Bilhares Convexos em Superfícies de Curvatura Constante

▲✉❝✐❛♥♦ ❈♦✉t✐♥❤♦ ❞♦s ❙❛♥t♦s

❇✐❧❤❛r❡s ❡♠ ❙✉♣❡r❢í❝✐❡s ❞❡ ❈✉r✈❛t✉r❛

❈♦♥st❛♥t❡

▲✉❝✐❛♥♦ ❈♦✉t✐♥❤♦ ❞♦s ❙❛♥t♦s

❇✐❧❤❛r❡s ❡♠ ❙✉♣❡r❢í❝✐❡s ❞❡ ❈✉r✈❛t✉r❛

❈♦♥st❛♥t❡

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❝✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ♠❛t❡♠át✐❝❛ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡ ❢❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r❛✿ ❙ô♥✐❛ P✐♥t♦ ❞❡ ❈❛r✈❛❧❤♦

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s

✷✵✶✹

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r♦✳

❙❡❣✉♥❞♦✳

✳✳✳

❘❡s✉♠♦

❆❜str❛❝t

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✷

✷ ▼♦❞❡❧♦s ❞❡ ●❡♦♠❡tr✐❛ ✻

✷✳✶ P❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✶✳✶ ❙♦❜r❡ ❝✉r✈❛s ❡♠E2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✷ ❊s❢❡r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷✳✶ ❙♦❜r❡ ❝✉r✈❛s ❡♠S2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✸ P❧❛♥♦ ❍✐♣❡r❜ó❧✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸✳✶ ❙♦❜r❡ ❝✉r✈❛s ❡♠H2 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷}

✸ ❇✐❧❤❛r❡s ✶✹

✸✳✶ ❆♣❧✐❝❛çã♦ ❞♦ ❇✐❧❤❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✸✳✶✳✶ ❋✉♥çã♦ ❣❡r❛❞♦r❛ ❡ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞♦ ❇✐❧❤❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✶✳✷ ❆ ❞❡r✐✈❛❞❛ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✶✳✸ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ ❯♠ ❡①❡♠♣❧♦ ✐♥t❡r❡ss❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✸ Pr♦♣r✐❡❞❛❞❡s ❚✇✐st ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✹ Ór❜✐t❛s P❡r✐ó❞✐❝❛s ✸✺

✹✳✶ Pr♦♣r✐❡❞❛❞❡s ❣❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✷ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ór❜✐t❛s P❡r✐ó❞✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✸ Ór❜✐t❛s ♣❡r✐ó❞✐❝❛s ❤✐♣❡r❜ó❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✸✳✶ P❡rt✉r❜❛çõ❡s ♥♦r♠❛✐s ❞❡ ♦✈❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✺ ❇✐❧❤❛r ❝✐r❝✉❧❛r ♣❡rt✉r❜❛❞♦ ✺✺

✺✳✶ ❈ír❝✉❧♦s ❣❡♦❞és✐❝♦s ♣❡rt✉r❜❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✷ ❈✉r✈❛s ✐♥✈❛r✐❛♥t❡s r❡ss♦♥❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✺✳✷✳✶ P♦t❡♥❝✐❛❧ ❘❛❞✐❛❧ ❞❡ ▼❡❧♥✐❦♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✻ ▼❡❞✐❞❛ ❞❡ ❍❛✉s❞♦r✛ ❞♦ ❝♦♥❥✉♥t♦ ❞❛ ór❜✐t❛s ❞❡ ♣❡rí♦❞♦ três ✻✼

✻✳✶ ❯♠ ❜✐t ❞❡ t❡♦r✐❛ ❞❛ ♠❡❞✐❞❛ ❞❡ ❍❛✉s❞♦r✛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✻✳✷ ❈♦♥❥✉♥t♦s s✲❞✐♠❡♥s✐♦♥❛✐s1< s≤2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵

✻✳✸ ❘❡s✉❧t❛❞♦s ❛✉①✐❧✐❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✻✳✹ Pr♦✈❛ ❞♦ t❡♦r❡♠❛ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✻✳✺ Pr♦✈❛ ❞♦ t❡♦r❡♠❛ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

◆♦ss♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ♥❡ss❡ tr❜❛❧❤♦ é ♦ ♣r♦❜❧❡♠❛ ❞♦ ❜✐❧❤❛r ❡♠ s✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡✳ ❖ ♣r♦❜❧❡♠❛ ❞♦ ❜✐❧❤❛r ♥♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❇✐r❦❤♦✛ ❬✶❪ ♥♦ ❝♦♠❡ç♦ ❞♦ sé❝✉❧♦ ❳❳✱ ❝♦♥s✐st❡ ♥♦ ❡st✉❞♦ ❞♦ ♠♦✈✐♠❡♥t♦ r❡t✐❧í♥❡♦ ❧✐✈r❡ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❞❡♥tr♦ ❞❡ ✉♠❛ r❡❣✐ã♦ ♣❧❛♥❛ ❧✐♠✐t❛❞❛ ♣♦r ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛✱ r❡✢❡t✐♥❞♦ ❡❧❛st✐❝❛♠❡♥t❡ ♥♦s ✐♠♣❛❝t♦s ❝♦♠ ♦ ❜♦r❞♦✳

❙❡❥❛S ✉♠❛ s✉♣❡r❢í❝✐❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ❝♦♠♣❧❡t❛ ❡ ❞❡ ❝✉r✈❛t✉r❛ ❣❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡K✳ ❖✉ s❡❥❛✱ S é✱ ♦✉ ♦ ♣❧❛♥♦ ❤②♣❡r❜ó❧✐❝♦H2 s❡_{K =−1}♦✉ ❛ ❡s❢❡r❛ ✉♥✐tár✐❛S2 s❡_{K = 1}✱ ♦✉ ♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦

M = E2 s❡ _{K = 0}✳ ❉❛❞❛ ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛ _Γ ❡♠ _S✱ ♦ ❜✐❧❤❛r ❝♦♥s✐st❡ ❞♦ ♠♦✈✐♠❡♥t♦ ❧✐✈r❡ ❞❡

✉♠❛ ♣❛rtí❝✉❧❛ ♥❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ♣♦r ❡ss❛ ❝✉r✈❛✳ ❚❛❧ ♠♦✈✐♠❡♥t♦ é r❡❛❧✐③❛❞♦ ❛tr❛✈és ❞❡ ❣❡♦❞és✐❝❛s ❡ ❛s ❝♦❧✐sõ❡s ❝♦♠ ❛ Γ s❡❣✉❡♠ ❛ ❧❡✐ â♥❣✉❧♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ ✐❣✉❛❧ ❛♦ â♥❣✉❧♦ ❞❡ r❡✢❡①ã♦✳

◗✉❛♥❞♦ ♦ ❜♦r❞♦ Γ é ✉♠❛ ♦✈❛❧ ✭✐✳❡✳ ✉♠❛ ❝✉r✈❛✱ ❢❡❝❤❛❞❛✱ r❡❣✉❧❛r✱ s✐♠♣❧❡s✱ ♦r✐❡♥t❛❞❛✱ ♣❡❧♦ ♠❡✲

♥♦s C2_{✱ ❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✱ ❡st❡ ♣r♦❜❧❡♠❛ ❞❡✜♥❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s ❝♦♥s❡r✈❛t✐✈♦s}

❜✐❞✐♠❡♥s✐♦♥❛✐s q✉❡ sã♦ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ s✐♠♣❧❡s✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❛ ❣❡♦♠❡tr✐❛ ❞♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉r❛çõ❡s ♣❡r♠✐t❡ s✐♠♣❧✐✜❝❛r ♠✉✐t❛s ❞❛s q✉❡stõ❡s ❞❛ ❞✐♥â♠✐❝❛✱ ❡ q✉❡ sã♦ ♥✉♠❡r✐❝❛♠❡♥t❡ tr❛✲ tá✈❡✐s✱ ♣❡r♠✐t✐♥❞♦ ✉♠❛ ❡①♣❧♦r❛çã♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ✐♠♣♦rt❛♥t❡ ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❞✐♥â♠✐❝❛✳ ❊st❛s ❝❛r❛❝t❡ríst✐❝❛s ♣❡r♠✐t❡♠ q✉❡ s❡❥❛♠ ✉s❛❞♦s ❝♦♠♦ ♠♦❞❡❧♦s s✐♠♣❧✐✜❝❛❞♦s ♥♦ ❡st✉❞♦ ❞❡ ❣❡♦❞é✲ s✐❝❛s✱ ❡♠ ót✐❝❛✱ ❛❝úst✐❝❛✱ ♠❡❝â♥✐❝❛ ❡st❛tíst✐❝❛✱ ❢ís✐❝❛ ❞❡ ♣❛rtí❝✉❧❛s✱ ❛str♦❢ís✐❝❛ ❡ s✐r✈❛♠ ❞❡ t❡rr❡♥♦ ✐❞❡❛❧ ♣❛r❛ ❝♦♠♣❛r❛çõ❡s ❡♥tr❡ ♠❡❝â♥✐❝❛ ❝❧áss✐❝❛ ❡ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳

❆❧é♠ ❞✐st♦✱ ♦s ❢❡♥ô♠❡♥♦s q✉❡ ♦❝♦rr❡♠ ♥♦s ❜✐❧❤❛r❡s✱ ❛♣❛r❡❝❡♠ ♥❛ ❝❧❛ss❡ ♠❛✐♦r ❞♦s s✐st❡♠❛s ❞✐♥â✲ ♠✐❝♦s ❝♦♥s❡r✈❛t✐✈♦s ❡ ♣♦rt❛♥t♦✱ ♦ s✉❝❡ss♦ ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❞✐♥â♠✐❝❛ ❞♦s ❜✐❧❤❛r❡s ♣♦ss✉✐ ✈❛❧♦r ✐♥trí♥s❡❝♦ ♣❛r❛ ❛ ár❡❛ ❡ ❝♦♥tr✐❜✉✐ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ♠❛✐s ❣❡r❛✐s ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s✳ ❉❡♥♦t❛r❡♠♦s ♣♦r S ✉♠❛ ❞❛s três s✉♣❡r❢í❝✐❡s✿ ♦ ♣❧❛♥♦ ❊✉❝❧✐❞✐❛♥♦ E✱ ✉♠ ❤❡♠✐s❢ér✐♦ ❞❛ ❡s❢❡r❛

✉♥✐tár✐❛ S2

+ ♦✉ ♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ H2✳ ❊ss❡ ♣r♦❜❧❡♠❛ é ♠♦❞❡❧❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❙❡❥❛ Γ

✉♠❛ ♦✈❛❧ ❝♦♥t✐❞❛ ❡♠ S✱ ✐st♦ é✱ ✉♠❛ ❝✉r✈❛ s✐♠♣❧❡s✱ ❢❡❝❤❛❞❛✱ ♣❡❧♦ ♠❡♥♦s C2_{✱ ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❡s✲}

tr✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ ❡ ❝♦♠ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛ ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛✳ ❙❡♥❞♦ S ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛ t❡♠♦s q✉❡ ❞❛❞♦Γ(s0)♥♦ tr❛ç♦ ❞❛ ❝✉r✈❛ Γ❡ ✉♠ ✈❡t♦r~v0 ✉♥✐tár✐♦ ❡♠ TΓ(s0)S ❛♣♦♥t❛♥❞♦ ♣❛r❛ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ♣♦r Γ✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❣❡♦❞és✐❝❛ γ q✉❡ ♣❛rt❡ ❞❡ γ(s0) t❡♠ ~v0

❝♦♠♦ ✈❡t♦r t❛♥❣❡♥t❡✳ ❙❡♥❞♦ Γ ✉♠❛ ❝✉r✈❛ ❢❡❝❤❛❞❛✱ ❛ ❣❡♦❞és✐❝❛ γ ✐♥t❡r❝❡♣t❛rá Γ ❡♠ ✉♠ ♥♦✈♦

