UNIVERSIDADE FEDERAL DE SÃO CARLOS CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA Teorema de Schur no Plano de Minkowski e caracterização de Hélices Inclinadas no Espaço de

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆ P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r ♥♦ P❧❛♥♦ ❞❡ ▼✐♥❦♦✇s❦✐ ❡

❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❍é❧✐❝❡s ■♥❝❧✐♥❛❞❛s ♥♦ ❊s♣❛ç♦ ❞❡

▼✐♥❦♦✇s❦✐✳

▲✉❝✐❛♥♦ ❞❡ ▼❡❧♦ ❘❛♠♦s

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ●✉✐❧❧❡r♠♦ ❆♥t♦♥✐♦ ▲♦❜♦s ❱✐❧❧❛❣r❛

UNIVERSIDADE FEDERAL DE SÃO CARLOS

CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA

Teorema de Schur no Plano de Minkowski e

caracterização de Hélices Inclinadas no Espaço de

Minkowski.

Luciano de Melo Ramos

Orientador: Prof. Dr. Guillermo A. Lobos Villagra

Dissertação apresentada ao

PPG-M da UFSCar como

parte dos requisitos para ob-tenção do título de Mestre em Matemática.

Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar

R175ts

Ramos, Luciano de Melo.

Teorema de Schur no plano de Minkowski e caracterização de hélices inclinadas no espaço de Minkowski / Luciano de Melo Ramos. -- São Carlos : UFSCar, 2013.

67 f.

Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2013.

1. Geometria. 2. Geometria de Minkowski. 3. Schur, Teorema de. 4. Hélices. I. Título.

❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✳ ❉❡❞✐❝♦ t❛♠❜é♠ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ♠ã❡✳

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r♦ ❛ ❉❡✉s ♣♦r t✉❞♦✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ♠ã❡ q✉❡ ❛❝r❡❞✐t♦✉ ❡♠ ♠✐♠ ❛té q✉❛♥❞♦ ❡✉ ♠❡s♠♦ ♥ã♦ ❛❝r❡❞✐t❛✈❛✳

❆♦s ♣r♦❢❡ss♦r❡s ●✉✐❧❧❡r♠♦ ❆♥t♦♥✐♦ ▲♦❜♦s ❱✐❧❧❛❣r❛ ❡ ❏♦ã♦ ◆✐✈❛❧❞♦ ❚♦♠❛③❡❧❧❛✱ ♣❡❧❛s ♦r✐❡♥t❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❡ ♣r❡❝✐s❛✈❛✱ ♣❡❧❛ ❤✉♠✐❧❞❛❞❡ ❝♦♠ q✉❡ ❡s❝✉t❛r❛♠ ❛s ♠✐♥❤❛s ✐❞é✐❛s s♦❜r❡ ❛ ❞✐ss❡rt❛çã♦ ❡ ♣❡❧❛s ♠✉✐t❛s ✈❡③❡s q✉❡ ❡♥t❡♥❞❡r❡♠ ♦s ♠❡✉s ♣r♦❜❧❡♠❛s ♣❡ss♦❛✐s ❡ ♠❡ ❛❥✉❞❛r❛♠ ♦✉ ❛❝♦♥s❡❧❤❛r❛♠✳

P♦r ✜♠✱ ❛♦s ❝♦❧❡❣❛s✱ ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈❛r✱ ❛♦ s❡♥❤♦r ▲✉ís ❱❡❧♦s♦ ❡ ❢❛♠í❧✐❛ ♣♦r t♦❞♦ ❛♣♦✐♦ ♥❡st❡s ❛♥♦s ❡♠ ❙ã♦ ❈❛r❧♦s✳

❘❡s✉♠♦

❯♠ r❡s✉❧t❛❞♦ ❝❧áss✐❝♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❝✉r✈❛s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ é ♦ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r✱ q✉❡ ♣r✐♠❡✐r♦ ❢♦✐ ♣r♦✈❛❞♦ ❡♠ ✶✾✷✶ ♣♦r ❆✳ ❙❝❤✉r ❡♠ ❬✸❪ ♥♦ ❝❛s♦ ❡♠ q✉❡ ❛s ❝✉r✈❛t✉r❛s ❞❛s ❝✉r✈❛s ❝♦✐♥❝✐❞❡♠ ♣♦♥t✉❛❧♠❡♥t❡✳ ❖ ❝❛s♦ ❣❡r❛❧ ❞♦ t❡♦r❡♠❛ ❢♦✐ ♣r♦✈❛❞♦ ❡♠ ✶✾✷✺ ♣♦r ❊✳ ❙❝❤♠✐❞t ❡♠ ❬✹❪✳ ❖ ♣r✐♠❡✐r♦ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❛♣r❡s❡♥t❛r ✉♠❛ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❙❤✉r ♣❛r❛ ♦ ♣❧❛♥♦ ❞❡ ▼✐♥❦♦✇s❦✐✳ ❊♠ s❡❣✉✐❞❛✱ ♠♦str❛r❡♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡ss❡ r❡s✉❧t❛❞♦ ❢❡✐t❛s ♣♦r ❘✳ ▲ó♣❡③ ❡♠ ❬✶❪✳ ◆♦ ❝❛s♦ ❞♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ ✈❡r❡♠♦s q✉❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r é ❢❛❧s♦✳ ❖ s❡❣✉♥❞♦ ♦❜❥❡t✐✈♦ é ♠♦str❛r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❤é❧✐❝❡s ✐♥❝❧✐♥❛❞❛s ♥♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ ♦❜t✐❞❛s ♣♦r ❆✳ ❚✳ ❆❧✐ ❡ ❘✳ ▲ó♣❡③ ❡♠ ❬✷❪✱ ❛ q✉❛❧ ❡st❡♥❞❡ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❤é❧✐❝❡s ✐♥❝❧✐♥❛❞❛s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ ♦❜t✐❞❛ ❡♠ ✷✵✵✹ ♣♦r ❙✳ ■③✉♠✐②❛ ❡ ◆✳ ❚❛❦❡✉❝❤✐ ❬✻❪✳ ❈♦♥❝❧✉í♠♦s ❡st❛ ❞✐ss❡rt❛çã♦ ♣r♦✈❛♥❞♦ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❤é❧✐❝❡s ✐♥❝❧✐♥❛❞❛s ♦❜t✐❞❛ ❡♠ ❬✷❪✳

