s) ˜ξi(β(t))

= d

dt|t=0γ0(s)˜g(β(t)) + d dt|t=0

k+1

X

i=1

γi(s) ˜ξi(β(t))

= γ0(s)˜g∗(x)X−

k

X

i=1

γi(s)˜g∗(x)Ag_ξ˜˜(x)X

= ˜g∗(x)(γ0(s)I−

k

X

i=1

γi(s)Ag_ξ˜˜(x))X

❡ ✭✶✳✶✳✶✹✮ s❡❣✉❡ ❞♦ ❢❛t♦ ❞❡ q✉❡ Ag_ξ˜_˜

i =A

g

ξi ♣❛r❛ t♦❞♦ 1≤i≤k✳ ❆❧é♠ ❞✐ss♦✱

˜

f∗(x, s) ∂ ∂s =

∂ ∂s

k+1

X

i=0

γi(s) ˜ξi =φx(γ′(s)).

❖s ✐t❡♥s ✭✐✐✮ ❡ ✭✐✐✐✮ s❡❣✉❡♠ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞♦ ✐t❡♠ (i).

P❛r❛ ♣r♦✈❛r ✭✶✳✶✳✶✾✮✱ ❞❛❞♦s ξ _∈ N₍f˜_x,s)M ❡ X _∈ TxN, s❡❥❛♠ β : J → Nm−1 ❡

βs :J →Mm ❝♦♠♦ ♥♦ ✐♥í❝✐♦ ❞❛ ❞❡♠♦♥str❛çã♦✳ ❊♥tã♦✱ ✉s❛♥❞♦ ✭✶✳✶✳✶✹✮ ♦❜t❡♠♦s

−f˜∗(x, s)A

˜

f

ξ(x, s)XH = ( ˜∇XHξ)T = (

d

dt|t=0ξ(βs(t)))

T _=−g˜

∗(x)Agξ˜(x)X

Pr❡❧✐♠✐♥❛r❡s ✶✷

✐st♦ é✱ ✈❛❧❡ ✭✶✳✶✳✶✾✮✳ P❛r❛ ♣r♦✈❛r ✭✶✳✶✳✷✶✮✱ ❞❛❞♦ ζ ∈Ek+2 ❝♦♠ _hζ, γ′_{(s)i= 0✱ ❡st❡♥❞❛ ζ ❛} ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ♥♦r♠❛❧ ♣❛r❛❧❡❧♦ ❛♦ ❧♦♥❣♦ ❞❡ γ✳ ❊♥tã♦

ζ′(s) = hζ

′_{(s), γ}′_(s)i

hγ′_{(s), γ}′_(s)iγ

′_{(s) = −}hγ′′(s), ζ(s)i

hγ′_{(s), γ}′_(s)iγ ′_(s)

❡

−f˜∗(x, s)A

˜

f

φx(ζ)(x, s)

∂

∂s =

˜

∇∂

∂sφx(ζ)

T

= (φx(ζ′(s)))T

= ₋hγ

′′_{(s), ζ(s)i}

hγ′_{(s), γ}′_(s)i(φx(γ ′_(s)))T

= −f˜∗(x, s)h

γ′′_{(s), ζ(s)i}

hγ′_{(s), γ}′_(s)i ∂ ∂s,

❡♠ q✉❡ ✉s❛♠♦s ✭✶✳✶✳✶✺✮ ♥❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡✳ ❆ ❡q✉❛çã♦ (1.1.21) s❡❣✉❡✳

◆♦ ❝❛s♦ ❡♠ q✉❡ ❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ f : Mm → Qn(ǫ)×R ♥❛ ❝❧❛ss❡ _A é ❞❛❞❛

❡♠ t❡r♠♦s ❞❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ g :Nm−1 _→Qn_{(ǫ)❝♦♠ ✜❜r❛❞♦ ♥♦r♠❛❧ ♣❧❛♥♦✱ ❝♦♠}

k =n−m+ 1✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛s ❛✜r♠❛çõ❡s ❞❛ Pr♦♣♦s✐çã♦ ✶✳✹ ❝♦♠♦ s❡❣✉❡✳

❈♦r♦❧ár✐♦ ✶✳✹✳ ◆❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✱ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