♣♦♥t♦ Γ(s1) q✉❡ é ú♥✐❝♦ ♣❡❧❛ ❝♦♥✈❡①✐❞❛❞❡ ❣❡♦❞és✐❝❛ ❞❡ Γ✳ ❙❡❥❛♠ 0 < θ0 < π ♦ â♥❣✉❧♦ ❞♦ ✈❡t♦r

t❛♥❣❡♥t❡γ′_(s

0)❛♦ ✈❡t♦r~v0 ❡0< θ1 < π♦ â♥❣✉❧♦✱ ❡♥tr❡ ♦ ✈❡t♦r t❛♥❣❡♥t❡Γ′(s1)❡ ♦ ✈❡t♦r t❛♥❣❡♥t❡

à ❣❡♦❞és✐❝❛ γ ❡♠ α(s1)✱ ❞❡♥♦t❛❞♦ ♣♦r γ′(s1)✳ ❉✐③❡♠♦s q✉❡ s1 ❡ θ1 sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦♥t♦

❞❡ s❛í❞❛ ❡ â♥❣✉❧♦ ❞❡ s❛í❞❛ ❡ ❞✐③❡♠♦s q✉❡ θ2 é ♦ â♥❣✉❧♦ ❞❡ ❜❛t✐❞❛ ✭♦✉ â♥❣✉❧♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛✮ ❞❛

♣❛rtí❝✉❧❛ ❡♠ Γ(s2)✳ ❉❡✜♥✐♠♦s ❛ tr❛❥❡tór✐❛ ❞❛ ♣❛rtí❝✉❧❛ ❛♣ós ❛ r❡✢❡①ã♦ ❝♦♠♦ s❡♥❞♦ ❛ ❣❡♦❞és✐❝❛

q✉❡ ♣❛rt❡ ❞❡ α(s1) ♥❛ ❞✐r❡çã♦ ❞❛❞❛ ♣❡❧♦ ✈❡t♦r ~v1 q✉❡ é ❛ r❡✢❡①ã♦ ❞❡ γ′(s1) ♣❡❧♦ ✈❡t♦r t❛♥❣❡♥t❡

Γ′_(s

1)✳ ❆ss✐♠✱ ♣♦r ❝♦♥str✉çã♦✱ t❡♠♦s q✉❡ ♦ â♥❣✉❧♦ ❞❡ α′(s1)❛~v1✱ ❞✐t♦ â♥❣✉❧♦ ❞❡ r❡✢❡①ã♦✱ é θ1✳

❈♦♠♦ ♥♦ ❝❛s♦ ❞♦ ♣r♦❜❧❡♠❛ ♦r✐❣✐♥❛❧❞❡ ❇✐r❦❤♦✛✱ t❡♠♦s ❛ss✐♠ ❞❡✜♥✐❞❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r F : [0, l)×(0, π)7→[0, l)×(0, π)✱ F(s0, ψ0) = (s1, ψ1)✳

◆❡ss❡ tr❛❜❛❧❤♦✱ s❡❣✉✐♥❞♦ ❛ tr❛❞✐çã♦ ✐♥✐❝✐❛❞❛ ♣♦r ❇✐r❦❤♦✛✱ ♠♦str❛♠♦s q✉❡ ❛♣❧✐❝❛çã♦ ❞❡ ❜✐❧❤❛r ❛❞♠✐t❡ ✉♠ tr❛t❛♠❡♥t♦ ✈❛r✐❛❝✐♦♥❛❧✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ r❡❧❛❝✐♦♥❛♠♦s ór❜✐t❛s ❞❛ ❛♣❧✐❝❛çã♦ ❛ ♣♦♥t♦s ❝rít✐❝♦s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❢✉♥❝✐♦♥❛❧✱ ❛ ❛çã♦ ❛ss♦❝✐❛❞❛ ❛♦ ♣r♦❜❧❡♠❛✳ ❆ ♣❛rt✐r ❞❡ss❛ ❞❡s❝r✐çã♦ ♣♦❞❡♠♦s ❡①♣❧♦r❛r ✈ár✐♦s ❛s♣❡❝t♦s ❞❛ ❞✐♥â♠✐❝❛ ❞♦ ❜✐❧❤❛r✳

❚r❛❥❡tór✐❛s ♣❡r✐ó❞✐❝❛s

❆s tr❛❥❡tór✐❛s ❞♦ ❜✐❧❤❛r sã♦ ♣♦❧✐❣♦♥❛✐s ❣❡♦❞és✐❝❛s ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ❞♦ ♠♦✈✐♠❡♥t♦✱ ❝♦♠ ✈ért✐❝❡s ♥❛ ❝✉r✈❛ Γ✳ ❚r❛❥❡tór✐❛s ♣❡r✐ó❞✐❝❛s✱ ❝♦♠ ♣❡rí♦❞♦ n ≥ 2✱ ♣♦r ✈♦❧t❛r❡♠ ❛♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❞❡♣♦✐s ❞❡

n ❜❛t✐❞❛s✱ s❡rã♦ ♣♦❧í❣♦♥♦s ✐♥s❝r✐t♦s✱ ❝♦♠ ❡①❛t❛♠❡♥t❡ n ✈ért✐❝❡s✳ ❈❤❛♠❛♠♦s ❞❡ (k, n)✲tr❛❥❡tór✐❛

à tr❛❥❡tór✐❛ ❞❡ ♣❡rí♦❞♦n q✉❡ ❞ák ✈♦❧t❛s ❡♠Γ✱ ❛♥t❡s ❞❡ ❢❡❝❤❛r✳ ▼♦str❛r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ??q✉❡

♣❛r❛ ❜✐❧❤❛r❡s ❡♠ S✱ ❛ss✐♠ ❝♦♠♦ ❇✐r❦❤♦✛ ❡♠ ❬✶❪ ♣❛r❛ ♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦✱ s❡ Γ é ✉♠❛ ♦✈❛❧ ❡♥tã♦

♣❛r❛ ❝❛❞❛ ♣❛r (k, n)✱ 0 < k/n < 1 ❡ ♠❞❝(k, n) = 1✱ ❡①✐st❡♠ ♣❡❧♦ ♠❡♥♦s ❞✉❛s (k, n)✲tr❛❥❡tór✐❛s✱

❝♦rr❡s♣♦♥❞❡♥❞♦ ❛ ♣♦❧í❣♦♥♦s ❞✐st✐♥t♦s✳ ◆♦ ❡st✉❞♦ ❞♦ ❝❛s♦ ❡s❢ér✐❝♦ ❡♥❝♦♥tr❛♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❜✐❧❤❛r q✉❡ ♣♦ss✉✐ ❛♣❡♥❛s ór❜✐t❛s ❞❡ ♣❡rí♦❞♦ ❞♦✐s✿ ♦ ❜✐❧❤❛r ❡♠ ✉♠ ❡q✉❛❞♦r✳ ❚❛❧ ❡①❡♠♣❧♦ é ✐♠♣♦s✲ sí✈❡❧ ❡♠ ❜✐❧❤❛r❡s ♣❧❛♥♦s ❡ s❡ ❞❡✈❡r à❡①✐stê♥❝✐❛ ❞❡ ♣♦♥t♦s ❝♦♥❥✉❣❛❞♦s ♥❛ ❡s❢❡r❛✳

❯s❛♥❞♦ ♦ P♦t❡♥❝✐❛❧ ❘❛❞✐❛❧ ❞❡ ▼❡❧♥✐❦♦✈✱ ❞❡✜♥✐❞♦ ❡♠ ❬❄❪✱ ❡ s✉❛s ❝♦♥s❡q✉ê♥❝✐❛s✱ ❡ ♣❛r❛ ❝❛❞❛

(k, n) ✜①❛❞♦✱ 0< k/n <1✱ ♠❞❝(k, n) = 1✱ ❡♠ ❬✻❪✱ P✐♥t♦✲❞❡✲❈❛r✈❛❧❤♦ ❡ ❘❛♠ír❡③✲❘♦s ❝♦♥str✉ír❛♠

♣❡rt✉r❜❛çõ❡s ❞❡ ❝ír❝✉❧♦s ♣❧❛♥♦s ❝♦♠ q✉❛❧q✉❡r ♥ú♠❡r♦ ♣❛r ❞❡ (k, n)✲tr❛❥❡tór✐❛s ❬❄❪ ❡ ❛ss✐♠ ❛♣r❡✲

s❡♥t❛♥❞♦ ❜✐❧❤❛r❡s ❛t✐♥❣✐♥❞♦ ♦ ♠í♥✐♠♦ ♣r❡✈✐st♦ ♣♦r ❇✐r❦❤♦✛✳ ❚❛♠❜é♠ ❝❧❛ss✐✜❝❛r❛♠ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡st❛s tr❛❥❡tór✐❛s✿ ♠❡t❛❞❡ ❞❡❧❛s é ❤✐♣❡r❜ó❧✐❝❛✱ ❡ ❧♦❣♦ ✐♥stá✈❡❧✱ ❡ ❛ ♦✉tr❛ ♠❡t❛❞❡ é ❧✐♥❡❛r♠❡♥t❡ ❡❧í♣t✐❝❛ ❬❄❪✱ ♠❛s ♥ã♦ ❡st✉❞❛♠ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡st❛s tr❛❥❡tór✐❛s ❡❧í♣t✐❝❛s✳

◆♦ ❝❛♣ít✉❧♦ ❝♦♥str✉í♠♦s ♦ P♦t❡♥❝✐❛❧ ❘❛❞✐❛❧ ❞❡ ▼❡❧♥✐❦♦✈ ♣❛r❛ ♣❡rt✉r❜❛çõ❡s ❞♦ ❜✐❧❤❛r ❝✐r❝✉❧❛r ❣❡♦❞és✐❝♦ ❡♠ S ❡ ♠♦str❛♠♦s q✉❡ ❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✻❪ ✈❛❧❡♠ ♣❛r❛ ❜✐❧❤❛r❡s ♦✈❛✐s ❡♠ S✳

❯♠❛ ♦✉tr❛ q✉❡stã♦ ❛ s❡r ❛❜♦r❞❛❞❛ ✈❡rs❛ s♦❜r❡ ❛ ♥✉❧✐❞❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ór❜✐t❛s ❝♦♠ ✉♠ ❞❛❞♦ ♣❡rí♦❞♦✳ ❊st✉❞❛♥❞♦ ❡ss❛ q✉❡stã♦ ♣❛r❛ ❜✐❧❤❛r❡s ♥♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦✱ ❲♦❥t❦♦✇✐s❦✱ ❡♠ [✸✸]✱ ❞❡♠♦♥str♦✉ ❛tr❛✈és ❞❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ✈✐❛ ❝❛♠♣♦s ❞❡ ❏❛❝♦❜✐ q✉❡✱ ♣❛r❛

♣❡rí♦❞♦ três ❛ r❡s♣♦st❛ à q✉❡stã♦ ❛❝✐♠❛ é ❛✜r♠❛t✐✈❛✳ ❇❧✉♠❡♥✱ ❑✐♠✱ ◆❛♥❝❡ ❡ ❩❤❛r♥✐ts❦② ❡♠ [✺]

❡st❡♥❞❡r❛♠ ❡ss❡ r❡s✉❧t❛❞♦ ♣❛r❛ S2

+ ❡ H2✳ ◆♦ ❡♥t❛♥t♦ ❡ss❡ ♣r♦❜❧❡♠❛ ♣❡r♠❛♥❡❝❡ ❛❜❡rt♦ ♣❛r❛ ✉♠

♣❡rí♦❞♦ n q✉❛❧q✉❡r✳ ❊st✉❞❛♥❞♦✱ ♦ ♠ét♦❞♦ ❞❡ P✐♥t♦✲❞❡✲❈❛r✈❛❧❤♦✱ ❑❛♠♣❤♦rst ❡ ❉✐❛s ❈❛r♥❡✐r♦ ♣❛r❛ ♣r♦♣r✐❡❞❛❞❡s ❣❡♥ér✐❝❛s ❞❡ ❜✐❧❤❛r❡s ♣❧❛♥♦s ❬✼❪ ❞❡s❝♦❜r✐♠♦s q✉❡ ❛s ❡q✉❛çõ❡s ✉t✐❧✐③❛❞❛s ❡♠ ❬✺❪ ♣❛r❛ ❞❡♠♦♥str❛r ♦ ❝❛s♦ n = 3 ❛♣❛r❡❝❡♠ ♣❛r❛ ✉♠ ♣❡rí♦❞♦ q✉❛❧q✉❡r✳ ❆ ♣❛rt✐r ❞❛í✱ ♣r❡t❡♥❞❡♠♦s