❆❜str❛❝t

❆ ❝❧❛ss✐❝❛❧ t❤❡♦r❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❣❡♦♠❡tr② ♦❢ ❝✉r✈❡s ✐♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✐s t❤❡ ❙❝❤✉r✬s ❚❤❡♦r❡♠✱ t❤❛t ✇❛s ♣r♦♦❢ ❜② ❆✳ ❙❝❤✉r ✐♥ ✶✾✷✶✱ ✇❤❡♥ ❜♦t❤ ❝✉r✈❛t✉r❡s ❛❣r❡❡ ♣♦✐♥t✇✐s❡ ❬✸❪✳ ❚❤❡ ♣r♦♦❢ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✇❛s ♣r♦✈❡❞ ✐♥ ✶✾✷✺ ❜② ❊✳ ❙❝❤♠✐❞t ✐♥ ❬✹❪✳ ❚❤❡ ✜rst ♦❜❥❡❝t✐✈❡ ✐♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s t♦ ♣r❡s❡♥t ▲♦r❡♥t③✐❛♥ ✈❡rs✐♦♥ ♦❢ ❙❝❤✉r✬s ❚❤❡♦r❡♠ ✐♥ t❤❡ ▼✐♥❦♦✇s❦✐ ♣❧❛♥❡✳ ❚❤❡♥ ✇❡ ✇✐❧❧ s❤♦✇ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ❞✉❡ t♦ ❘✳ ▲ó♣❡③ ❬✶❪✳ ■♥ t❤❡ ▼✐♥❦♦✇s❦✐ s♣❛❝❡ ✇❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ❙❝❤✉r✬s ❚❤❡♦r❡♠ ✐s ❢❛❧s❡✳ ❚❤❡ s❡❝♦♥❞ ♦❜❥❡❝t✐✈❡ ✐s s❤♦✇ ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ s❧❛♥t ❤❡❧✐❝❡s ✐♥ t❤❡ ▼✐♥❦♦✇s❦✐ s♣❛❝❡ ♦❜t❛✐♥❡❞ ❜② ❆✳ ❚✳ ❆❧✐ ❛♥❞ ❘✳ ▲ó♣❡③ ✐♥ ❬✷❪✱ ✇❤✐❝❤ ❡①t❡♥❞s ♥❛t✉r❛❧❧② ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ s❧❛♥t ❤❡❧✐❝❡s ✐♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ♦❜t❛✐♥❡❞ ✐♥ ✷✵✵✹ ❜② ❙✳ ■③✉♠✐②❛ ❆♥❞ ◆✳ ❚❛❦❡✉❝❤✐ ❬✻❪✳ ❲❡ ❝♦♥❝❧✉❞❡ ✇✐t❤ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ t❤❛t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ s❧❛♥t ❤❡❧✐❝❡s ❬✷❪✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✸

✶✳✶ ❖ ❊s♣❛ç♦ ❞❡ ▲♦r❡♥t③✲▼✐♥❦♦✇s❦✐ En1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✶✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸ Pr♦❞✉t♦ ❱❡t♦r✐❛❧ ❡♠ E31✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✶✳✹ ●❡♦♠❡tr✐❛ ❞❡ ❈✉r✈❛s ❡♠ En1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✶✳✺ ❚❡♦r✐❛ ❧♦❝❛❧ ❞❛s ❈✉r✈❛s P❧❛♥❛s ❡♠ E21✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✶✳✻ ❚❡♦r✐❛ ▲♦❝❛❧ ❞❛s ❈✉r✈❛s ❊s♣❛❝✐❛✐s ❡♠ E31✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✷ ❖ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r ♥♦ P❧❛♥♦ ❞❡ ▼✐♥❦♦✇s❦✐ ✹✶

✷✳✶ ❖ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r ♥♦ P❧❛♥♦ ❞❡ ▼✐♥❦♦✇s❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷ ❆♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸ ❯♠❛ ❈❛r❛❝t❡r✐③❛çã♦ ❞❡ ❍é❧✐❝❡s ■♥❝❧✐♥❛❞❛s ❡♠ E31✳ ✺✺

✸✳✶ ❍é❧✐❝❡s ■♥❝❧✐♥❛❞❛s ❡♠ E31✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✸✳✷ ■♥❞✐❝❛tr✐③❡s ❞❡ ✉♠❛ ❍é❧✐❝❡ ■♥❝❧✐♥❛❞❛ ♥ã♦ ♥✉❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✻

■♥tr♦❞✉çã♦

◆❡st❛ ❞✐ss❡rt❛çã♦ ❡st✉❞❛♠♦s ♦s tr❛❜❛❧❤♦s ❞❡ ▲ó♣❡③ ❬✶❪ ❡ ❆❧✐✲▲ó♣❡③ ❬✷❪✱ ♣✉❜❧✐❝❛❞♦s ❡♠ ✷✵✵✹ ❡ ✷✵✶✶✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆♣r❡s❡♥t❛❞♦s r❡s✉❧t❛❞♦s ❞❡ ❝✉r✈❛s ♥♦ ♣❧❛♥♦ ❡ ♥♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ q✉❡ ❡st❡♥❞❡♠ r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ✈á❧✐❞♦s ♣❛r❛ ❝✉r✈❛s ♥♦ ♣❧❛♥♦ ❡ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ❆ss✐♠ ✐♥✐❝✐❛♠♦s ♦ ❝❛♣ít✉❧♦ ✶ ✐♥tr♦❞✉③✐♥❞♦ ♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③✲▼✐♥❦♦✇s❦✐ ❞❡ ❞✐♠❡♥sã♦n✱En1✱ q✉❡ é ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧Rn ♠✉♥✐❞♦

❞❡ ✉♠ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞❡ í♥❞✐❝❡ ✶✳ ❉❡st❛❝❛♠♦s ❛s ♣r✐♥❝✐♣❛✐s ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ t❛♠❜é♠ ♠♦str❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❝✉r✈❛s ❡♠ En1✱ ❝♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❞✐st✐♥❣✉✐✲❧❛ ❞❛ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ❝✉r✈❛s ♥♦ ❡s♣❛ç♦