(i) ❆ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ f˜é ❞❛❞❛ ♣♦r

˜

f∗(x, s)XH= ˜g∗(x)(γ0(s)I−

n−_Xm+1

i=1

γi(s)Ag_ξi(x))X ✭✶✳✶✳✷✸✮

♣❛r❛ t♦❞♦ X ∈TxN✱ ❡♠ q✉❡ I é ♦ ❡♥❞♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ TxN✱ ❡

˜

f∗(x, s) ∂

∂s =φx(γ

′_{(s)). ✭✶✳✶✳✷✹✮}

(ii) ❆ ❛♣❧✐❝❛çã♦ f˜✭❡✱ ♣♦rt❛♥t♦✱ f✮ é ✉♠❛ ✐♠❡rsã♦ ❡♠ (x, s) s❡✱ ❡ s♦♠❡♥t❡ s❡✱

Ps(x) :=γ0(s)I −

n−_Xm+1

i=1

γi(s)Agξi(x) =−A

˜

g

φx(γ(s))=−A

˜

g

φx(γ(s)), ✭✶✳✶✳✷✺✮

❡♠ q✉❡ γ(s) = (γ0(s), ..., γn−m+1(s),0), é ✉♠ ❡♥❞♦♠♦r✜s♠♦ ✐♥✈❡rsí✈❡❧ ❞❡ TxN.

(iii) ❙❡ f˜é ✉♠❛ ✐♠❡rsã♦ ❡♠ (x, s)✱ ❡♥tã♦

Pr❡❧✐♠✐♥❛r❡s ✶✸

(iv) ❙❡ f˜é ✉♠❛ ✐♠❡rsã♦ ❡♠ (x, s)✱ ❡♥tã♦

αf˜(XH, YH) =αg˜(Ps(x)X, Y)− h

α˜g_(P

s(x)X, Y), φx(γ′(s))i

hγ′_{(s), γ}′_(s)i φx(γ

′_{(s)). ✭✶✳✶✳✷✼✮}

❡ αf˜_(XH_, ∂

∂s) = 0, ♣❛r❛ q✉❛✐sq✉❡r X, Y ∈TxN, ❡

αf˜( ∂

∂s, ∂

∂s) = φx(γ

′′_{(s)). ✭✶✳✶✳✷✽✮}

❉❡♠♦♥str❛çã♦✿ ❖s ✐t❡♥s (i)✱ (ii) ❡ (iii) s❡❣✉❡♠ ❞✐r❡t❛♠❡♥t❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✸✳ P❛r❛

♣r♦✈❛r♠♦s ♦ ✐t❡♠ (iv)✱ ♦❜s❡r✈❡ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ξ_∈N₍f˜_x,s)M✱ t❡♠♦s

hαf˜_(XH_{, Y}H_{), ξi = hA}f˜

ξXH, YHif˜=hf˜∗A

˜

f

ξXH,f˜∗YHi

(1.1.19),(1.1.23)

=

= _hg˜∗Aξg˜X,g˜∗Ps(x)Yi=hA˜gξX, Ps(x)Yi˜g

= _hα˜g(Ps(x)X, Y), ξi,

❡ ✭✶✳✶✳✷✼✮ s❡❣✉❡ ❞❡ ✭✶✳✶✳✷✻✮✳ ❆❧é♠ ❞✐ss♦✱

hαf˜( ∂

∂s, ∂

∂s), φx(ζ)i=hA

˜

f φx(ζ)

∂ ∂s,

∂ ∂si

(1.1.21)

= _hγ′′(s), ζ_i

s❡ ζ ∈Ek+2_{, hζ, γ}′_{(s)i= 0, ❡ ✭✶✳✶✳✷✽✮ s❡❣✉❡✳}

❖❜s❡r✈❛çã♦ ✶✳✺✳ ❉❡❝♦rr❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞♦ ❈♦r♦❧ár✐♦ ✶✳✹ q✉❡✱ s❡ {X1, ..., Xm−1} é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ TxNm−1 ❞❡ ❞✐r❡çõ❡s ♣r✐♥❝✐♣❛✐s ❞❡ g˜✱ ❡♥tã♦

∂ ∂s, X

H

1 , ..., XmH−1

é ✉♠❛ ❜❛s❡ ♦rt♦❣♦♥❛❧ ❞❡ T₍f˜_x,s)M ❞❡ ❞✐r❡çõ❡s ♣r✐♥❝✐♣❛✐s ❞❡ f˜✳

✶✳✶✳✷ ❙✉❜✈❛r✐❡❞❛❞❡s ❞❡ r♦t❛çã♦ ❡♠

Q

(

ǫ

)