❛♣r✐♠♦r❛r ♦ ♠ét♦❞♦ ♣❛r❛ ♦❜t❡r♠♦s ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ r❡s✉❧t❛❞♦ t❛♥t♦ ♣❛r❛S2

+❡H2 q✉❛♥t♦

♣❛r❛ E ❡ ♣❛r❛ ✉♠ ♣❡rí♦❞♦ _n q✉❛❧q✉❡r✳

■♥t❡❣r❛❜✐❧✐❞❛❞❡ ❡ q✉❡❜r❛ ❞❡ ❝✉r✈❛s ✐♥✈❛r✐❛♥t❡s ❇✐r❦❤♦✛ ♣r♦✈♦✉ q✉❡ ♦s ❜✐❧❤❛r❡s ❝✐r❝✉❧❛r✱ ❡♠E✱

é ✐♥t❡❣rá✈❡❧✳ ◆ã♦ é ❞✐❢í❝✐❧ ♣r♦✈❛r q✉❡ ♦ ❜✐❧❤❛r ♥✉♠ ❝ír❝✉❧♦ ❣❡♦❞és✐❝♦ é ✐♥t❡❣rá✈❡❧ ❡♠ S2

+ ❡ H2✳

P❛r❛ ❡st❡s ❜✐❧❤❛r❡s✱ ❛ ❡str✉t✉r❛ ❞❡ r❡t❛s ✐♥✈❛r✐❛♥t❡s é ♠❛✐s ✜♥❛✿ ❝♦♠♦ ❡❧❡ é ✐♥t❡❣rá✈❡❧✱ s❡✉ ❡s♣❛ç♦ ❞❡ ❢❛s❡ é ❢♦❧❤❡❛❞♦ ♣♦r r❡t❛s ❤♦r✐③♦♥t❛✐s ✐♥✈❛r✐❛♥t❡s✳ ❊ ♣❛r❛ ❝❛❞❛ ♣❛r (k, n) ❞❛❞♦✱ ❤á ✉♠❛ r❡t❛

❤♦r✐③♦♥t❛❧ ✐♥✈❛r✐❛♥t❡ r❡ss♦♥❛♥t❡✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r α(k,n)✱ ❢♦r♠❛❞❛ ♣♦r (k, n)✲tr❛❥❡tór✐❛s✳

❉❛❞❛ ✉♠❛ ♣❡rt✉r❜❛çã♦ ❞♦ ❝ír❝✉❧♦ ❡♠ E✱ ❘❛♠ír❡③✲❘♦s ❡♠ ❬❄❪ ❞❡t❡r♠✐♥❛ ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ❛

r❡t❛ α(k,n)✱ q✉❡ ❡r❛ ✐♥✈❛r✐❛♥t❡✱ ♥ã♦ s❡❥❛ ♣r❡s❡r✈❛❞❛✳ ❆ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ✉t✐❧✐③❛❞❛ é ♦ P♦t❡♥❝✐❛❧

❘❛❞✐❛❧ ❞❡ ▼❡❧♥✐❦♦✈✳ ❯s❛♥❞♦ ❛s ♠❡s♠❛s té❝♥✐❝❛s✱ ❡♠ ♠❡✉ tr❛❜❛❧❤♦ ❞❡ ❞♦✉t♦r❛♠❡♥t♦✱ ♦❜t✐✈❡✲ ♠♦s ❝♦♥❞✐çõ❡s s♦❜r❡ ❛s ♣❡rt✉r❜❛çõ❡s ❞❡ ❝ír❝✉❧♦s ❣❡♦❞és✐❝♦s q✉❡ ❣❛r❛♥t❡♠ ❛ q✉❡❜r❛ ❞❡ ❝✉r✈❛s r❡ss♦♥❛♥t❡s✳

❇✐r❦❤♦✛ t❛♠❜é♠ ♣r♦✈♦✉ q✉❡ ♦ ❜✐❧❤❛r ♥✉♠❛ ❡❧í♣s❡ ❡♠ E é ✐♥t❡❣rá✈❡❧✳ ❘❛♠ír❡③✲❘♦s ❡ P✐♥t♦✲❞❡✲

❈❛r✈❛❧❤♦ ❞❡t❡r♠✐♥❛r❛♠ ❝♦♥❞✐çõ❡s s♦❜r❡ ❛s ♣❡rt✉r❜❛çõ❡s ❞❛ ❡❧í♣s❡ ❡♠ E q✉❡ ❣❛r❛♥t❡♠ ❛ q✉❡❜r❛

❞❡ ❝✉r✈❛s r❡ss♦♥❛♥t❡s ❝♦♠ ❝á✉st✐❝❛ ❡❧í♣t✐❝❛ ✭❬❄❪✮✳

❆♣r❡s❡♥t❛♠♦s ❛q✉✐ ♣r♦♣r✐❡❞❛❞❡s ❞✐♥â♠✐❝❛s ❞♦ ❜✐❧❤❛r ❝✐r❝✉❧❛r ❝♦♠♦ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡✱ ❡①✐stê♥❝✐❛ ❞❡ ❝❛✉st✐❝❛s✱ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s ❡ ♣r♦♣r✐❡❞❛❞❡ t✇✐st✳ ❚❛✐s ♣r♦♣r✐❡❞❛❞❡s✱ ❝♦♥❤❡❝✐❞❛s ♣❛r❛ ♦ ❜✐❧❤❛r ♥♦ ❝ír❝✉❧♦ ♣❧❛♥♦ ❢♦r❛♠ ♣r♦✈❛❞❛s ❛q✉✐✳ ❯t✐❧✐③❛♠♦s ✉♠ ♣r♦❣r❛♠❛ ❞❡ ❣❡♦♠❡tr✐❛ ♥ã♦ ❡✉❝❧✐❞❡❛♥❛✱ ♥♦♥❡✉❝❧✐❞ ✭✻✮✱ ♣❛r❛ ✉♠❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ t❛✐s ♣r♦♣r✐❡❞❛❞❡s ♥♦ ❡s♣❛ç♦ ❞❡ ❝♦♥✜❣✉rçã♦ ❞♦ ❜✐❧❤❛r ♥♦ ❝ír❝✉❧♦ ❤✐♣❡r❜ó❧✐❝♦ ❡ ♦ ♠❛♣❧❡ ♣❛r❛ ♦ ❝❛s♦ ❡s❢ér✐❝♦✳

❱✐♠♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ t❡♦r❡♠❛3✱ q✉❡ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r ❡♠ ♥✉♠❛ ♦✈❛❧ Γ ❞❡S ♣♦ss✉✐ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ór❜✐t❛s ♣❡r✐ó ♦❞✐❝❛s ❞❡ q✉❛❧q✉❡r ♣❡rí♦❞♦ n ≥ 2✳ P♦❞❡♠♦s ❢♦r♠✉❧❛r ❛s ❞✉❛s

s❡❣✉✐♥t❡s q✉❡stõ❡s ❡♠ t♦r♥♦ ❞❛ q✉❛✐s é ❞❡s❡♥✈♦❧✈✐❞♦ ♦ ❝❛♣ít✉❧♦ é ❞❡s❡♥✈♦❧✈✐❞♦✳

➱ ♣♦ssí✈❡❧ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ t❛✐s ór❜✐t❛s❄ ❊ ♦ q✉❡ s❡ ♣♦❞❡ s♦❜r❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡❧❛s ♣❛r❛ ❝❛❞❛ ♣❡rí♦❞♦ n ❞❛❞♦❄

◆❡ss❡ ❝❛♣ít✉❧♦✱ ✉s❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ▼❛❝❦❛②✲▼❡✐ss✱ ✈❛♠♦s ♠♦str❛r q✉❡ ✉♠❛ ❞❛s ór❜✐t❛s ♣❡r✲ r✐ó❞✐❝❛s ❞♦ t❡♦r❡♠❛ 3é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛ ♦✉ ♣❛r❛❜ó❧✐❝❛✱ r❡s✉❧t❛❞♦ ❥á ❝♦♥❤❡❝✐❞♦ ♣❛r❛ ♦

❝❛s♦ S = E2✱ ✈❡r_[✶✸_]✳ ❈♦♠ r❡❧❛çã♦ à s❡❣✉♥❞❛ q✉❡stã♦✱ ❡st❡♥❞❡r❡♠♦s ♣❛r❛_S ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦

❞❡ ❬✼❪✿

●❡♥❡r✐❝❛♠❡♥t❡ ❡①✐st❡ ❛♣❡♥❛s ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r ♣❛r❛ ❝❛❞❛ ♣❡rí♦❞♦ n ❡ ❡❧❛s sã♦ t♦❞❛s ❤✐♣❡r❜ó❧✐❝❛s✳

❉❡♠♦♥str❛r❡♠♦s t❛❧ r❡s✉❧t❛❞♦ ✉t✐❧✐③❛♥❞♦ ❛s ✐❞é✐❛s ❞❡s❡♥✈♦❧✈✐❞❛s ♣♦r P✐♥t♦✲❞❡✲❈❛r✈❛❧❤♦✱ ❑❛♠♣❤♦rst ❡ ❉✐❛s ❈❛r♥❡✐r♦ ♣❛r❛ ❛ ♣r♦✈❛ ❞♦ ❝❛s♦ S =E2✱ ❝❛❜❡ r❡ss❛❧t❛r ♥♦ ❡♥t❛♥t♦ q✉❡ ❛❧❣✉♠s ❛❞❛♣t❛çõ❡s

s❡rã♦ ♥❡❝❡ssár✐❛s✳ ❊♠[✷✶]✱ ❘❛♠ír❡③✲❘♦s ❡st✉❞❛♥❞♦ ❜✐❧❤❛r❡s ♥♦ ❝ír❝✉❧♦ ❊✉❝❧✐❞❡❛♥♦ ♣❡rt✉r❜❛❞♦

❡st❛❜❡❧❡❝❡✉ ❝♦♥❞✐çõ❡s s♦❜r❡ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ❋♦✉r✐❡r ❞❡ss❛s ♣❡rt✉r❜❛çõ❡s ♣❛r❛ q✉❡ ❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r ♥ã♦ ♣♦ss✉❛ r❡t❛s ✐♥✈❛r✐❛♥t❡s ❝♦♠ ❞❛❞♦ ♥ú♠❡r♦ ❞❡ r♦t❛çã♦ r❛❝✐♦♥❛❧✳ ➱ ❢❛t♦ ❝♦♥❤❡❝✐❞♦ q✉❡ t❛✐s r❡t❛s✱ t❛♠❜é♠ ❝❤❛♠❛❞❛s ❞❡ t♦r♦s ✐♥✈❛r✐❛♥t❡s✱ sã♦ ❞❡tr✉í❞❛s ♣♦r q✉❛s❡ t♦❞❛s ♣❡rt✉r❜❛çõ❡s ❞♦ ❜✐❧❤❛r ❡ q✉❡❜r❛♠✲s❡ ♥✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ór❜✐t❛s ♣❡r✐ó ❞✐❝❛s✳

◆♦ss♦ ♦❜❥❡t✐✈♦ ♥❛ ♣r✐♠❡✐r❛ s❡çã♦ ❞❡ss❡ ❝❛♣ít✉❧♦ é ❡st❡♥❞❡r ❡ss❡ r❡s✉❧t❛❞♦ ❞❡t❡r♠✐♥❛♥❞♦ ❝♦♥❞✐✲ çõ❡s ♣❛r❛ ♣❡rs✐stê♥❝✐❛ ❞❛s r❡t❛s ✐♥✈❛r✐❛♥t❡s r❡ss♦♥❛♥t❡s ❞♦ ❜✐❧❤❛r ♥♦ ❝ír❝✉❧♦ ❣❡♦❞és✐❝♦ ♣❡rt✉r❜❛❞♦ t❛♥t♦ ❡♠ S2