❡✉❝❧✐❞✐❛♥♦✳ ❊st❛s ❞✐❢❡r❡♥ç❛s ♦❝♦rr❡♠ ❞❡✈✐❞♦ ❛♦ ❝❛rát❡r ❝❛✉s❛❧ ❞❛ ❝✉r✈❛ ♥♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③✲▼✐♥❦♦✇s❦✐✳ P♦r ❡①❡♠♣❧♦✱ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ♦ ❚❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ❝✉r✈❛s ❞✐③ q✉❡✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠❡tr✐❛s✱ t♦❞❛ ❝✉r✈❛ ❡st❛ ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ s✉❛ ❝✉r✈❛t✉r❛ ❡ t♦rçã♦✳ ◆♦ ❡♥t❛♥t♦✱ ♥♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ ❡①✐st❡♠ ❝✉r✈❛s q✉❡ ♣♦ss✉❡♠ s♦♠❡♥t❡ t♦rçã♦✳ ❚❛❧ ❢❛t♦✱ ♦❝♦rr❡ ❝♦♠ ❝✉r✈❛s t✐♣♦✲❧✉③ ❡ ❝✉r✈❛s t✐♣♦✲❡s♣❛ç♦ ❝♦♠ ✈❡t♦r ❛❝❡❧❡r❛çã♦ t✐♣♦✲❧✉③✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❡♥✉♥❝✐❛♠♦s ♦ r❡s✉❧t❛❞♦ ❝❧áss✐❝♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ♣r♦✈❛❞♦ ♣♦r ❆✳ ❙❝❤✉r✱ ❡♠ ✶✾✷✶✱ ❬✸❪✱ q✉❛♥❞♦ ❛s ❝✉r✈❛t✉r❛s sã♦ ✐❣✉❛✐s ♣♦♥t♦ ❛ ♣♦♥t♦✳ ❖ ❝❛s♦ ❣❡r❛❧ ❢♦✐ ♣r♦✈❛❞♦ ♣♦r ❡ ❊✳ ❙❝❤♠✐❞t ❡♠ ✶✾✷✺✱ ❬✹❪✳ ❊♠ s❡❣✉✐❞❛✱ ♠♦str❛♠♦s ❛tr❛✈és ❞❡ ❝♦♥tr❛✲❡①❡♠♣❧♦s q✉❡ ♥ã♦ ❡①✐st❡ ✉♠❛ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r ♥♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐✱ ❞❡✈✐❞♦ ❛ ❘✳ ▲ó♣❡③ ❡♠ ❬✶❪✳ P♦st❡r✐♦r♠❡♥t❡✱ ❞❡♠♦♥str❛♠♦s ♣❛r❛ ♦ ♣❧❛♥♦ ❡✉❝❧✐❞✐❛♥♦✱ ❛ s❡❣✉✐♥t❡ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r ♣❛r❛ ❞✉❛s ❝✉r✈❛s ❝♦♥✈❡①❛s✱ q✉❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✺❪✳

❚❡♦r❡♠❛ ❙❡❥❛♠ α1, α2 : [0, l] → R2 ❞✉❛s ❝✉r✈❛s ❝♦♥✈❡①❛s✱ ♣❛r❛♠❡tr✐③❛❞❛s ♣❡❧♦

❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦✳ ❉❡♥♦t❡♠♦s ♣♦r k1✱ k2 ❛s ❢✉♥çõ❡s ❝✉r✈❛t✉r❛s ❞❡ α1 ❡ α2✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♥s✐❞❡r❡ d1(s) = d(α1(0), α1(s)) ❡ d2(s) = d(α2(0), α2(s))✱ ♦♥❞❡

d(., .) é ❛ ❞✐stâ♥❝✐❛ ❡✉❝❧✐❞✐❛♥❛ ❞❡ R2✳ ❙❡ k1(s) ≥ k2(s), ❡♥tã♦✱ d1(s) ≤ d2(s)✱

s _∈ [0, l]✳ ❆❧é♠ ❞✐ss♦✱ d1(s) = d2(s) ♣❛r❛ t♦❞♦ s ∈ [0, l] s❡✱ ❡ s♦♠❡♥t❡ s❡✱ α1 ❡ α2

sã♦ ❝♦♥❣r✉❡♥t❡s✳

❚❡r♠✐♥❛♠♦s ♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❝♦♠ ❛ ✈❡rsã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❙❝❤✉r ♥♦ ♣❧❛♥♦ ❞❡ ▼✐♥❦♦✇s❦✐ ❞❡ ❘✳ ▲ó♣❡③ ❬✶❪✳

❚❡♦r❡♠❛ ❙❡❥❛♠ α1, α2 : [0, l] →R2 ❞✉❛s ❝✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦

❞❡ ❛r❝♦ ❝♦♥✈❡①❛s✳ ❉❡♥♦t❡♠♦s ♣♦r k1✱ k2 ❛s ❢✉♥çõ❡s ❝✉r✈❛t✉r❛s ❞❡ α1 ❡ α2✱ r❡s♣❡❝✲

t✐✈❛♠❡♥t❡✳ ❙❡❥❛♠ d1(s) = d(α1(0), α1(s)) ❡ d2(s) = d(α2(0), α2(s))✱ ♦♥❞❡ d(., .) é ❛

❞✐stâ♥❝✐❛ ❡✉❝❧✐❞✐❛♥❛ ❞❡ R2✳ ❙❡ k1(s)≥ k2(s), ❡♥tã♦✱ d1(s)≤ d2(s)✱ s ∈ [0, l]. ❆❧é♠

❞✐ss♦✱ d1(s) =d2(s) ♣❛r❛ t♦❞♦ s ∈[0, l] s❡✱ ❡ s♦♠❡♥t❡ s❡✱ α1 ❡ α2 sã♦ ❝♦♥❣r✉❡♥t❡s✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ sã♦ ❛♣r❡s❡♥t❛❞❛s ❤é❧✐❝❡s ❡ ❤é❧✐❝❡s ✐♥❝❧✐♥❛❞❛s ♥♦ ❡s♣❛ç♦s ❡✉✲ ❝❧✐❞✐❛♥♦ ❡ ❞❡ ▼✐♥❦♦✇s❦✐✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆s ❞❡✜♥✐çõ❡s ♥♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ sã♦ ❜❛s❡❛❞❛s ♥❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❞❡ ❤é❧✐❝❡s ❡ ❤é❧✐❝❡s ✐♥❝❧✐♥❛❞❛s ❢❡✐t❛s ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡✉❝❧✐❞✐❛♥♦✱ ✉♠❛ ✈❡③ q✉❡✱ ❛ ❞❡✜♥✐çã♦ ❞❡ â♥❣✉❧♦ ❡♥tr❡ ✈❡t♦r❡s ♥♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③✲▼✐♥❦♦✇s❦✐ ♥ã♦ ❡①✐st❡✳ ❆♣r❡s❡♥t❛♠♦s ❡ ❞❡♠♦♥str❛♠♦s ❛ s❡❣✉✐♥t❡ ❝❛r❛❝t❡r✐✲ ③❛çã♦ ♣❛r❛ ❤é❧✐❝❡s ✐♥❝❧✐♥❛❞❛s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ❢❡✐t❛ ♣♦r ❙✳ ■③✉♠②✐❛ ❡ ◆✳ ❚❛❦❡✉❝❤✐ ❬✻❪✳

❚❡♦r❡♠❛ ❙❡❥❛ α:I _→R3 ✉♠❛ ❝✉r✈❛ ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ❝♦♠