_×

R

◆❡st❛ s❡çã♦ ❞❡✜♥✐r❡♠♦s✱ ❝♦♠ ❜❛s❡ ❡♠ ❬✶✻❪✱ ❛s s✉❜✈❛r✐❡❞❛❞❡s ❞❡ r♦t❛çã♦ ❞❡ ❞✐♠❡♥sã♦ m ❡♠ Qn(ǫ)_×Rq✉❡ tê♠ ❝✉r✈❛s ❝♦♠♦ ❣❡r❛tr✐③❡s✳ ❚❛❧ ❞❡✜♥✐çã♦ ❡st❡♥❞❡ ❛q✉❡❧❛ ❞❛❞❛ ❡♠ ❬✶✶❪

♣❛r❛ ♦ ❝❛s♦ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s✳

❙❡❥❛♠ (x0, ..., xn+1) ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s ❡♠ En+2 ❝♦♠ r❡s♣❡✐t♦ às q✉❛✐s ❛

♠étr✐❝❛ ❞❡ En+2 é ❡s❝r✐t❛ ❝♦♠♦

Pr❡❧✐♠✐♥❛r❡s ✶✹

❈♦♥s✐❞❡r❡ En+1 ❝♦♠♦

En+1 =_{(x0, ..., xn+1)∈En+2 :xn+1 = 0}

❡

Qn(ǫ) =_{(x0, ..., xn)∈En+1 :ǫx20+x21+...+x2n=ǫ} (x0 >0s❡ǫ=−1).

❙❡❥❛Pn−m+3 _{✉♠ s✉❜❡s♣❛ç♦ ❞❡E}n+2 _{❞❡ ❞✐♠❡♥sã♦n−m+ 3q✉❡ ❝♦♥té♠ ♦s ✈❡t♦r❡s}

e0 ❡ en+1, ❡♠ q✉❡ {e0, ..., en+1} é ❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❞❡ En+2✳ ❊♥tã♦

(Qn(ǫ)_×R)_∩Pn−m+3 =Qn−m+1(ǫ)_×R.

❉❡♥♦t❡ ♣♦rI ♦ ❣r✉♣♦ ❞❡ ✐s♦♠❡tr✐❛s ❡♠En+2q✉❡ ✜①❛ ♦s ♣♦♥t♦s ❞❡ ✉♠ s✉❜❡s♣❛ç♦Pn−m+2 _⊂

Pn−m+3 _{q✉❡ ❝♦♥té♠ ❛ ❞✐r❡çã♦ e}

n+1. ❈♦♥s✐❞❡r❡ ✉♠❛ ❝✉r✈❛ r❡❣✉❧❛rγ ❡♠ Qn−m+1(ǫ)×R⊂

Pn−m+3 _{s✐t✉❛❞❛ ❡♠ ✉♠ ❞♦s ❞♦✐s ❤✐♣❡r♣❧❛♥♦s ❞❡ P}n−m+3 _{❞❡t❡r♠✐♥❛❞♦s ♣♦r P}n−m+2_.

❉❡✜♥✐çã♦ ✶✳✻✳ ❯♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ r♦t❛çã♦ ❡♠ Qn_{(ǫ)×R ❝♦♠ ❝✉r✈❛ ❣❡r❛tr✐③γ ❡ ❡✐①♦}

Pn−m+2 _{é ❛ ór❜✐t❛ ❞❡γ s♦❜ ❛ ❛çã♦ ❞❡I✳}

❙❡❣✉❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ q✉❡ t❛❧ ✈❛r✐❡❞❛❞❡ ❞❡ r♦t❛çã♦ ♣♦ss✉✐ ❞✐♠❡♥sã♦ m✳

❙✉♣♦♥❤❛♠♦s q✉❡Pn−m+3 _{s❡❥❛ ❣❡r❛❞♦ ♣♦re}

0, em, ..., en+1✳ ◆♦ ❝❛s♦ǫ= 1 s✉♣♦♥❤❛✱

t❛♠❜é♠✱ q✉❡ Pn−m+2 _{s❡❥❛ ❣❡r❛❞♦ ♣♦r e}

m, ..., en+1✳ ❊s❝r❡✈❡♥❞♦ ❛ ❝✉r✈❛γ ❝♦♠♦

γ(s) =γ0(s)e0+

n

X

i=m

γ(s)ei−m+1+h(s)en+1, ✭✶✳✶✳✷✾✮

❝♦♠

n−_Xm+1

i=0

γ2

i = 1,❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ r♦t❛çã♦✱ ❞❡ ❞✐♠❡♥sã♦m✱ ❡♠Sn×R❝♦♠ ❝✉r✈❛ ♣❡r✜❧