+ q✉❛♥t♦ ❡♠ H2✳

❖ ❝♦♥❥✉♥t♦ Pn✱ ❞❡ t♦❞❛s ór❜✐t❛s ♣❡r✐ó❞✐❝❛s ❞❡ ♣❡rí♦❞♦ n ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r ❡♠ E2✱ S2₊ ❡

H2 _{t❡♠ s✐❞♦ ♦❜❥❡t♦ ❞❡ ♠✉✐t❛s q✉❡stõ❡s✳ ❯♠❛ ❞❡ss❛s q✉❡stõ❡s ✈❡rs❛ s♦❜r❡ ♦ q✉ã♦ s✐❣♥✐✜❝❛t✐✈♦ é}

❡ss❡ ❝♦♥❥✉♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡✱ ♦✉ s❡❥❛✱ t❡r✐❛ ❡ss❡ ❝♦♥❥✉♥t♦ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛❄ P❛r❛ tr❛❜❛❧❤❛r♠♦s ❡ss❛ q✉❡stã♦ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❝❛❞❛ ♣❡rí♦❞♦ s❡♣❛r❛❞❛♠❡♥t❡✳

❙❡n = 2 t❡♠♦s ♥♦s ♣❧❛♥♦s ❊✉❝❧✐❞❡❛♥♦ ❡ ❤✐♣❡r❜ó❧✐❝♦✱ ❞❡✈✐❞♦ ❛♦ ♣r✐♥❝í♣✐♦ ❞❡ ▼❛✉♣❡rt✉✐s✱ q✉❡

♦ ❝♦♥❥✉♥t♦ ❞❛s ór❜✐t❛s ❞❡ ♣❡rí♦❞♦ ✷✱ ♣♦r ❡st❛r ❝♦♥t✐❞♦ ♥❛ r❡t❛ ψ = π

2 ♣♦ss✉✐ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡

♥✉❧❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❝♦♥✈❡①✐❞❛❞❡ ❞♦ ❜♦r❞♦ ❞♦ ❜✐❧❤❛r✳ ◆♦ ❝❛s♦ ❡s❢ér✐❝♦✱ ❡♥tr❡t❛♥t♦✱ ♣♦❞❡♠♦s t❡r ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❞❡ P2 ♣♦s✐t✐✈❛✳ ❉❡ ❢❛t♦ ❝♦♠♦ ✈✐st♦ ♥♦ ❝❛♣ít✉❧♦ 2 s❡ ♦ ❜♦r❞♦ ❞♦ ❜✐❧❤❛r

❢♦r ♦ ❡q✉❛❞♦r ❡①✐st❡♠ ❛♣❡♥❛s ór❜✐t❛s ❞❡ ♣❡rí♦❞♦ ✷✳ ▼❛s ❝♦♠♦ ♦ ❡q✉❛❞♦r ♥ã♦ é ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦ ❡ss❡ ❢❡♥ô♠❡♥♦ ♥❛ ❡s❢❡r❛ ♥ã♦ ❝♦♥tr❛❞✐③ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

❖ ❝♦♥❥✉♥t♦ P2 ❞❛s ór❜✐t❛s ❞❡ ♣❡rí♦❞♦ ✷ ❆ ❛♣❧✐❝❛çã♦ ❞❡ ❜✐❧❤❛r ❡♠ ✉♠❛ ♦✈❛❧ ♥✉♠❛ s✉♣❡r❢í❝✐❡

❞❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ S ♣♦ss✉✐ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛✳

❊♠ ❬✷✸❪✱ ❘✐❝❤❧✐❦✱ ♠♦str♦✉ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s ór❜✐t❛s ❞❡ ♣❡rí♦❞♦ três ♣♦ss✉✐ t❛♠❜é♠ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛✱ ✉t✐❧✐③❛♥❞♦✲s❡ ❞❡ ✐♥❢♦r♠❛çõ❡s ❛❞q✉✐r✐❞❛s ♣♦r ❛❧❣✉♠❛s s✐♠✉❧❛çõ❡s ♥ú♠❡r✐❝❛s✳ ❊♠ ❬✸✸❪ ❡ss❡ ♠❡s♠♦ r❡s✉❧t❛❞♦ ❢♦✐ ❞❡♠♦♥str❛❞♦ ♣♦r ❲♦❥t❦♦✇s❦✐ ♣❛r❛ ❜✐❧❤❛r❡s ♥♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦ ✉t✐✲ ❧✐③❛♥❞♦ ❞❡ss❛ ✈❡③ ✉♠❛ ❛❜♦r❞❛❣❡♠ ✈✐❛ ❝❛♠♣♦s ❞❡ ❏❛❝♦❜✐✳ ❯t✐❧✐③❛♥❞♦ ❛ ❛❜♦r❞❛❣❡♠ ✈✐❛ ❝❛♠♣♦s ❞❡ ❏❛❝♦❜✐ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❲♦❥t❦♦✇✐s❦✱ ❬✺❪✱ ❇❧✉♠❡♥✱ ❑✐♠✱ ◆❛♥❝❡ ❡ ❩❤❛r♥✐ts❦② ♠♦str❛♠ q✉❡ P3 t❛♠✲

❜é♠ ♣♦ss✉✐ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛ ♣❛r❛ ❜✐❧❤❛r❡s ❡str✐t❛♠❡♥t❡ ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❝♦♥✈❡①♦s✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❡s ❝♦♥s❡❣✉✐r❛♠ ❝♦♥str✉✐r ✉♠ ❜✐❧❤❛r ❛♣❡♥❛s ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❝♦♥✈❡①♦ ♦♥❞❡ P3 t❡♠ ✐♥t❡r✐♦r

♥ã♦ ✈❛③✐♦✱ ♠♦str❛♥❞♦ ❝♦♠ ✐ss♦ q✉❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥✈❡①✐❞❛❞❡ ❣❡♦❞és✐❝❛ ❡str✐t❛ é ♥❡❝❡ssár✐❛✳ ❖✉tr❛ q✉❡stã♦ ♣♦ssí✈❡❧✱ ❛❜♦r❞❛❞❛ ❡♠ ❬✸✶❪ ♣❛r❛ ❜✐❧❤❛r❡s ♣❧❛♥♦s ♣♦r ❩❤❛r♥✐ts❦② ❡ ▼❡r❡♥❦♦✈✱ é q✉❛❧ ❛ ✧❢♦r♠❛✧♣♦ssí✈❡❧ ♣❛r❛ ❡ss❡ ❝♦♥❥✉♥t♦❄ ◆♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦✱ ❡❧❡s ❞❡♠♦♥str❛r❛♠ q✉❡ ❡ss❡ ❝♦♥❥✉♥t♦ t❡♠ ♠❡❞✐❞❛ ❞❡ ❍❛✉s❞♦r✛ ♥♦ ♠á①✐♠♦ ✶ ❡ s❡ ❡ss❡ ✈❛❧♦r ❡①tr❡♠♦ ❢♦r ❛t✐♥❣✐❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ór❜✐t❛s ❞❡ ♣❡rí♦❞♦ ✸ ♣♦ss✉✐ r❡t❛ t❛♥❣❡♥t❡ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ♥♦ s❡♥t✐❞♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ❍❛✉s❞♦r✛✳ ❊ss❛ ❝♦♥❝❧✉sã♦ é ✐♥t❡r❡ss❛♥t❡ ❡ t❡♠ s✐♠✐❧❛r✐❞❛❞❡ ❝♦♠ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

❯♠❛ ❝✉r✈❛ r♦t❛❝✐♦♥❛❧ ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r é ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ▲✐♣s❝❤✐t③ ❡ ❛ss✐♠ ♣♦ss✉✐ r❡t❛ t❛♥❣❡♥t❡ ❡♠ q✉❛s❡ t♦❞♦ ♣♦♥t♦ ♥♦ s❡♥t✐❞♦ ❞❡ ▲❡❜❡s❣✉❡✳

❈❛♣ít✉❧♦ ✷

▼♦❞❡❧♦s ❞❡ ●❡♦♠❡tr✐❛

◆♦ss♦ ♦❜❥❡t✐✈♦ ♥❡ss❡ ❝❛♣ít✉❧♦ é ❧✐st❛r r❡s✉❧t❛❞♦s s♦❜r❡ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ❝✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ S ❝♦♠♣❧❡t❛ ❞❡ ❝✉r✈❛t✉r❛ ❝♦♥st❛♥t❡ K✳ ➱ ❞❡♠♦♥str❛❞♦ ❡♠ ❬✾❪ q✉❡ s❡r ❝♦♠♣❧❡t❛ ❡ ❞❡ ❝✉r✈❛t✉r❛ ❣❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ ✐♠♣❧✐❝❛ q✉❡ S é ✐s♦♠❡tr✐❝❛ ❛♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦ E2✱ à ❡s❢❡r❛

✉♥✐tár✐❛ S2 ♦✉ ❛♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ H2 s❡ r❡s♣❡❝t✐✈❛♠❡♥t❡ _{K = 0}✱ _{K = 1} ♦✉ _{K =−1} ✳ ❆ss✐♠✱

❞❛q✉✐ ♣♦r ❞✐❛♥t❡ S =E2✱ _S=S2 ♦✉_{S =}H2✳

❊ss❡ ❡st✉❞♦ s❡rá ❝♦♥❝❡♥tr❛❞♦ ♥❛s ♣r♦♣r✐❡❞❛❞❡s ❣❡♦♠étr✐❝❛s ❡ ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ❝✉r✈❛s ♥❛s r❡s♣❡❝✲ t✐✈❛s s✉♣❡r❢í❝✐❡s✳ ❆♣❡♥❛s ♣❛r❛ ✜♥s ❞❡ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦s ❝❛s♦s ❡s❢ér✐❝♦ ❡ ❤✐♣❡r❜ó❧✐❝♦ ❛♣r❡s❡♥t❛r❡♠♦s ♣❛r❛ ♦ ❝❛s♦ ❊✉❝❧✐❞❡❛♥♦ ♦ q✉❡ t❛♠❜é♠ s❡rá ❞❡s❝r✐t♦ ♣❛r❛ ♦s ♦✉tr♦s ❞♦✐s ❝❛s♦s✳

✷✳✶ P❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦

❈♦♥s✐❞❡r❡♠♦s ♦ ❡s♣❛ç♦ R2 ♠✉♥✐❞♦ ❝♦♠ ❛ ♠étr✐❝❛ ❊✉❝❧✐❞❡❛♥❛✱ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r ❁✱❃✳ ❉❡♥♦✲

t❡♠♦s ♣♦r E2 ❡ss❛ s✉♣❡r❢í❝✐❡ ❞❡ R3✳

❆s ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s ❡♠ R2

x(r, θ) = (rcosθ, rsenθ,0)

❝♦♠ 0< r ❡ 0< θ <2π ❞❡✜♥❡♠ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞❡E2✳

❉❡♥♦t❛r❡♠♦s ♣♦r < ·,· > ♣❛r❛ ❞❡♥♦t❛r ❛ ♠étr✐❝❛ ❊✉❝❧✐❞❡❛♥❛ ❡♠ E2 q✉❡ é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦

❝❛♥ô♥✐❝♦ ❞❡ R3 ❞❛❞♦ ♣♦r_{<(x1, y1, z1),(x2, y2, z2)>= x1x2+y1y2+z1z2} r❡str✐t♦ ❛♦ ♣❧❛♥♦✳

❖❜s❡r✈❛çã♦✿ ❯♠ ✈❡t♦r t❛♥❣❡♥t❡_V~ _❛E2 ❡♠ ✉♠ ♣♦♥t♦ _P é ✉♠ ✈❡t♦r ❞❡ _V~ _∈R3 q✉❡ s❛t✐s❢❛③❡♠

< ~V ,(0,0,1)>= 0✳

❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❝❛r❛❝t❡r✐③❛ ❣❡♦❞és✐❝❛s ❡♠ E2✳