❝✉r✈❛t✉r❛ k(s)₆= 0✱ s_∈ I ❡ t♦rçã♦ τ✳ ❊♥tã♦ α é ❤é❧✐❝❡ ✐♥❝❧✐♥❛❞❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❢✉♥çã♦

g(s) = k

2

(k2_+τ2₎3/2 _τ

k

′

(s)

é ❝♦♥st❛♥t❡ ❡♠ t♦❞♦ ♣♦♥t♦ ❞♦ ✐♥t❡r✈❛❧♦ ❞❡ ❞❡✜♥✐çã♦ ❞❛ ❝✉r✈❛✳

❆✐♥❞❛ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ♠♦str❛♠♦s✱ ❝♦♥❢♦r♠❡ ❘✳ ▲ó♣❡③ ❡♠ ❬✷❪✱ q✉❡ ✐♥❞❡✲ ♣❡♥❞❡♥t❡ ❞♦ ❝❛rát❡r ❝❛✉s❛❧✱ ❛s ❤é❧✐❝❡s ✐♥❝❧✐♥❛❞❛s ♥♦ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐ t❛♠❜é♠ ♣♦ss✉❡♠ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦✱ ❛s q✉❛✐s ✉s❛♠♦s ♣❛r❛ ♠♦str❛r q✉❡ ❛s ✐♥❞✐❝❛tr✐③❡s t❛♥✲ ❣❡♥t❡ ❡ ❜✐♥♦r♠❛❧ ❞❡ ❤é❧✐❝❡s ✐♥❝❧✐♥❛❞❛s ♥ã♦ ♥✉❧❛s ❝♦♠ ✈❡t♦r ♥♦r♠❛❧ ♥ã♦ ♥✉❧♦ sã♦ ❤é❧✐❝❡s ✉♥✐tár✐❛s ❡♠ E31✳

❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

✶✳✶ ❖ ❊s♣❛ç♦ ❞❡ ▲♦r❡♥t③✲▼✐♥❦♦✇s❦✐

E

❆♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛♠♦s ✉♠ ♣r♦❞✉t♦ ❡s❝❛❧❛r✱ ❝♦♠♦ s❡♥❞♦ ✉♠❛ ❛♣❧✐✲ ❝❛çã♦ ❜✐❧✐♥❡❛r✱ s✐♠étr✐❝❛ ❡ ♥ã♦ ❞❡❣❡♥❡r❛❞❛ s♦❜r❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳ ❯♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ✉♠ ♣r♦❞✉t♦ ❡s❝❛❧❛r ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✳

❊①❡♠♣❧♦ ✶✳✶✳ ❆ ●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧ ❈❧áss✐❝❛ é ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ ❊s♣❛ç♦ ❊✉✲ ❝❧✐❞✐❛♥♦ ❞❡ ❞✐♠❡♥sã♦ n✱ ❝♦♥st✐t✉í❞♦ ❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ Rn ♠✉♥✐❞♦ ❝♦♠ ♦ ♣r♦❞✉t♦

✐♥t❡r♥♦h,_i:_Rn_×

Rn →R✱ q✉❡ ♣❛r❛ ♦s ✈❡t♦r❡su✱v _∈Rn _{❞❡ ❝♦♦r❞❡♥❛❞❛s(x}

1, ..., xn)✱

(y1, ..., yn)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♠ r❡❧❛çã♦ à ❜❛s❡ ❝❛♥ô♥✐❝❛ é ❞❡✜♥✐❞♦ ♣♦r✱

hu, v_i=_h(x1, ..., xn),(y1, ..., yn)i=x1y1+...+xnyn.

❊①❡♠♣❧♦ ✶✳✷✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ ♣r♦❞✉t♦ ❡s❝❛❧❛r q✉❡ ♥ã♦ é ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ❞❛❞♦ ♣❡❧❛ ❛♣❧✐❝❛çã♦ : R

n

×Rn → R✱ q✉❡ ❛ss♦❝✐❛ ♣❛r❛ ♦s ✈❡t♦r❡s u = (x1, ..., xn) ❡ v = (y1, ..., yn) ♦ ✈❛❧♦r✱

uv = (x1, ..., xn)(y1, ..., yn) = x1y1+...+xn−1yn−1−xnyn.

◆♦t❡ q✉❡ ❛ ❜✐❧✐♥❡❛r✐❞❛❞❡ ❡ ❛ s✐♠❡tr✐❛ ❞❛ ❛♣❧✐❝❛çã♦ sã♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞❛ ❞✐s✲ tr✐❜✉t✐✈✐❞❛❞❡ ❡ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦ ❝♦r♣♦ R✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆❧é♠ ❞✐ss♦✱ s❡

u = (x1, ..., xn) ∈ Rn é ♥ã♦ ♥✉❧♦✱ ❛♦ ♠❡♥♦s ✉♠ xi é ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ❡♥tã♦ u ei = ǫixi✱ ♦♥❞❡ ǫi = 1 s❡ i ∈ {1, ..., n −1} ♦✉ ǫi = −1✱ s❡ i = n✱ ♣♦rt❛♥t♦✱ é ♥ã♦✲❞❡❣❡♥❡r❛❞❛✳ ❊st❡ ♣r♦❞✉t♦ ❡s❝❛❧❛r ♥ã♦ é ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✱ ♣♦✐s✱ enen=−1✳

❖ í♥❞✐❝❡ ❞❡ ✉♠ ♣r♦❞✉t♦ ❡s❝❛❧❛r g : Rn × Rn → R é ❛ ❞✐♠❡♥sã♦ ❞♦ ♠❛✐♦r

s✉❜❡s♣❛ç♦W _⊂Rn _{t❛❧ q✉❡g(w, w)<0✱ ♣❛r❛ t♦❞♦w∈W ✭✈❡❥❛ ♠❛✐s s♦❜r❡ ♦ í♥❞✐❝❡} ❞❡ ✉♠ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❡♠ ❬✺❪ ♣❛❣✳ ✹✼✮✳ ❚♦❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ t❡♠ í♥❞✐❝❡ ✵✳ ❖ í♥❞✐❝❡ ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞♦ ❊①❡♠♣❧♦ ✶✳✷ é 1✳

❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧Rn✱ ♠✉♥✐❞♦ ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❧♦r❡♥t③✐❛♥♦ ❞❡✜♥✐❞♦ ♥♦ ❊①❡♠♣❧♦

✶✳✷ é ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③✲▼✐♥❦♦✇s❦✐ ❞❡ ❞✐♠❡♥sã♦ n ❡ é ❞❡♥♦t❛❞♦ ♣♦r En1✳