γ ❡ ❡✐①♦ Pn−m+2_{✱ ♣♦❞❡ s❡r ♣❛r❛♠❡tr✐③❛❞❛ ♣♦r}

˜

f(s, t) = (γ0(s)ϕ1(t), ..., γ0(s)ϕm(t), γ1(s), ..., γn−m+1(s), h(s)), ✭✶✳✶✳✸✵✮

❡♠ q✉❡ t = (t1, ..., tm−1) ❡ ϕ= (ϕ1, ..., ϕm) ♣❛r❛♠❡tr✐③❛Sm−1 ⊂Rm.

P❛r❛ ǫ = ₋1✱ t❡♠♦s três ♣♦ss✐❜✐❧✐❞❛❞❡s ❞✐st✐♥t❛s ❛ ❝♦♥s✐❞❡r❛r✱ ❝♦♥❢♦r♠❡

Pn−m+2 _{s❡❥❛ ▲♦r❡♥t③✐❛♥♦✱ ❘✐❡♠❛♥♥✐❛♥♦ ♦✉ ❞❡❣❡♥❡r❛❞♦✱ ❡ ❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ r♦t❛çã♦ s❡rá}

Pr❡❧✐♠✐♥❛r❡s ✶✺

❝❛s♦✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ Pn−m+2 _{s❡❥❛ ❣❡r❛❞♦ ♣♦r e}

0, em+1,· · · , en+1✱ ❡ q✉❡ γ s❡❥❛ ❞❛❞❛

♣♦r(1.1.29)✱ ❝♦♠₋γ₀2+

n−_Xm+1

i=1

γ_i2 =₋1.❊♥tã♦✱ ❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ r♦t❛çã♦ ❞❡Hn_×R❝♦♠

❝✉r✈❛ ♣❡r✜❧ γ ❡ ❡✐①♦ Pn−m+2 _{♣♦❞❡ s❡r ♣❛r❛♠❡tr✐③❛❞❛ ♣♦r}

˜

f(s, t) = (γ0(s), γ1(s)ϕ1(t),· · · , γ1(s)ϕm(t), γ2(s),· · · , γn−m+1(s), h(s)), ✭✶✳✶✳✸✶✮

❡♠ q✉❡ t = (t1, ..., tm−1) ❡ ϕ= (ϕ1, ..., ϕm) ♣❛r❛♠❡tr✐③❛Sm−1 ⊂Rm.

◆♦ s❡❣✉♥❞♦ ❝❛s♦✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ Pn−m+2 _{s❡❥❛ ❣❡r❛❞♦ ♣♦r e}

m,· · · , en+1✳

❊♥tã♦✱ ❝♦♠ ❛ ❝✉r✈❛ γ t❛♠❜é♠ ❞❛❞❛ ♣♦r (1.1.29) ❝♦♠ −γ₀2+

n−_Xm+1

i=1

γ_i2 =−1✱ ❛

♣❛r❛♠❡tr✐③❛çã♦ é t❛♠❜é♠ ❞❛❞❛ ♣♦r (1.1.31)✱ s❡♥❞♦ q✉❡ ♥❡st❡ ❝❛s♦ ϕ = (ϕ1, ..., ϕm)

♣❛r❛♠❡tr✐③❛ Hm−1 _⊂Lm.

❋✐♥❛❧♠❡♥t❡✱ q✉❛♥❞♦ Pn−m+2 _{é ❞❡❣❡♥❡r❛❞♦✱ ❡s❝♦❧❤❛ ✉♠❛ ❜❛s❡ ♣s❡✉❞♦✲♦rt♦♥♦r♠❛❧}

ˆ

e0 =

1

√

2(−e0+en), eˆn= 1

√

2(e0+en), eˆj =ej, ✭✶✳✶✳✸✷✮

♣❛r❛ j _{∈ {}1,_{· · ·}, n₋1, n+ 1_}✱ ❡ s✉♣♦♥❤❛ q✉❡Pn−m+2 _{s❡❥❛ ❣❡r❛❞♦ ♣♦reˆ}

m,· · · ,ˆen+1✳ ◆♦t❡

q✉❡ heˆ0,eˆ0i= 0 =heˆn,eˆni ❡ hˆe0,eˆni= 1✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ♣❛r❛♠❡tr✐③❛rγ ♣♦r