▲❡♠❛ ✶✳ ✶✳ ❉❛❞♦s ✉♠ ♣♦♥t♦ A ∈E2 ❡ ✉♠ ✈❡t♦r _T~ t❛♥❣❡♥t❡ ✉♥✐tár✐♦ ❡♠ _A✱ ❛ ❣❡♦❞és✐❝❛ _γ(t)

q✉❡ ♣❛ss❛ ♣♦r A ♥❛ ❞✐r❡çã♦ T~ ♣♦ss✉✐ ❡q✉❛çã♦✿

γ(t) = A+T t~

✷✳ ❆ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ g(A, B) é✿

g2(A, B) =< A−B, A−B > ✭✷✳✶✮

❙♦❜r❡ ❛s ✐s♦♠❡tr✐❛s ❞❡E2 t❡♠♦s✿

Pr♦♣♦s✐çã♦ ✶✳ ❆s ✐s♦♠❡tr✐❛s ❞❡ E2 sã♦ ❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ♦rt♦❣♦♥❛✐s ❞❡ R3 q✉❡ ♣r❡s❡r✲

✈❛♠ ♦ ♣❧❛♥♦✳

❙♦❜r❡ â♥❣✉❧♦s ❡♠E2 t❡♠♦s

❉❡✜♥✐çã♦ ✶✳ ❉❛❞♦s ❞♦✐s ✈❡t♦r❡s_V~ _❡U~ _{✉♥✐tár✐♦s t❛♥❣❡♥t❡s ❡♠ ✉♠ ♣♦♥t♦P ∈}E2 q✉❛❧q✉❡r✱ t❡♠♦s

q✉❡ ♦ â♥❣✉❧♦✱ ∠(V , ~~ U)✱ ❡♥tr❡ ❡❧❡s é✿

cos∠(V , ~~ U) =< ~V , ~U > ❈♦♠♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r t❡♠♦s✿

Pr♦♣♦s✐çã♦ ✷✳ ❙❡❥❛ ♦ tr✐â♥❣✉❧♦ ❣❡♦❞és✐❝♦ ❡♠ S2 _{❝♦♠ ✈ért✐❝❡s A✱ B ❡ C✱ ♦s ❧❛❞♦sAB✱ BC ❡ CA}

♠❡❞✐♥❞♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ a✱ b ❡ c✳ ❉❡♥♦t❛♥❞♦ ♣♦r Aˆ✱ Bˆ ❡ Cˆ ♦s â♥❣✉❧♦s ❞♦s r❡s♣❡❝t✐✈♦s ✈ért✐❝❡s A✱ B ❡ C t❡♠♦s ❛s r❡❧❛çõ❡s✿

• ▲❡✐ ❞♦s ❝♦ss❡♥♦s

a2 =b2+c2−2bccos ˆA

• ▲❡✐ ❞♦s s❡♥♦s

sin ˆA

a =

sin ˆB

b =

sin ˆC c

❆ ♣r♦♣r✐❡❞❛❞❡ ❛❜❛✐①♦✱é ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❛ ♣r♦♣♦s✐çã♦✱ ❡ ✈❛❧❡ t❛♠❜é♠ ♣❛r❛ S2 ❡H2

❈♦r♦❧ár✐♦ ✶✳ ❚r✐â♥❣✉❧♦s ❣❡♦❞és✐❝♦s ✐sós❝❡❧❡s ♣♦ss✉❡♠ â♥❣✉❧♦s ❞❛ ❜❛s❡ ✐❣✉❛✐s✳

✷✳✶✳✶ ❙♦❜r❡ ❝✉r✈❛s ❡♠

E

❉❡✜♥✐çã♦ ✷✳ ❯♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❡♠ E2 é ✉♠❛ ❛♣❧✐❝❛çã♦ _{Γ : I−→}R2 ❞❡ ❝❧❛ss❡_Cq_{✱ q≥0}

❞❛❞❛ ♣♦r✿

Γ(t) :=x(r(t), θ(t))

♦♥❞❡ r(t) ❡ θ(t) sã♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ ❝❧❛ss❡ Cq_✳

❖❜s❡r✈❛çã♦✿ ❙❡rá út✐❧✱ ♣❛r❛ ✜♥s ❞❡ ❛♥❛❧♦❣✐❛✱ ❝♦♥s✐❞❡r❛rΓ(s)✉♠❛ ❝✉r✈❛ ❡♠ R3 _{❛❝r❡s❝❡♥t❛♥❞♦✲}

s❡ ❛ ❡❧❛ ❛ t❡r❝❡✐r❛ ❝♦♦r❞❡♥❛❞❛ ♥✉❧❛✳

❉❡✜♥✐çã♦ ✸✳ ❯♠❛ ❝✉r✈❛ ❡♠E2 é ❞✐t❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱_s✱ s❡_<Γ′_(s),Γ′_(s)>=

1✱ ♦♥❞❡ Γ′_{(s) é ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❛ Γ ❡♠ s ❡} ′ _{é ❛ ❞❡r✐✈❛❞❛ ❝♦♠ r❡❧❛çã♦ ❛ s✳}

❙❡Γ ❡stá ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ t❡♠♦s

<Γ′′(s),Γ′(s)>= 0

❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s q✉❡

Γ′′(s) =κ(s)N~(s)

♦♥❞❡ _N~_{(s) = (0,0,1)×Γ}′_{(s) ❡ κ sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♦ ✈❡t♦r ♥♦r♠❛❧ ✭♥❛ ♦r✐❡♥t❛çã♦ ♣♦s✐t✐✈❛ ❞❡} R3✮ ❡ ❛ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛ ❞❡ _Γ ❡♠ _s✳ ❉❛í✿

κ(s) =<Γ′′(s),(0,0,1)×Γ′(s)> ✭✷✳✷✮

◆♦ss♦ ♣ró①✐♠♦ ♦❜❥❡t✐✈♦ á❜♦r❞❛r ❛ ♥♦çã♦ ❞❡ ❝✉r✈❛ ❝♦♥✈❡①❛✳

❉❡✜♥✐çã♦ ✹ ✭❈♦♥✈❡①✐❞❛❞❡ ●❡♦❞és✐❝❛✮✳ ❯♠❛ ❝✉r✈❛ r❡❣✉❧❛r Γ é ❞✐t❛ ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❡str✐t❛♠❡♥t❡

❝♦♥✈❡①❛ s❡ q✉❛❧q✉❡r ❣❡♦❞és✐❝❛ t❛♥❣❡♥t❡ ❛ Γ ❛ ✐♥t❡r❝❡♣t❛ ❡♠ ♥♦ ♠á①✐♠♦ ✉♠ ♣♦♥t♦✳

❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♥♦s ❝✉❥❛ ♣r♦✈❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷❪ ♥♦s ❞✐③ q✉❡ ❝✉r✈❛t✉r❛ ❣❡♦✲ ❞és✐❝❛ ♣♦s✐t✐✈❛ é ✉♠❛ ❝♦♥❞✐ã❝♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ✉♠❛ ❝✉r✈❛s s✐♠♣❧❡s ❢❡❝❤❛❞❛s s❡r ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳

▲❡♠❛ ✷✳ ❯♠❛ ❝✉r✈❛ r❡❣✉❧❛r Γ s✐♠♣❧❡s ❡ ❢❡❝❤❛❞❛ ❝✉❥❛ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛ é ♣♦s✐t✐✈❛ é ❣❡♦❞❡s✐❝❛✲

♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳

✷✳✷ ❊s❢❡r❛

❈♦♥s✐❞❡r❡♠♦s ♦ ❡s♣❛ç♦ R3 ♠✉♥✐❞♦ ❝♦♠ ❛ ♠étr✐❝❛ ❊✉❝❧✐❞❡❛♥❛✱ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r ❁✱❃✳ ❉❡♥♦✲

t❡♠♦s ♣♦r S2

+ ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s V =x~i+y~j+z~k❡♠ R3 s❛t✐s❢❛③❡♥❞♦ x2+y2+z2 = 1✳

❆ ❛♣❧✐❝❛çã♦

x(φ, θ) = (senφcosθ,senφsenθ,cosφ)

❝♦♠ 0< φ < pi ❡ 0< θ <2π✱ t♦r♥❛ S2 ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡R3 ❝♦♠ ♣r✐♠❡✐r❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧

I(V~) = a2+ sen2φb2

♦♥❞❡ _V~ _{= axρ +bxθ é ✉♠ ✈❡t♦r ♥♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❛} S2

+ ❡♠ x(φ, θ) ❡ xφ✱ xθ sã♦ ❛s ❞✐r❡çõ❡s

♣r✐♥❝✐♣❛✐s ♥♦ ♣❧❛♥♦ t❛♥❣❡♥t❡✳

❙❡♠ ❝❤❛♥❝❡ ❞❡ ❝♦♥❢✉sã♦ ✈❛♠♦s ❡s❝r❡✈❡r <·,·>t❛♠❜é♠ ♣❛r❛ ❞❡♥♦t❛r ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ❡♠

S2✳

❖❜s❡r✈❛çã♦✿ ❯♠ ✈❡t♦r t❛♥❣❡♥t❡_V~ _❛S2 ❡♠ ✉♠ ♣♦♥t♦ _P é ✉♠ ✈❡t♦r ❞❡_V~ _∈R3 q✉❡ s❛t✐s❢❛③❡♠

< ~V , P >= 0✳

❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❝❛r❛❝t❡r✐③❛ ❣❡♦❞és✐❝❛s ❡♠ S2✳

▲❡♠❛ ✸✳ ✶✳ ❆s ❣❡♦❞és✐❝❛s ❞❡ S2

+ sã♦ ♦s ❝ír❝✉❧♦s ♠á①✐♠♦s✱ ❝✉r✈❛s ♦❜t✐❞❛s ♣❡❧❛ ✐♥t❡rs❡çã♦ ❞❡ S2 ♣♦r ♣❧❛♥♦s ❞❡ R3 q✉❡ ♣❛ss❛♠ ♣❡❧❛ ♦r✐❣❡♠❀

✷✳ ❉❛❞♦s ✉♠ ♣♦♥t♦ A ∈ S2 ❡ ✉♠ ✈❡t♦r _T t❛♥❣❡♥t❡ ✉♥✐tár✐♦ à ❡s❢❡r❛ ❡♠ _A✱ ❛ ❣❡♦❞és✐❝❛ _γ(t)

q✉❡ ♣❛ss❛ ♣♦r A ♥❛ ❞✐r❡çã♦ T ♣♦ss✉✐ ❡q✉❛çã♦✿

γ(t) =Acost+T sint

✸✳ ❆ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ g(A, B) é✿

cos g(A, B) =< A, B > ✭✷✳✸✮

❙♦❜r❡ ❛s ✐s♦♠étr✐❛s ❞❡S2 t❡♠♦s✿

Pr♦♣♦s✐çã♦ ✸✳ ❆s ✐s♦♠❡tr✐❛s ❞❡ S2 sã♦ ❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ♦rt♦❣♦♥❛✐s ❞❡ R3 q✉❡ ❛ ♣r❡✲

s❡r✈❛♠✳

❙♦❜r❡ â♥❣✉❧♦s ❡♠S2 t❡♠♦s

❉❡✜♥✐çã♦ ✺✳ ❉❛❞♦s ❞♦✐s ✈❡t♦r❡s _V~ _{❡ U}~ _{✉♥✐tár✐♦s t❛♥❣❡♥t❡s ❛} S ❡♠ ✉♠ ♣♦♥t♦ _{P ∈} S2 _{q✉❛❧q✉❡r✱}

t❡♠♦s q✉❡ ♦ â♥❣✉❧♦✱ ∠(V , ~~ U)✱ ❡♥tr❡ ❡❧❡s é✿

cos∠(V , ~~ U) =< ~V , ~U >

❈♦♠♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r t❡♠♦s✿

Pr♦♣♦s✐çã♦ ✹✳ ❙❡❥❛ ♦ tr✐â♥❣✉❧♦ ❣❡♦❞és✐❝♦ ❡♠ S2 ❝♦♠ ✈ért✐❝❡s _A✱ _B ❡ _C✱ ♦s ❧❛❞♦s_AB✱ _BC ❡ _CA