❖s ❡s♣❛ç♦s E21 ❡ E31✱ sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣❧❛♥♦ ❞❡ ▼✐♥❦♦✇s❦✐ ❡ ❡s♣❛ç♦ ❞❡ ▼✐♥❦♦✇s❦✐✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ♥♦r♠❛ ❧♦r❡♥t③✐❛♥❛ ❞❡ ✉♠ ✈❡t♦r v ❡♠ En1 é ♦ ♥ú♠❡r♦ r❡❛❧kvk= p

|vv|✳

❯♠ ✈❡t♦r v _∈En1 é ❝❤❛♠❛❞♦ t✐♣♦✲❡s♣❛ç♦ q✉❛♥❞♦v é ♦ ✈❡t♦r ♥✉❧♦ ♦✉ vv >0✱ é

❝❤❛♠❛❞♦ t✐♣♦✲t❡♠♣♦ s❡ vv < 0 ❡ t✐♣♦✲❧✉③ s❡ v ₆= 0 ❡ vv = 0✳ ❖ ❈❛rát❡r ❈❛✉s❛❧

❞❡ ✉♠ ✈❡t♦r ❡♠ En1 é ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s❡r t✐♣♦✲❡s♣❛ç♦✱ t✐♣♦✲t❡♠♣♦ ♦✉ t✐♣♦✲❧✉③✳ ❖s

✈❡t♦r❡s t✐♣♦✲❧✉③ ❡stã♦ ♥♦ ❤✐♣❡r❝♦♥❡ C =_{(x1, ..., xn)∈Rn;x12+...+x2n−1−x2n = 0}✳ ❖ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ✈❡t♦r❡s t✐♣♦✲❧✉③ é ♦ ❝♦♥❥✉♥t♦Cn−1 _=C

−{0_}✱ ❝❤❛♠❛❞♦ ❈♦♥❡

❞❡ ▲✉③ ❞❡ ❊✐♥st❡✐♥✳

❋✐❣✉r❛ ✶✳✶✿ ❈♦♥❡ ❞❡ ▲✉③ ❞❡ ❊✐♥st❡✐♥ ♥♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③✲▼✐♥❦♦✇s❦✐✳

❚♦❞♦s ♦s ✈❡t♦r❡s ❞♦ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ D =< e1, ..., en−1 >✱ ♦♥❞❡ ♦s ✈❡t♦r❡s

ei ∈En1✱i= 1, ..., nsã♦ ♦s ✈❡t♦r❡s ❞❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❞❡ Rn✱ sã♦ t✐♣♦ ❡s♣❛ç♦✳

❖ ❝❛rát❡r ❝❛✉s❛❧ ❞♦ ❛♥tí♣♦❞❛ ❡ ❞♦s ♠ú❧t✐♣❧♦s ❞❡ ✉♠ ✈❡t♦r sã♦ ♣r❡s❡r✈❛❞♦s✱ ♣♦✐s

ww=uu ❡ ww=λ2_u

u.

◗✉❛♥❞♦ ❛ ú❧t✐♠❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ ✉♠ ✈❡t♦r❡ t✐♣♦✲t❡♠♣♦ ♦✉ t✐♣♦✲❧✉③ é ♣♦s✐t✐✈❛ ✭r❡s♣✳ ♥❡❣❛t✐✈❛✮ ❞✐③❡♠♦s q✉❡ ❡❧❡ t❡♠ ♣❛r✐❞❛❞❡ ♣♦s✐t✐✈❛ ✭r❡s♣✳ ♥❡❣❛t✐✈❛✮✳

❖ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r T ❡ ♦ ❈♦♥❡ ❞❡ ▲✉③

sã♦ ❝♦♥❥✉♥t♦s q✉❡ ♣♦ss✉❡♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s✱ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✳

❚❡♦r❡♠❛ ✶✳✶✳ ❙❡❥❛♠ u ❡ v ✈❡t♦r❡s ♥ã♦ t✐♣♦✲❡s♣❛ç♦ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ❡♠ En1✳

❊♥tã♦uv ≤0❡ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱u❡v sã♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s

t✐♣♦✲❧✉③✳

Pr♦✈❛✿ ❙❡❥❛♠ u= (x1, ..., xn)❡v = (y1, ..., yn)✈❡t♦r❡s ❞❡ En1 ♥ã♦ t✐♣♦✲❡s♣❛ç♦ ❞❡

♠❡s♠❛ ♣❛r✐❞❛❞❡✳

❈♦♥s✐❞❡r❛♥❞♦ ❛s n₋1 ♣r✐♠❡✐r❛s ❝♦♦r❞❡♥❛❞❛s ❞♦s ✈❡t♦r❡s u❡ v ❝♦♠♦ ♦s ✈❡t♦r❡s u= (x1, ..., xn−1)❡v = (y1, ..., yn−1)❡♠ Rn−1 ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✉s✉❛❧✱ s❡❣✉❡ q✉❡✱

uu =hu, ui −x

2

n =kuk2 − |xn|2 ≤0, vv =hv, vi −y

2

n =kvk2− |yn|2 ≤0.

▲♦❣♦✱

ku_kkv_{k ≤ |}xnyn|=xnyn,

✉♠❛ ✈❡③ q✉❡✱ xn ❡ yn ♣♦ss✉❡♠ ♦ ♠❡s♠♦ s✐♥❛❧✱ ♣♦r ❤✐♣ót❡s❡✳ P♦rt❛♥t♦✱ ✉s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱

uv =hu, vi −xnyn ≤ |hu, vi| −xnyn≤ kukkvk −xnyn≤0. ✭✶✳✶✮

❘❡st❛ ♠♦str❛r q✉❡uv = 0s❡✱ ❡ s♦♠❡♥t❡ s❡✱u❡vsã♦ ✈❡t♦r❡s t✐♣♦✲❧✉③ ❧✐♥❡❛r♠❡♥t❡

♦ ♣r♦❞✉t♦ ❡♥tr❡ ♦s ❞♦✐s ✈❡t♦r❡s é ♥✉❧♦✳ P❛r❛ ❛ r❡❝í♣r♦❝❛✱ ♦❜s❡r✈❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ q✉❡ s❡ uv = 0 ❡♥tã♦ |hu, vi| = kukkvk✱ ❧♦❣♦✱ u = λv✳ Pr❡❝✐s❛♠♦s ♠♦str❛r ❛✐♥❞❛

q✉❡ u ❡v sã♦ t✐♣♦✲❧✉③ ❡ xn=λyn✳

❙✉♣♦♥❞♦✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ u é t✐♣♦✲t❡♠♣♦✱ t❡♠♦s✱ x