γ(s) =γ0(s)ˆe0+

n

X

i=m

γi−m+1(s)ˆei+h(s)ˆen+1, ✭✶✳✶✳✸✸✮

❝♦♠ 2γ0(s)γn−m+1(s) + Pn_i=1−mγi2(s) = −1✱ ❡ ❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❝♦rr❡s♣♦♥❞❡♥t❡

s✉❜✈❛r✐❡❞❛❞❡ ❞❡ r♦t❛çã♦ ♥❛s ❝♦♦r❞❡♥❛❞❛s ♣s❡✉❞♦✲♦rt♦♥♦r♠❛✐s ❡s❝♦❧❤✐❞❛s é

˜

f(s, t) = γ0, γ0t1,· · · , γ0tm−1, γ1,· · · , γn−m, γn−m+1−

γ0

2

m_X−1

i=1

t2_i, h

!

, ✭✶✳✶✳✸✹✮

❡♠ q✉❡ t = (t1, ..., tm−1) ♣❛r❛♠❡tr✐③❛ Rm−1✱ γi =γi(s)✱ 0≤i≤n−m+ 1✱ ❡h=h(s).

❚❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❞❡ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s s✉❜✈❛r✐❡❞❛❞❡s ❞❡ r♦t❛çã♦ ❡♠

Qn(ǫ)_×R✿

❚❡♦r❡♠❛ ✶✳✼✳ ❬✶✻❪ ❙❡❥❛ f :Mm _→Qn_{(ǫ)×R✱ ǫ ∈ {−1,1}✱ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ t❛❧}

q✉❡ ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s T✱ ❞❡✜♥✐❞♦ ❡♠(1.1.1)✱ ♥ã♦ s❡ ❛♥✉❧❡ ❡♠ ♥❡♥❤✉♠ ♣♦♥t♦✳ ❊♥tã♦ ❛s

s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

Pr❡❧✐♠✐♥❛r❡s ✶✻

t♦t❛❧♠❡♥t❡ ❣❡♦❞és✐❝❛ Qn−m+1(ǫ)×R_⊂Qn(ǫ)×R❀

(ii) f é ❞❛❞❛ ❝♦♠♦ ❡♠ ✭✶✳✶✳✶✸✮ ❡♠ t❡r♠♦s ❞❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ✉♠❜í❧✐❝❛ g :Nm−1 _→Qn_(ǫ)❀

(iii) ❡①✐st❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ♥♦r♠❛✐s ζ ❛♦ ❧♦♥❣♦ ❞❡ f t❛❧ q✉❡

Af_ξX =< ζ, ξ > X ✭✶✳✶✳✸✺✮

♣❛r❛ q✉❛✐sq✉❡r X _{∈ {}T_}⊥ _{❡ ξ ∈Γ(N}f_M)✳

✶✳✷ ❍✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ r♦t❛çã♦ ❞❡

Q

(c)

✳

❉❡♥♦t❛r❡♠♦s ♣♦r Qn_s(c) ✉♠❛ ❢♦r♠❛ ❡s♣❛❝✐❛❧ ♣s❡✉❞♦✲❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❞✐♠❡♥sã♦ n✱ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ c❡ í♥❞✐❝❡s∈ {0,1}✱ ✐st♦ é✱Qn_s(c)é ✉♠❛ ❢♦r♠❛ ❡s♣❛❝✐❛❧ ❘✐❡♠❛♥♥✐❛♥❛

♦✉ ▲♦r❡♥t③✐❛♥❛ ❝♦♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ c✱ ❝♦♥❢♦r♠❡ s❡❥❛ s = 0 ♦✉ s = 1✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ➱ ✉♠ ❢❛t♦ ❝♦♥❤❡❝✐❞♦ q✉❡Qn_s(c)❛❞♠✐t❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ✉♠❜í❧✐❝❛

❡♠ Rn_s++1_ǫ0✱ ❡♠ q✉❡ R

n+1

s+ǫ0 ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ♣s❡✉❞♦✲❘✐❡♠❛♥♥✐❛♥♦ ❞❡ ❞✐♠❡♥sã♦ n+ 1 ❝✉❥❛