♠❡❞✐♥❞♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ a✱ b ❡ c✳ ❉❡♥♦t❛♥❞♦ ♣♦r Aˆ✱ Bˆ ❡ Cˆ ♦s â♥❣✉❧♦s ❞♦s r❡s♣❡❝t✐✈♦s ✈ért✐❝❡s A✱ B ❡ C t❡♠♦s ❛s r❡❧❛çõ❡s✿

• ▲❡✐ ❞♦s ❝♦ss❡♥♦s ✶

cos( ˆA) = cosbcosc−cosa senbsenc

• ▲❡✐ ❞♦s ❝♦ss❡♥♦s ✷

cosa= cos ˆBcos ˆC−cos ˆA sen ˆBsen ˆC

• ▲❡✐ ❞♦s s❡♥♦s

sin ˆA

sena =

sin ˆB

senb =

sin ˆC

senc

➱ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✿

❈♦r♦❧ár✐♦ ✷✳ ❚r✐â♥❣✉❧♦s ❣❡♦❞és✐❝♦s ✐sós❝❡❧❡s ♣♦ss✉❡♠ ❛♥❣✉❧♦s ❞❛ ❜❛s❡ ✐❣✉❛✐s✳

✷✳✷✳✶ ❙♦❜r❡ ❝✉r✈❛s ❡♠

S

❊ss❛ s❡çã♦ ❡stá ❜❛s❡❛❞❛ ❡♠ [✾]✳

❉❡✜♥✐çã♦ ✻✳ ❯♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❡♠ S2

+ é ✉♠❛ ❛♣❧✐❝❛çã♦ Γ : I−→R3 ❞❡ ❝❧❛ss❡ Cq✱ q≥0

s❛t✐s❢❛③❡♥❞♦

<Γ(t),Γ(t)>= 1

❡ s❡rá ❞❛❞❛ ♣♦r✿

Γ(t) :=x(φ(t), θ(t))

♦♥❞❡ φ(t) ❡ θ(t) sã♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ ❝❧❛ss❡ Cq_✳

❖❜s❡r✈❛çã♦✿ ❱✐st♦ ❝♦♠♦ ✈❡t♦r✱ Γ(s)✱ é ♥♦r♠❛❧ à ❡s❢❡r❛ ♥♦ ♣♦♥t♦ Γ(s)✳

❉❡✜♥✐çã♦ ✼✳ ❯♠❛ ❝✉r✈❛ ❡♠S2 é ❞✐t❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱_s✱ s❡_<Γ′_(s),Γ′_(s)>=

1✱ ♦♥❞❡ Γ′_{(s) é ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❛ Γ ❡♠ s ❡} ′ _{é ❛ ❞❡r✐✈❛❞❛ ❝♦♠ r❡❧❛çã♦ ❛ s✳}

❙❡Γ ❡stá ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧❛ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ t❡♠♦s

< DΓ

′

ds (s),Γ

′_{(s)>= 0}

♦♥❞❡ D

ds é ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ❡♠S 2

+✳ ❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s q✉❡

DΓ′

ds (s) = κ(s)N~(s)

♦♥❞❡ _N~_{(s) = Γ(s)×Γ}′_{(s)é ♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ ❞❡ Γ ❡♠ s ❡ κ é ❛ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛✳ ❉❛í✿}

κ(s) =< DΓ

′

ds ,Γ(s)×Γ

′_{(s)> ✭✷✳✹✮}

◆♦ss♦ ♣ró①✐♠♦ ♦❜❥❡t✐✈♦ á❜♦r❞❛r ❛ ♥♦çã♦ ❞❡ ❝✉r✈❛ ❝♦♥✈❡①❛ ♣❛r❛ ❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❙❛♥t❛❧ó [✷✺]t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❞❡ ❝✉r✈❛s ❝♦♥✈❡①❛s ♥❛ ❡s❢❡r❛✿

❉❡✜♥✐çã♦ ✽✳ ❬❈♦♥✈❡①✐❞❛❞❡ ●❡♦❞és✐❝❛❪ ❯♠❛ ❝✉r✈❛ r❡❣✉❧❛r Γ ♥❛ ❡s❢❡r❛ é ❞✐t❛ ❣❡♦❞❡s✐❝❛♠❡♥t❡

❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ s❡ q✉❛❧q✉❡r ❣❡♦❞és✐❝❛ t❛♥❣❡♥t❡ ❛ Γ ❛ ✐♥t❡r❝❡♣t❛ ❡♠ ♥♦ ♠á①✐♠♦ ✉♠ ♣♦♥t♦✳

▲❡♠❜r❛♥❞♦ q✉❡ ❛s ❣❡♦❞és✐❝❛s ❞❡S2 sã♦ ♦s ❝ír❝✉❧♦s ♠á①✐♠♦s✱ s❡❣✉❡ ❞❡ss❛ ❞❡✜♥✐çã♦ q✉❡ ❛ àr❡❛

❞❡ ✉♠❛ ❝✉r✈❛ Γ ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❝♦♥✈❡①❛ é ♠❡♥♦r q✉❡ 2π ❛❧é♠ ❞♦ ❢❛t♦ ❞❡ q✉❡ ❡❧❛ ❞❡✈❡ ❡st❛r ❝♦♥t✐❞❛ ❡♠ ✉♠ ❤❡♠✐s❢ér✐♦✳

❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ ❝✉r✈❛s s✐♠♣❧❡s ❢❡❝❤❛❞❛s ❣❡♦❞❡s✐❝❛✲ ♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛s✳

▲❡♠❛ ✹✳ ❯♠❛ ❝✉r✈❛ r❡❣✉❧❛r Γ s✐♠♣❧❡s ❡ ❢❡❝❤❛❞❛ ❝✉❥❛ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛ é ♣♦s✐t✐✈❛ é ❣❡♦❞❡s✐❝❛✲

♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳

❆♣❡s❛r ❞❡ ❛✐♥❞❛ ♥ã♦ t❡r♠♦s ❡♥❝♦♥tr❛❞♦ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ ❢❛t♦ ❡①✐st❡♠ ♥❛ ❧✐t❡r❛t✉r❛✱ ❬✷✽❪ ♣❛❣✳ ✶✼✵✱ ❛✜r♠❛t✐✈❛s q✉❡ ♥♦s ❧❡✈❛♠ ❛ ❝r❡r q✉❡ ❡①✐st❡ ✉♠❛ ❞❡♠♦♥str❛çã♦✳ P♦❞❡♠♦s ♥♦ ❡♥t❛♥t♦ ♠♦str❛r ✉♠ r❡s✉❧t❛❞♦ ♠❛✐s ❢r❛❝♦✿

❈✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛ ♣♦s✐t✐✈❛ ❡♠ t♦❞♦ ♣♦♥t♦ ✐♠♣❧✐❝❛ q✉❡ Γ é ❧♦❝❛❧♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❣❡♦❞❡s✐✲

❝❛♠❡♥t❡ ❝♦♥✈❡①❛✳

✷✳✸ P❧❛♥♦ ❍✐♣❡r❜ó❧✐❝♦

❖ ♠♦❞❡❧♦ ❞♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ ♦❜❥❡t♦ ❞❡ tr❛❜❛❧❤♦ ❛q✉✐ é ♦ ❞❛ ❢♦❧❤❛ ❞♦ ❤✐♣❡r❜♦❧♦✐❞❡ ❡ss❛ s❡çã♦ ❡stá ❜❛s❡❛❞❛ ❡♠ [✷✷]✱ [✶✼]❡ [✷✾]✳

❉❡♥♦t❡♠♦s ♣♦r M3 _{♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ R}3 _{♠✉♥✐❞♦ ❝♦♠ ❛ ❢♦r♠❛ q✉❛❞rát✐❝❛ q(x, y, z) = x}2₊

y2_−z2_{✳ ❖ ❡s♣❛ç♦R}3 _{❝♦♠ ❡ss❛ ♠étr✐❝❛ ❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐✳ ❙❡V}~ _{= (v}

1, v2, v3)

❡ _U~ _{= (u1, u2, u3) sã♦ ✈❡t♦r❡s ❡♠} R3 ❞❡✜♥✐♠♦s✱ ♦ ♣r♦❞✉t♦

≪V , ~~ U ≫= 1

2q(U~ +V~)−q(V~)−q(U~)

✱ ✐✳❡✱

≪V, U ≫= v1u1 +v2u2−v3u3

❊ss❡ ♣r♦❞✉t♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠ ♠étr✐❝❛ ❞❡ ▲♦r❡♥t③ ❡ ❛♣❛r❡❝❡ ♥❛t✉r❛❧♠❡♥t❡ ❡♠ r❡❧❛t✐✈✐❞❛❞❡✳ ❊❧❡ ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ➱ ❜✐❧✐♥❡❛r✱ s✐♠étr✐❝♦ ❡ ♥ã♦ ❞❡❣❡♥❡r❛❞♦✱ ♦✉ s❡❥❛ s❡ ≪

~

U , ~V ≫= 0 ♣❛r❛ t♦❞♦ U~ ❡♥tã♦ V~ = 0✳ ▼❡s♠♦ ❞❡✜♥✐♥❞♦ ✉♠❛ ♠étr✐❝❛ q✉❡ ♥ã♦ é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛

♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ ♣♦ss✉✐ t❛♠❜é♠ ❛s ♥♦çõ❡s ❞❡ ❝♦♥❡①ã♦✱ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡✱ ❣❡♦❞és✐❝❛s✱ ✐s♦♠étr✐❛s✱ ❡t❝✳ ❯♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❡ss❡ ❡s♣❛ç♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❄❄

❈❛❞❛ ✈❡t♦r ❡♠ M ♣♦❞❡ s❡r três t✐♣♦s ❝❛✉s❛✐s✿

✶✳ ❚✐♣♦ ❡s♣❛ç♦✿ ❙❡ ≪V , ~~ V ≫>0

✷✳ ❚✐♣♦ t❡♠♣♦✿ ❙❡ ≪V , ~~ V ≫<0

✸✳ ❚✐♣♦ ❧✉③✿ ❙❡ ≪V , ~~ V ≫= 0

❉❡♥♦t❡♠♦s ♣♦r H2 ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s _{V =x~i+y~j+z~k} ❡♠ M3 s❛t✐s❢❛③❡♥❞♦ _q(V~_{) = −1}

❡ z >0❝♦♠ ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ❞❡ M3✳ ●r❛✜❝❛♠❡♥t❡✱ H2 ❝♦rr❡s♣♦♥❞❡ à ❢♦❧❤❛ ❞♦ ❤✐♣❡r❜♦❧ó✐❞❡ ❞❡

❞✉❛s ❢♦❧❤❛s ❞❛❞❛ ♣♦r z = p1 +x2_+y2

❆ ❛♣❧✐❝❛çã♦

x(ρ, θ) = (senhρcosθ,senhρsenθ,coshρ)

❝♦♠ 0< ρ ❡0< θ <2π✱ t♦r♥❛ H2 ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡_M3 ❝♦♠ ♣r✐♠❡✐r❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧

I(V~) = a2+ senh2ρb2

♦♥❞❡ _V~ _{= axρ +bxθ é ✉♠ ✈❡t♦r ♥♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❛} H2 _{❡♠ x(ρ, θ) ❡ x}

ρ✱ xθ sã♦ ❛s ❞✐r❡çõ❡s

♣r✐♥❝✐♣❛✐s ♥♦ ♣❧❛♥♦ t❛♥❣❡♥t❡✳

❙❡♠ ❝❤❛♥❝❡ ❞❡ ❝♦♥❢✉sã♦ ✈❛♠♦s ❡s❝r❡✈❡r ≪ ·,· ≫ t❛♠❜é♠ ♣❛r❛ ❞❡♥♦t❛r ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ❡♠

H2_✳

❖❜s❡r✈❛çã♦✿ ❱❡t♦r❡s t❛♥❣❡♥t❡s ❛ H2 ❡♠ ✉♠ ♣♦♥t♦ _p sã♦ ♦s ✈❡t♦r❡s ❞❡ _V~ _∈ E3 q✉❡ s❛t✐s❢❛③❡♠