2

n

ku_k2 −1 > 0✳ P♦❞❡♠♦s

t❡r u = 0 ♦✉ u ₆= 0✳ P❛r❛ u = 0✱ t❡♠♦s✱ u = (0, ...0, xn)✱ ❝♦♠ xn 6= 0✱ ❧♦❣♦✱

0 = uv =−xnyn✱ ✐♠♣❧✐❝❛♥❞♦ yn = 0 ❡ v ∈ D✱ ❝♦♥tr❛❞✐③❡♥❞♦ ❛ ❤✐♣ót❡s❡ ❞❡ v ♥ã♦

s❡r t✐♣♦✲❡s♣❛ç♦✳

❆♥❛❧✐s❡♠♦s ❛❣♦r❛ q✉❛♥❞♦u é ♥ã♦ ♥✉❧♦✳ P♦r ❤✐♣ót❡s❡✱uv =kukkvk −xnyn = 0✱

❞❡ ♦♥❞❡✱ kv_k=xnyn/kuk✳ ❉✐st♦✱

v v =kvk

2

−y_n2 = (x2_ny2_n/_ku_k2)₋y_n2 =y_n2

x2

n

ku_k2 −1

>0

❝♦♥tr❛❞✐③❡♥❞♦✱ ♥♦✈❛♠❡♥t❡✱ v ♥ã♦ s❡r t✐♣♦✲❡s♣❛ç♦✳ P♦rt❛♥t♦✱ u ❞❡✈❡ s❡r t✐♣♦✲❧✉③✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ v é ✈❡t♦r t✐♣♦✲❧✉③✳

P♦r ú❧t✐♠♦✱ ❝♦♠♦ u❡ v sã♦ t✐♣♦✲❧✉③✱ t❡♠♦s _ku_k2 =x2_n✱ _kv_k2 =y_n2 ❡

0 =uv =hu, vi −xnyn=λkvk

2

−xnyn=λy2n−xnyn ⇒ yn(λyn−xn) = 0

❝♦♠ yn6= 0 ✭v é t✐♣♦✲❧✉③✮✱ ♣♦rt❛♥t♦✱ xn =λyn ❡u=λv✳

❚❡♦r❡♠❛ ✶✳✷✳ ❙❡❥❛♠ u❡ v ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ♦✉ t✐♣♦✲❧✉③ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ❞❡_En₁ ❡ t ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✱ ❡♥tã♦ ♦ ✈❡t♦r tv t❡♠ ❛ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ❞❡ v ❡ ♦ ✈❡t♦r u+v ❤❡r❞❛ ♦ t✐♣♦ ❡ ❛ ♣❛r✐❞❛❞❡ ❞♦s ✈❡t♦r❡s u ❡v✳ ❆❧é♠ ❞✐ss♦✱u+v s❡rá t✐♣♦✲❧✉③ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u ❡ v ❢♦r❡♠ ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s t✐♣♦✲❧✉③✳

Pr♦✈❛✿ ➱ ❝❧❛r♦ q✉❡ ♦ ✈❡t♦rtv ❤❡r❞❛ ♦ t✐♣♦ ❡ ♣❛r✐❞❛❞❡ ❞♦ ✈❡t♦rv✳ ❖ ✈❡t♦ru+v✱ t❡♠ ❛ ú❧t✐♠❛ ❝♦♦r❞❡♥❛❞❛ ❞❛❞❛ ♣❡❧❛ s♦♠❛ ❞❛s ú❧t✐♠❛s ❝♦♦r❞❡♥❛❞❛s ❞❡u❡v q✉❡ t❡♠ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ♣♦r ❤✐♣ót❡s❡✱ ❧♦❣♦ ❛ ♣❛r✐❞❛❞❡ ❞❡ u+v é ❛ ♠❡s♠❛ ❞❡ u ❡ v✳ ❆❧é♠ ❞✐ss♦✱

(u+v)(u+v) =u_|{z}u

≤0

+2u_|{z}v

≤0

+_|{z}vv

≤0

≤0,

❧♦❣♦✱ u+v ♥ã♦ é t✐♣♦✲❡s♣❛ç♦ ❡ (u+v)(u+v) = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ uv = 0

❞❡♣❡♥❞❡♥❞❡s✳

❈♦r♦❧ár✐♦ ✶✳✶✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ é ✉♠ s✉❜✲ ❝♦♥❥✉♥t♦ ❝♦♥❡①♦ ❞❡ E₁n✳

Pr♦✈❛✿ ❉❡ ❢❛t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✷✱ ❞❛❞♦s u ❡ v ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ❞❡ E₁n✱ ♣❛r❛ t _∈ [0,1]✱ tu ❡ (1₋t)v sã♦ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ q✉❡ u ❡ v✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆♦✈❛♠❡♥t❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✷✱ ♦ ✈❡t♦r tu+ (1₋t)v t❛♠❜é♠ é t✐♣♦✲t❡♠♣♦ ❡ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ q✉❡ u ❡ v✳ P♦rt❛♥t♦✱ t♦❞♦ ✈❡t♦r ❞♦ s❡❣♠❡♥t♦[u, v]é t✐♣♦✲t❡♠♣♦ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ q✉❡ u❡v✱ ♣r♦✈❛♥❞♦ ♦ q✉❡ q✉❡rí❛♠♦s✳

❊st❡s r❡s✉❧t❛❞♦s✱ ♥♦s ♠♦str❛♠ q✉❡ ♦ ❝♦♥❥✉♥t♦T t❡♠ ❞✉❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s✱

❝❤❛♠❛❞❛s ❞❡ ❝♦♥❡s t✐♣♦✲t❡♠♣♦✳ ❖ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ ♦ ❝♦♥❡ ❞❡ ❧✉③Cn−1_{✱ ♦❜s❡r✈❛♥❞♦}

q✉❡ ❞❛❞♦s ❞♦✐s ✈❡t♦r❡s t✐♣♦✲❧✉③ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ♣♦❞❡♠♦s ❧✐❣❛r ❡st❡s ❞♦✐s ✈❡t♦r❡s ❛tr❛✈és ❞❡ ✉♠ ❝❛♠✐♥❤♦ ❥✉st❛♣♦st♦ ♥♦ ❝♦♥❡✱ ❢♦r♠❛❞♦ ♣♦r ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❡ ✉♠ ❛r❝♦ ❞❡ ❝✐r❝✉♥❢êr❡♥❝✐❛✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❉✐③❡♠♦s q✉❡ ♦ ✈❡t♦rué ▲♦r❡♥t③ ♦rt♦❣♦♥❛❧ ❛♦ ✈❡t♦rv ❡♠ _En

1 q✉❛♥❞♦

uv = 0✳

❈♦r♦❧ár✐♦ ✶✳✷✳ ❖ ♣r♦❞✉t♦ ❧♦r❡♥t③✐❛♥♦ ❞❡ ❞♦✐s ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ♥✉♥❝❛ s❡ ❛♥✉❧❛✳ Pr♦✈❛✿ ❙❡u❡v sã♦ ❞♦✐s ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦✱ ❡♥tã♦₋u❡₋v t❛♠❜é♠ sã♦ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦✱ ♣♦ré♠✱ ❝♦♠ ♣❛r✐❞❛❞❡s ❞✐❢❡r❡♥t❡s ❞❡u❡v✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡st❡ ♠♦❞♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶ t❡♠♦s✱

(

uv <0, s❡ ✉ ❡ ✈ t❡♠ ♠❡s♠❛ ♣❛r✐❞❛❞❡✱

uv >0, s❡ ✉ ❡ ✈ t❡♠ ♣❛r✐❞❛❞❡ ❞✐st✐♥t❛s.