♠étr✐❝❛ ♣♦ss✉✐ ❝✉r✈❛t✉r❛ ♥✉❧❛ ❡ í♥❞✐❝❡ s+ǫ0✱ ❡♠ q✉❡ǫ0 = 0 ♦✉1✱ ❝♦♥❢♦r♠❡ s❡❥❛ c >0♦✉

c < 0✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P❛r❛ c= 0✱ ❛ Qn_s(c) ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ Rn ♦✉ ♦ ❡s♣❛ç♦

❞❡ ▲♦r❡♥t③ Ln✱ q✉❛♥❞♦ s= 0 ♦✉ s= 1✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

◆❡st❛ s❡çã♦ ❞❡✜♥✐r❡♠♦s ❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ r♦t❛çã♦ f : M3 _{→ Q}4

s(c)

s♦❜r❡ ✉♠❛ ❝✉r✈❛✱ ❡st❡♥❞❡♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ r♦t❛çã♦ ❞❡ ❢♦r♠❛s ❡s♣❛❝✐❛✐s ❘✐❡♠❛♥♥✐❛♥❛s ❞❛❞❛ ❡♠ ❬✶✷❪✳ ❆❧é♠ ❞✐ss♦✱ ❡①✐❜✐r❡♠♦s ♣❛r❛♠❡tr✐③❛çõ❡s ❞❡ t❛✐s ❤✐♣❡rs✉♣❡r❢í❝✐❡s✳ ❚❛✐s ♣❛r❛♠❡tr✐③❛çõ❡s s❡rã♦ út❡✐s ♥♦ ❈❛♣ít✉❧♦ ✹✳

❙❡❥❛P3 _{✉♠ s✉❜❡s♣❛ç♦ ❞❡R}5

s+ǫ0 ⊃Q 4

s(c)❞❡ ❞✐♠❡♥sã♦3t❛❧ q✉❡P3∩Q4s(c)6=∅✱ ❡♠

q✉❡ǫ0 = 0 ♦✉1✱ ❝♦♥❢♦r♠❡ s❡❥❛c > 0♦✉c <0✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♥♦t❡ ♣♦r I ♦ s✉❜❣r✉♣♦

❞♦ ❣r✉♣♦ ❞❡ ✐s♦♠❡tr✐❛s ❞❡ R5_s+ǫ0 q✉❡ ✜①❛ ♦s ♣♦♥t♦s ❞♦ s✉❜❡s♣❛ç♦ P2 _{⊂ P}3_{. ❈♦♥s✐❞❡r❡}

✉♠❛ ❝✉r✈❛ r❡❣✉❧❛r γ ❡♠ Q2_s(c) = P3_∩Q4

s(c)✱ ❝♦♥t✐❞❛ ❡♠ ✉♠ ❞♦s ❞♦✐s s❡♠✐✲❡s♣❛ç♦s ❞❡ P3

❞❡t❡r♠✐♥❛❞♦s ♣♦r P2_.

❉❡✜♥✐çã♦ ✶✳✽✳ ❯♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❞❡ r♦t❛çã♦ f : M3 _{→ Q}4

s(c) ❝♦♠ ❝✉r✈❛ ❣❡r❛tr✐③ γ ❡

❡✐①♦ P2 _{é ❛ ór❜✐t❛ ❞❡γ s♦❜ ❛ ❛çã♦ ❞❡I.}

❙❡P2 _{é ♥ã♦✲❞❡❣❡♥❡r❛❞♦✱ ❡♥tã♦ f ♣♦❞❡ s❡r ♣❛r❛♠❡tr✐③❛❞❛ ♣♦r}

Pr❡❧✐♠✐♥❛r❡s ✶✼

❝♦♠ r❡s♣❡✐t♦ ❛ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧{e1,· · · , e5}❞❡R5s+ǫ0 s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s(i)♦✉

(ii)❛❜❛✐①♦✱ ❝♦♥❢♦r♠❡ ❛ ♠étr✐❝❛ ❡♠P2 _{♣♦ss✉❛ í♥❞✐❝❡s+ǫ}

0 ♦✉s+ǫ0−1✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✿

✭✐✮ hei, eii = 1 ♣❛r❛ 1 ≤ i ≤ 3✱ he3+j, e3+ji = ǫj ♣❛r❛ 1 ≤ j ≤ 2✱ ❝♦♠ (ǫ1, ǫ2) ✐❣✉❛❧ ❛