≪V , p~ ≫= 0 ❡ ❡❧❡s sã♦ ❞♦ t✐♣♦ ❡s♣❛ç♦✱ ✐st♦ é✱ I(·)é ❞❡✜♥✐❞❛ ♣♦s✐t✐✈❛✳

❈♦♠ t✉❞♦ ✐ss♦ t❡♠♦s q✉❡ H2 _{é ✉♠ s✉♣❡r❢í❝✐❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐✳ ❊ss❡ é ♦}

♠♦❞❡❧♦ ❞❡ ❑❧❡✐♥ ♣❛r❛ ♦ ♣❧❛♥♦ ❍✐♣❡r❜ó❧✐❝♦✳ ▲❡♠❛ ✺✳ ❬✷✷❪

✶✳ ❆s ❣❡♦❞és✐❝❛s ❞❡ H2 sã♦ ❝✉r✈❛s ♦❜t✐❞❛s ♣❡❧❛ ✐♥t❡rs❡çã♦ ❞❡ H2 ♣♦r ♣❧❛♥♦s ❞❡ R3 q✉❡ ♣❛ss❛♠

♣❡❧❛ ♦r✐❣❡♠❀

✷✳ ❉❛❞♦s ✉♠ ♣♦♥t♦ A ∈ H2 ❡ ✉♠ ✈❡t♦r _T t❛♥❣❡♥t❡ ✉♥✐tár✐♦ ❛ H2 ❡♠ _A✱ ❛ ❣❡♦❞és✐❝❛ _γ(t) q✉❡

♣❛ss❛ ♣♦r A ♥❛ ❞✐r❡çã♦ T ♣♦ss✉✐ ❡q✉❛çã♦✿

γ(t) =Acosht+T sinht

✸✳ ❆ ❞✐stâ♥❝✐❛ ❣❡♦❞és✐❝❛ g(A, B) é✿

cosh g(A, B) = − ≪A, B ≫ ✭✷✳✺✮

❙♦❜r❡ â♥❣✉❧♦s ❡♠H2 t❡♠♦s

❉❡✜♥✐çã♦ ✾✳ ❬✷✷❪ ❉❛❞♦s ❞♦✐s ✈❡t♦r❡s _V~ _{❡ U}~ _{t❛♥❣❡♥t❡s ❛} H2 ❡♠ ✉♠ ♣♦♥t♦ _{P ∈} H2 q✉❛❧q✉❡r✱

t❡♠♦s q✉❡ ♦ â♥❣✉❧♦✱ ∠(V , ~~ U)✱ ❡♥tr❡ ❡❧❡s é✿

cos∠(V , ~~ U) =≪V , ~~ U ≫

❈♦♠♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞♦ ❝❧á❧❝✉❧♦ ❞❡ â♥❣✉❧♦ ❡♠H2 t❡♠♦s✿

Pr♦♣♦s✐çã♦ ✺ ✭❬✷✹❪✮✳ ❈♦♥s✐❞❡r❡ ✉♠ tr✐â♥❣✉❧♦ ❣❡♦❞és✐❝♦ ❡♠ H2 ❝♦♠ ✈ért✐❝❡s _A✱ _B ❡ _C✱ ♦s ❧❛❞♦s

AB✱ BC ❡ CA ♠❡❞✐♥❞♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ a✱ b ❡ c✳ ❉❡♥♦t❛♥❞♦ ♣♦r Aˆ✱ Bˆ ❡ Cˆ ♦s â♥❣✉❧♦s ❞♦s r❡s♣❡❝t✐✈♦s ✈ért✐❝❡s A✱ B ❡ C t❡♠♦s ❛s r❡❧❛çõ❡s✿

• ▲❡✐ ❞♦s ❝♦ss❡♥♦s ✶

cos( ˆA) = coshbcoshc−cosha senhbsenhc

• ▲❡✐ ❞♦s ❝♦ss❡♥♦s ✷

cosha= cosh ˆBcos ˆC−cos ˆA sen ˆBsen ˆC

• ▲❡✐ ❞♦s s❡♥♦s

sin ˆA

senha =

sin ˆB

senhb =

sin ˆC

senhc

✷✳✸✳✶ ❙♦❜r❡ ❝✉r✈❛s ❡♠

H

❊ss❛ s❡çã♦ ❡stá ❜❛s❡❛❞❛ ♥♦ ❧✐✈r♦ [✶✼]❡ ❡♠ [✷✾]✳

❉❡✜♥✐çã♦ ✶✵✳ ❯♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ❡♠ H2 é ✉♠❛ ❛♣❧✐❝❛çã♦ _{Γ : I −→} M3 ❞❡ ❝❧❛ss❡ _Cq_✱

q ≥0 s❛t✐s❢❛③❡♥❞♦

≪Γ(t),Γ(t)≫=−1

❡ s❡rá ❞❛❞❛ ♣♦r✿

Γ(t) := x(ρ(t), θ(t))

♦♥❞❡ ρ(t) ❡ θ(t) sã♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ ❝❧❛ss❡ Cq_✳

❖❜s❡r✈❛çã♦✿ ❈✉r✈❛s ❡♠ H2 sã♦ ❞♦ t✐♣♦ t❡♠♣♦✳ ❖ ✈❡t♦r _Γ(s) é ♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ ❛♦

❤✐♣❡r❜♦❧♦✐❞❡ ♥♦ ♣♦♥t♦ Γ(s)✳

❉❡✜♥✐çã♦ ✶✶✳ ❯♠❛ ❝✉r✈❛ ❡♠ H2 _{é ❞✐t❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱ s✱ s❡ ≪}

Γ′_(s),Γ′_{(s)≫= 1✱ ♦♥❞❡Γ}′_{(s) é ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❛ Γ ❡♠ s ❡} ′ _{é ❛ ❞❡r✐✈❛❞❛ ❝♦♠ r❡❧❛çã♦ ❛ s✳}

❙❡Γ ❡stá ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧❛ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ t❡♠♦s

≪ DΓ

′

ds (s),Γ

′_{(s)≫= 0}

♦♥❞❡ D

ds é ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ❡♠H

2_{✳ ❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s q✉❡}

DΓ′

ds (s) = κ(s)N~(s)

♦♥❞❡ _N~_{(s) = Γ(s)∧Γ}′_{(s)é ♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ ❞❡ Γ ❡♠ s ❡ κ é ❛ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛✳ ❉❛í✿}

κ(s) =≪ DΓ

′

ds ,Γ(s)∧Γ

′_{(s)≫ ✭✷✳✻✮}

❖♥❞❡∧ é ♦ ♣r♦❞✉t♦ ❡①t❡r✐♦r ❡♠ M3✳

❯♠❛ ❞❡♠♦♥tr❛çã♦ ♣❛r❛ ❡ss❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❝✉r✈❛s ❣❡♦❞❡s✐❝❛♠❡♥t❡ ✭❡str✐t❛♠❡♥t❡✮ ❝♦♥✈❡①❛s ♥♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✹❪✳

◆♦ss♦ ♣ró①✐♠♦ ♦❜❥❡t✐✈♦ á❜♦r❞❛r ❛ ♥♦çã♦ ❞❡ ❝✉r✈❛ ❝♦♥✈❡①❛ ♣❛r❛ ♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❙❛♥t❛❧ó [✷✻]t❡♠♦s ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❞❡ ❝✉r✈❛s ❝♦♥✈❡①❛s ♥❛ ❡s❢❡r❛✿

❉❡✜♥✐çã♦ ✶✷ ✭❈♦♥✈❡①✐❞❛❞❡ ●❡♦❞és✐❝❛✮✳ ❯♠❛ ❝✉r✈❛ r❡❣✉❧❛r Γ ♥♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ é ❞✐t❛ ❣❡♦❞❡✲

s✐❝❛♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛ s❡ q✉❛❧q✉❡r ❣❡♦❞és✐❝❛ t❛♥❣❡♥t❡ ❛ Γ ❛ ✐♥t❡r❝❡♣t❛ ❡♠ ♥♦ ♠á①✐♠♦

✉♠ ♣♦♥t♦✳

❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✱ ❝✉❥❛ ♣r♦✈❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✹❪✱ ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ❝♦♥❞✐çã♦ s✉✜✲ ❝✐❡♥t❡ ♣❛r❛ q✉❡ ❝✉r✈❛s s✐♠♣❧❡s ❢❡❝❤❛❞❛s s❡❥❛♠ ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛s✳

▲❡♠❛ ✻✳ ❯♠❛ ❝✉r✈❛ r❡❣✉❧❛r Γ s✐♠♣❧❡s ❡ ❢❡❝❤❛❞❛ ❝✉❥❛ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛ é ♣♦s✐t✐✈❛ ❡♠ t♦❞♦

♣♦♥t♦ é ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✳

❈❛♣ít✉❧♦ ✸

❇✐❧❤❛r❡s

❇✐❧❤❛r❡s ♣❧❛♥♦s tê♠ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❡st✉❞❛❞♦s ❞❡s❞❡ ♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳✳ ◆❛ ❧✐t❡r❛t✉r❛ ❡①✐st❡♠ ✈ár✐❛s r❡❢❡rê♥❝✐❛s s♦❜r❡ t❛✐s ❜✐❧❤❛r❡s ♣♦r ❡①❡♠♣❧♦ [✶✺]✱ [✶✸]✱ [✷✼]✱ [✶✷] ❡♥tr❡ ♦✉tr❛s✳ ◆♦ss♦

♦❜❥❡t✐✈♦ ♥❡ss❡ ❝❛♣ít✉❧♦ ❡ ❡st✉❞❛r ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❡ ❜✐❧❤❛r❡s ♥❛ ❡s❢❡r❛ ✉♥✐tár✐❛ S2 ❡ ♥♦ P❧❛♥♦

❍✐♣❡r❜ó❧✐❝♦ H2✳ ❈♦♥str✐r❡♠♦s ❡ ❞❡♠♦♥str❛r❡♠♦s ✈❡rsõ❡s ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❜✐❧❤❛r❡s ♥♦

♣❧❛♥♦ ❡✉❝❧✐❞❡❛♥♦ ♣❛r❛ ❜✐❧❤❛r❡s ❡♠ S ❡ ❡♠ H2✳ ❈♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❡ ❡♥①✉❣❛r ❛ ❡s❝r✐t❛ ❡ ✈✐s❛♥❞♦

❛ ❝❡❧❡r✐❞❛❞❡ t❡①t✉❛❧ ✉t✐❧✐③❛r❡♠♦s ♦ sí♠❜♦❧♦ S ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦ E2✱ ❛ ❡s❢❡r❛

✉♥✐tár✐❛ S2

+ ❡ ♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ H2 ❝✉❥♦s ♠♦❞❡❧♦s ❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣✐t✉❧♦ ✷✳

❯♠ ❞♦s ♣r✐♠❡✐r♦s ❡ ❢✉♥❞❛♠❡♥t❛❧ r❡s✉❧t❛❞♦ ♣❛r❛ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r é ♦ s❡✲ ❣✉✐♥t❡✿

❆ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r ❡♠ ✉♠❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛B ❝♦♥✈❡①❛ ❞♦ ♣❧❛♥♦ ❊✉❝❧✐❞❡❛♥♦ é ✉♠ ❞✐❢❡♦♠♦r✲ ✜s♠♦ ❚✇✐st q✉❡ ♣r❡s❡r✈❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡ ♣♦ss✉✐ ❝❧❛ss❡ ❞❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ✉♠❛ ❛ ♠❡♥♦s q✉❡ ❛ ❝❧❛ss❡ ❞❛ ❢♦r♥t❡✐r❛ ❞❡ B✳

❊st❡ r❡s✉❧t❛❞♦ ❢♦✐ ♣r✐♠❡✐r♦ ♣r♦✈❛❞♦ ♣♦r ❇✐r❦❤♦✛✱ ♣❛r❛ ❜✐❧❤❛r❡s ❡♠ E2✳ ❊ss❡ s❡rá t❛♠❜é♠