◆❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ✉s❛♠♦s u−v =−(uv)<0.

❈♦r♦❧ár✐♦ ✶✳✸✳ ❙❡❥❛ u ✉♠ ✈❡t♦r t✐♣♦✲t❡♠♣♦ ❞❡ En1✳ ❙❡ v ∈ En1 é ▲♦r❡♥t③ ♦rt♦❣♦♥❛❧

❛ u✱ ❡♥tã♦ v é t✐♣♦✲❡s♣❛ç♦✳

Pr♦✈❛✿ ❇❛st❛ ♦❜s❡r✈❛r♠♦s q✉❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✱ v ♥ã♦ ♣♦❞❡ s❡r t✐♣♦✲t❡♠♣♦ ♥❡♠ t✐♣♦✲❧✉③✳

❙❡ U _⊂ En1 é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ m✱ ❛ r❡str✐çã♦ ❞♦ ♣r♦❞✉t♦

❡s❝❛❧❛r ❧♦r❡♥t③✐❛♥♦ ❛♦ s✉❜❡s♣❛ç♦ U é ❛✐♥❞❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r✳ ❙❡ ♦ s✉❜❡s♣❛ç♦ t❡♠ ❞✐♠❡♥sã♦ ✶✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ v _∈ U✱ ❝♦♥s✐❞❡r❛♥❞♦ ✉♠ ✈❡t♦r vi ∈ U✱ t❡♠♦s v =λvi ❡

vv =λ

2_v

ivi.

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t♦❞♦s ♦s ✈❡t♦r❡s t❡rã♦ ♦ ♠❡s♠♦ ❝❛rát❡r ❝❛✉s❛❧ ❡ ❛ ♠❛tr✐③ s❡rá ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦ ✭r❡s♣✳ ♥❡❣❛t✐✈♦✱ ♥✉❧♦✮ s❡ ♦ ✈❡t♦r vi é t✐♣♦✲❡s♣❛ç♦ ✭r❡s♣✳ t✐♣♦✲ t❡♠♣♦✱ t✐♣♦✲❧✉③✮✳ ❱❛♠♦s ❛♥❛❧✐s❛r ♦ ❝❛s♦ ❡♠ q✉❡ ❛ ❞✐♠❡♥sã♦ ❞♦ s✉❜❡s♣❛ç♦U é ♠❛✐♦r q✉❡ ✶✱ ♠❛s ❛♥t❡s ❞❡✜♥✐♠♦s ♦ q✉❡ ✈❡♠ ❛ s❡r ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❡♠En1✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❯♠❛ ❜❛s❡B=_{v1, ..., vn}❞❡En1 é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③

s❡vnvn =−1❡vivj =δij✱ ❝❛s♦ ❝♦♥trár✐♦✳ ❯♠ ✈❡t♦rv ∈E

n

1 é ❞✐t♦ ✉♥✐tár✐♦ q✉❛♥❞♦

vv =±1✳

❊①❡♠♣❧♦ ✶✳✸✳ ❆ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❞❡ En1 é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③✳

❚♦❞❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③ ❞✐❛❣♦♥❛❧✐③❛ ❛ ♠❛tr✐③ ❞♦ ♣r♦❞✉t♦ ❧♦r❡♥t③✐❛♥♦✱ ❝♦♠ 1 ♥❛s n₋1 ♣r✐♠❡✐r❛s ❡♥tr❛❞❛s ❞❛ ❞✐❛❣♦♥❛❧ ❡ ₋1 ♥❛ ú❧t✐♠❛✳

❖❜s❡r✈❛♠♦s q✉❡ ❡♠ ✉♠ s✉❜❡s♣❛ç♦U _⊂En1✱ ❛ r❡str✐çã♦ ❞♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❧♦r❡♥t✲

③✐❛♥♦ ❛✐♥❞❛ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❧✐♥❡❛r ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣r♦❞✉t♦ ❡s❝❛❧❛r ❞❡✈✐❞♦ ❛ ❞❡❣❡♥❡r❛❧✐❞❛❞❡✮ ❡ s✉❛ ♠❛tr✐③ t❡♠ ♦r❞❡♠ ✐♥❢❡r✐♦r ❛ n✳ ❇✉s❝❛♠♦s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❛ ♠❛tr✐③ ❞❡st❛ r❡str✐çã♦ ❛♦ s✉❜❡s♣❛ç♦U✳ P❛r❛ t❛❧✱ ✉s❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❞❛ ➪❧❣❡❜r❛ ▲✐♥❡❛r✿ ❚♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♠✉♥✐❞♦ ❞❡ ✉♠ ♣r♦❞✉t♦ ❡s❝❛❧❛r ♣♦ss✉✐ ✉♠❛ ❜❛s❡ ♦rt♦❣♦♥❛❧✳

Pr✐♠❡✐r❛♠❡♥t❡✱ ❛♥❛❧✐s❛♠♦s ♦ ❝❛s♦ ❡♠ q✉❡ U ♣♦ss✉✐ ✉♠ ✈❡t♦r t✐♣♦✲t❡♠♣♦ v✳ ◗✉❛♥❞♦ ✐st♦ ❛❝♦♥t❡❝❡✱ ♣♦❞❡♠♦s t♦♠❛r ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③ ♣❛r❛ U q✉❡ ❝♦♥t❡♥❤❛ ♦ ✈❡t♦r v

kv_k✱ ♣♦✐s✱ ✉♠❛ ❜❛s❡ ♦rt♦❣♦♥❛❧ ❞❡ U ❢♦r♠❛❞❛ ♣♦r ✈❡t♦r❡s ✉♥✐tár✐♦s q✉❡ ❝♦♥té♠ ✉♠ ✈❡t♦r t✐♣♦✲t❡♠♣♦ é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③ ♣❛r❛ U✳ P♦rt❛♥t♦✱ ✉♠❛ ♠❛tr✐③ ❞❡ U ♥❡st❡ ❝❛s♦ é ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❝♦♠ 1 ♥❛s m₋1