(1,1)✱ (1,₋1) ♦✉(₋1,₋1)✱ ❝♦♥❢♦r♠❡ s❡❥❛ s+ǫ0 ❢♦r 0✱ 1♦✉ 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

✭✐✐✮ he1, e1i = −1✱ hei, eii = 1 ♣❛r❛ 2 ≤ i ≤ 4 ❡ he5, e5i = ¯ǫ✱ ❡♠ q✉❡ ¯ǫ = 1 ♦✉ ¯ǫ = −1✱

❝♦♥❢♦r♠❡ s+ǫ0 s❡❥❛ ✐❣✉❛❧ ❛ 1 ♦✉2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ t❡♠♦s q✉❡ P2 _{= span{e}

4, e5}✱ P3 = span{e1, e4, e5}✱ u = (u1, u2)✱

γ(s) = (γ1(s), γ4(s), γ5(s)) é ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❡♠

Q2

s(c) ⊂ P3 ❡ ϕ(u) = (ϕ1(u), ϕ2(u), ϕ3(u)) é ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ♦rt♦❣♦♥❛❧ ❞❛ ❡s❢❡r❛

✉♥✐tár✐❛ S2 _⊂ (P2₎⊥ _{♥♦ ❝❛s♦ (i) ❡ ❞♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ H}2 _{⊂ (P}2₎⊥ _{♥♦ ❝❛s♦ (ii)✳ ❉✐③✲s❡} q✉❡ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ é✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞♦ t✐♣♦ ❡s❢ér✐❝♦ ♦✉ ❤✐♣❡r❜ó❧✐❝♦✳

❙❡P2 _{é ❞❡❣❡♥❡r❛❞♦✱ ❡♥tã♦f é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❞❡ r♦t❛çã♦ ❞♦ t✐♣♦ ♣❛r❛❜ó❧✐❝♦✱}

q✉❡ ♣♦❞❡ s❡r ♣❛r❛♠❡tr✐③❛❞❛ ♣♦r

f(s, u) = (γ1(s), γ1(s)u1, γ1(s)u2, γ4(s)−

1

2γ1(s)(u

2

1+u22), γ5(s)), ✭✶✳✷✳✷✮

❝♦♠ r❡s♣❡✐t♦ ❛ ✉♠❛ ❜❛s❡ ♣s❡✉❞♦✲♦rt♦♥♦r♠❛❧ {e1, . . . , e5} ❞❡ Rs5+ǫ0 t❛❧ q✉❡ he1, e1i = 0 = he4, e4i✱ he1, e4i = 1✱ he2, e2i = 1 = he3, e3i ❡ he5, e5i = ¯¯ǫ := −2(s +ǫ0) + 3✱ ❡♠ q✉❡

γ(s) = (γ1(s), γ4(s), γ5(s)) é ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❡♠

Q2_s(c)⊂P3 _=span{e

1, e4, e5}✳

◆❛ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ❝❛❧❝✉❧❛♠♦s ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í✲ ❝✐❡ ❞❡ r♦t❛çã♦ f :M3 _→Q4

s(c)✱ c6= 0.

Pr♦♣♦s✐çã♦ ✶✳✾✳ ❙❡❥❛ f :M3 _→Q4

s(c)✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❞❡ r♦t❛çã♦✳ ❊♥tã♦ ♦s ❝❛♠♣♦s

❝♦♦r❞❡♥❛❞♦s ∂

∂ui✱ 1≤i≤2 ❡

∂

∂s sã♦ ❞✐r❡çõ❡s ♣r✐♥❝✐♣❛✐s✳ ❆❧é♠ ❞✐ss♦✱

(i) ❙❡ f é ❞♦ t✐♣♦ ❡s❢ér✐❝♦✱ ❡♥tã♦ ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s λ1 = λ2 := λ ❡ λ3

❝♦rr❡s♣♦♥❞❡♥t❡s ❛s ❞✐r❡çõ❡s ∂

∂ui✱ 1≤i≤2 ❡

∂

∂s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦✿

λ =        p

1−cγ2 1 −γ1′2

γ1

, s❡ ǫ1 =ǫ2 = 1

p

−1 +cγ2 1 +γ1′2

γ1

, s❡ ǫ1 6=ǫ2

❡ λ3 =

       − γ ′′

1 +cγ1

p

1−cγ2 1 −γ1′2

, s❡ ǫ1 =ǫ2 = 1

γ′′

1 +cγ1

p

−1 +cγ2 1 +γ1′2