♥♦ss♦ ♣r✐♠❡✐r♦ ❡ ✐♥é❞✐t♦ ♦❜❥❡t✐✈♦ ♣❛r❛ ❜✐❧❤❛r❡s ❝♦♥✈❡①♦s ❡♠ S✿

❆ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r ❡♠ ✉♠❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ B ❝♦♠ ❜♦r❞♦ ♣❡❧♦ ♠❡♥♦s C2 _{❣❡♦❞❡s✐❝❛♠❡♥t❡}

❝♦♥✈❡①♦ ❝♦♥✈❡①❛ ❡♠ S é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❚✇✐st q✉❡ ♣r❡s❡r✈❛ ❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ❡ ♣♦ss✉✐ ❝❧❛ss❡ ❞❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ♣❡❧♦ ♠❡♥♦s C1_✳

◆❛ ❧✐t❡r❛t✉r❛[✺]❡ [✸] ❡ss❡ r❡s✉❧t❛❞♦ é ❛✜r♠❛❞♦ ♠❛s ♥❡♥❤✉♠❛ ♣r♦✈❛ é ❝♦♥❤❡❝✐❞❛✳ ◆♦ tr❛❜❛❧❤♦ [✸]✱ ✈✐s❛♥❞♦ ❛ ❡①t❡♥sã♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❲♦❥t❦♦✇s❦✐ s♦❜r❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❝ír❝✉❧♦s ❣❡♦❞é✲

s✐❝♦s✱ ❇✐❛❧② ✉t✐❧✐③❛ ❛s ❡q✉❛çõ❡s ❞♦ ❧❡♠❛ 10 q✉❡ s ❛♦ ❢✉♥❞❛♠❡♥t❛✐s ♥♦ ♥♦ss♦ r❡s✉❧t❛❞♦s✳ ❖ ♥♦ss♦

❧❡♠♠❛10é ❞❡♠♦♥str❛❞♦ ✉t✐❧✐③❛♥❞♦ ♦ ❡st✉❞♦ ❞♦s ♠♦❞❡❧♦s ❞❡ ❣❡♦♠❡tr✐❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♥♦ ❝❛♣ít✉❧♦ ✷✳

❈♦♠♦ ♥♦ss♦ ♦❜❥❡t♦ ❞❡ tr❛❜❛❧❤♦ s❡rã♦ ♦s ❜✐❧❤❛r❡s ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❝♦♥✈❡①♦s✱ ❡♠ ✈✐st❛ ❞❛ ❞❡✜✲ ♥✐çã♦ 8✱ ♦ ❡st✉❞♦ ❞❡ ❜✐❧❤❛r❡s ♥❛ ❡s❢❡r❛ s❡rá ❢❡✐t♦ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ❛♣❡♥❛s ✉♠ ❤❡♠✐s❢ér✐♦ q✉❡ s❡rá

❞❡♥♦t❛❞♦ ♣♦r S =S2 +✳

❚❡♠✐♥❛♠♦s ❡ss❛ ✐♥tr♦❞✉çã♦ ❝♦♠ ✉♠❛ ♥♦t❛ ❞❡ ♦r❣❛♥✐③❛çã♦✳ ◆❛ s❡çã♦3.1❝♦♥str✉ír❡♠♦s ❛ ❛♣❧✐✲

❝❛çã♦ ❞❡ ❜✐❧❤❛r ❡ ❞❡♠♦♥str❛r❡♠♦s q✉❡ ❜✐❧❤❛r❡s ❣❡♦❞❡s✐❝❛♠❡♥t❡ ❝♦♥✈❡①♦s ❡♠S sã♦ ❞✐❢❡♦♠♦r✜s♠♦s t✇✐st ❝♦♥s❡r✈❛t✐✈♦s ❡ ♦❜t❡r❡♠♦s ❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ❞❡r✐✈❛❞❛ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ❜✐❧❤❛r ❡♠S✱ ❞❡ ❢♦r♠❛ ✐♥é❞✐t❛ ♣❛r❛ S2

+, H2✳ ❆♣r❡s❡♥t❛r❡♠♦s ♥❛ s❡çã♦ 3.2 ✉♠ ♣r✐♠❡✐r♦ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ❜✐❧❤❛r✿

❖ ❜✐❧❤❛r ❡♠ ❝ír❝✉❧♦s ❣❡♦❞és✐❝♦s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞✐♥â♠✐❝❛s✳ ❋✐♥❛❧✐③❛r❡♠♦s ❡ss❡ ❝❛♣ít✉❧♦ ❝♦♠ ❛ ❡①t❡♥sã♦ ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❜✐❧❤❛r❡s ♣❧❛♥♦s

♣❛r❛ ❜✐❧❤❛r❡s ♥❛ s❡♠✐❡s❢❡r❛ ❡ ♥♦ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ q✉❡ ❞❡r✐✈❛♠ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ t✇✐st✳

✸✳✶ ❆♣❧✐❝❛çã♦ ❞♦ ❇✐❧❤❛r

+ ♦✉H2 ❡ Γ ✉♠❛ ♦✈❛❧ ❡♠ S✱ ✐st♦ é✱ ❞❡ ❝❧❛ss❡ Cq ❝♦♠ q ≥ 2✱ s✐♠♣❧❡s✱ ❢❡❝❤❛❞❛✱

♦r✐❡♥t❛❞❛ ♣♦s✐t✐✈❛♠❡♥t❡ ❡ ❝♦♠ ❝✉r✈❛t✉r❛ ❣❡♦❞és✐❝❛ ♣♦s✐t✐✈❛ ✭κ > 0✮✳ ❉❡♥♦t❛r❡♠♦s ♣♦r |Γ| ♦ ❝♦♠♣r✐♠❡♥t♦ t♦t❛❧ ❞❛ ❝✉r✈❛ Γ✳

❙✉♣♦♥❤❛♠♦sΓ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✱ ✐✳❡✱kΓ′_{(s)k = 1✱ ♦♥❞❡k · kr❡♣r❡s❡♥t❛}

❛ ♥♦r♠❛ ❊✉❝❧✐❞❡❛♥❛ s❡ S = E2✱ ❛ ❍✐♣❡r❜ó❧✐❝❛ ♥♦ ❝❛s♦_{S =} H2 ❡ ❛ ❡s❢ér✐❝❛ ♣❛r❛ _{S =} S2

+✳ ❆q✉✐ ′

é ❛ ❞❡r✐✈❛❞❛ ❝♦♠ r❡❧❛çã♦ ❛ s✳ ❉❛ ♣❛r❛♠❡tr✐③❛çã♦ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ t❡♠♦s q✉❡ s❡♠ r✐s❝♦ ❞❡ ❝♦♥❢✉sã♦ ♣♦❞❡♠♦s ♥♦s r❡❢❡r✐r ❛ ✉♠ ♣♦♥t♦ Γ(s) ❛♣❡♥❛s ❝♦♠♦ s✳

❱❛♠♦s ❝♦♥str✉✐r ✉♠❛ ❛♣❧✐❝❛çã♦ ❛ t❡♠♣♦ ❞✐s❝r❡t♦ q✉❡ ♥♦s ❛❥✉❞❛rá ❛ ❡st✉❞❛r ♦ s❡❣✉✐♥t❡ ❢❡♥ô♠❡♥♦✿ ❯♠❛ ♣❛rtí❝✉❧❛ s❡ ♠♦✈❡ s♦❜r❡ ❛ s✉♣❡r❢í❝✐❡ S s❡❣✉✐♥❞♦ ✉♠❛ tr❛❥❡tór✐❛ ❣❡♦❞és✐❝❛ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛ ✉♠✱ ♥♦ ✐♥t❡r✐♦r ❞❡ ✉♠❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛✱ ❝♦♠ ❜♦r❞♦ s✉❛✈❡ ❡ s♦❢r❡♥❞♦ ❝♦❧✐sõ❡s ❡❧ást✐❝❛s ❝♦♠ t❛❧ ❜♦r❞♦✳

P♦❞❡♠♦s ❞❡s❝r❡✈❡r ❡ss❡ ♣r♦❜❧❡♠❛ ❞❛ ❢♦r♠❛ s❡❣✉✐♥t❡✿ ❈♦♠♦ S ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛ t❡♠♦s q✉❡ ❞❛❞♦s1 ♥❛ ❝✉r✈❛Γ❡ ✉♠ ✈❡t♦r~v1 ✉♥✐tár✐♦ ❡♠Ts1S ❛♣♦♥t❛♥❞♦ ♣❛r❛ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ♣♦r Γ✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❣❡♦❞és✐❝❛ γ q✉❡ ♣❛rt❡ ❞❡s1 t❡♠~v1 ❝♦♠♦ ✈❡t♦r t❛♥❣❡♥t❡✳ ❙❡♥❞♦Γ ✉♠❛

❝✉r✈❛ ❢❡❝❤❛❞❛✱ ❛ ❣❡♦❞és✐❝❛ γ ✐♥t❡r❝❡♣t❛rá Γ ❡♠ ✉♠ ♥♦✈♦ ♣♦♥t♦ s2 q✉❡ é ú♥✐❝♦ ♣❡❧❛ ❝♦♥✈❡①✐❞❛❞❡

❣❡♦❞és✐❝❛ ❞❡ Γ✳ ❙❡❥❛♠ 0 < ψ1 < π ♦ â♥❣✉❧♦ ❞♦ ✈❡t♦r t❛♥❣❡♥t❡ Γ′(s1) ❛♦ ✈❡t♦r ~v1 ❡ 0 < ψ2 < π

♦ â♥❣✉❧♦✱ ❡♥tr❡ ♦ ✈❡t♦r t❛♥❣❡♥t❡ Γ′_(s

2) ❡ ♦ ✈❡t♦r t❛♥❣❡♥t❡ à ❣❡♦❞és✐❝❛ γ ❡♠ s2✱ ❞❡♥♦t❛❞♦ ♣♦r

γ′_(s

2)✳ ❉✐③❡♠♦s q✉❡s1 ❡ψ1 sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦♥t♦ ❞❡ s❛í❞❛ ❡ â♥❣✉❧♦ ❞❡ s❛í❞❛ ❡ ❞✐③❡♠♦s q✉❡

ψ2 é ♦ â♥❣✉❧♦ ❞❡ ❜❛t✐❞❛ ✭♦✉ â♥❣✉❧♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛✮ ❞❛ ♣❛rtí❝✉❧❛ ❡♠ s2✳ ❉❡✜♥✐♠♦s ❛ tr❛❥❡tór✐❛ ❞❛

♣❛rtí❝✉❧❛ ❛♣ós ❛ r❡✢❡①ã♦ ❝♦♠♦ s❡♥❞♦ ❛ ❣❡♦❞és✐❝❛ q✉❡ ♣❛rt❡ ❞❡ s2 ♥❛ ❞✐r❡çã♦ ❞❛❞❛ ♣❡❧♦ ✈❡t♦r~v2

q✉❡ é ❛ r❡✢❡①ã♦ ❞❡ γ′_(s

2)♣❡❧♦ ✈❡t♦r t❛♥❣❡♥t❡ Γ′(s2)✳ ❆ss✐♠✱ ♣♦r ❝♦♥str✉çã♦✱ t❡♠♦s q✉❡ ♦ â♥❣✉❧♦

❞❡ Γ′_(s

2) ❛~v2✱ ❞✐t♦ â♥❣✉❧♦ ❞❡ r❡✢❡①ã♦✱ é ψ2✳

❙❡❥❛ T ₌ R_/kΓkZ ❡ _{M =} T_{×(0, π)}✳ ❉❛❞♦s _{s1 ∈} T ❡ _{ψ1 ∈ (0, π)} t❡♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦✱ q✉❡

❞❡♥♦t❛r❡♠♦s ♣♦r F

F : M −→ M

(s1, ψ1) 7→ (s2(s1, ψ1), ψ2(s1, ψ1))

q✉❡ ♠♦❞❡❧❛ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❇✐❧❤❛r✳ ❆♦ ❝♦♥❥✉♥t♦ M❞❡♥♦♠✐♥❛♠♦s ❡s♣❛ç♦ ❞❡ ❢❛s❡ ❞♦ ❜✐❧❤❛r✳

❋✐❣✉r❛ ✸✳✶✿ ❆♣❧✐❝❛çã♦ ❞♦ ❇✐❧❤❛r