♣r✐♠❡✐r❛s ❡♥tr❛❞❛s ❞❛ ❞✐❛❣♦♥❛❧ ❡ −1 ♥❛ ú❧t✐♠❛ ❡♥tr❛❞❛✳ ❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡

♥ã♦ ❤❛❥❛ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ♥❡♠ ✈❡t♦r❡s t✐♣♦✲❧✉③ ❡♠ U✳ ❊♥tã♦ U é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♦♥❞❡ t♦❞❛s ❜❛s❡s t❡rã♦ ❛♣❡♥❛s ✈❡t♦r❡s t✐♣♦✲❡s♣❛ç♦ ❡ ❛ss✐♠ ✉♠❛ ♠❛tr✐③ ❞♦ ♣r♦❞✉t♦ ❝♦♠ r❡❧❛çã♦ ❛ ✉♠❛ ❜❛s❡ ♦rt♦❣♦♥❛❧ ❞❡ U s❡rá ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ♦♥❞❡

❡♥❝♦♥tr❛♠♦s 1 ❡♠ t♦❞❛s ❛ m ❡♥tr❛❞❛s ❞❛ ❞✐❛❣♦♥❛❧✳ ❖ ú❧t✐♠♦ ❝❛s♦ q✉❡ ♥♦s r❡st❛ s✉♣♦r é q✉❡U ♥ã♦ ♣♦ss✉❛ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ♠❛s ♣♦ss✉❛ ✉♠ ✈❡t♦r t✐♣♦✲❧✉③u✳ ◆❡st❡ ❝❛s♦✱ ✉♠❛ ❜❛s❡ ❞❡ U q✉❡ ❝♦♥t❡♥❤❛ ♦ ✈❡t♦r u ♥ã♦ ♣♦❞❡ ♣♦ss✉✐r ♦✉tr♦s ✈❡t♦r❡s t✐♣♦✲ ❧✉③ ❡♠ ✈✐rt✉❞❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✶✱ ❧♦❣♦ ❛ ♠❛tr✐③ ❞♦ ♣r♦❞✉t♦ ❧♦r❡♥t③✐❛♥♦ é ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❝♦♠ 1 ♥❛s m₋1 ♣r✐♠❡✐r❛s ❡♥tr❛❞❛s ❞❛ ❞✐❛❣♦♥❛❧ ❡ 0 ♥❛ ú❧t✐♠❛✳

❉❡✜♥✐çã♦ ✶✳✸✳ ❙❡❥❛A ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❞❛ r❡str✐çã♦ ❞♦ ♣r♦❞✉t♦ ❧♦r❡♥t③✐❛♥♦ ❛♦ s✉❜❡s♣❛ç♦ U✱ ❝♦♠ r❡❧❛çã♦ ❛ ✉♠❛ ❜❛s❡ ❞❡st❡ s✉❜❡s♣❛ç♦✳

✶✳ ◗✉❛♥❞♦A♣♦ss✉✐ ✉♠❛ ❡♥tr❛❞❛ ♥❡❣❛t✐✈❛✱ ❞✐③❡♠♦s q✉❡ ♦ s✉❜❡s♣❛ç♦ é t✐♣♦✲t❡♠♣♦❀ ✷✳ ◗✉❛♥❞♦ A ♣♦ss✉✐ ✉♠❛ ❡♥tr❛❞❛ ♥✉❧❛✱ ❞✐③❡♠♦s q✉❡ ♦ s✉❜❡s♣❛ç♦ é t✐♣♦✲❧✉③❀ ✸✳ ◗✉❛♥❞♦ A t❡♠ ❛♣❡♥❛s ❡♥tr❛❞❛s ♣♦s✐t✐✈❛s✱ ❞✐③❡♠♦s q✉❡ ♦ s✉❜❡s♣❛ç♦ é t✐♣♦✲

❡s♣❛ç♦✳

❖❜s❡r✈❛çã♦ ✶✳✶✳ ◆♦t❛♠♦s q✉❡ s❡ ✉♠ s✉❜❡s♣❛ç♦ U é t✐♣♦✲❡s♣❛ç♦✱ ❡♥tã♦ t♦❞♦s ♦s ✈❡t♦r❡s ❞❡ U t❛♠❜é♠ ♦ sã♦✳ ❆❧é♠ ❞✐ss♦✱ s✉❜❡s♣❛ç♦s t✐♣♦✲❧✉③ ❡ t✐♣♦✲❡s♣❛ç♦ ♥ã♦ ♣♦s✲ s✉❡♠ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦✳ P♦rt❛♥t♦✱ ✉♠ s✉❜❡s♣❛ç♦ é t✐♣♦✲t❡♠♣♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣♦ss✉✐ ✉♠ ✈❡t♦r t✐♣♦✲t❡♠♣♦ ❡ ✉♠ s✉❜❡s♣❛ç♦ é t✐♣♦✲❧✉③ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣♦ss✉✐ ✉♠ ✈❡t♦r t✐♣♦✲❧✉③ ♠❛s ♥❡♥❤✉♠ t✐♣♦✲t❡♠♣♦✳

❚❡♦r❡♠❛ ✶✳✸✳ ❙❡❥❛♠ U _⊂ En1 ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ n − 1 ❡ ~n ♦

❣❡r❛❞♦r ❞♦ s✉❜❡s♣❛ç♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ U⊥ _{❞❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛✳ ❖ s✉❜❡s♣❛ç♦}

U é t✐♣♦✲❡s♣❛ç♦ ✭r❡s♣✳ t✐♣♦✲t❡♠♣♦✱ t✐♣♦✲❧✉③✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ✈❡t♦r ~n é ✉♠ ✈❡t♦r t✐♣♦✲t❡♠♣♦ ✭r❡s♣✳ t✐♣♦✲❡s♣❛ç♦✱ t✐♣♦✲❧✉③✮✳

Pr♦✈❛✿ ❉❡ ❢❛t♦✱ ♦s ✈❡t♦r❡s~n= (a1, ..., an)❡w= (a1, ...,−an)♣♦ss✉❡♠ ♦ ♠❡s♠♦ ❝❛rát❡r ❝❛✉s❛❧ ❡ ♣❛r❛ q✉❛❧q✉❡r x= (x1, ..., xn)∈U✱

0 = _h~n, x_i=a1x1+...+anxn=a1x1+...−(−an)xn =wx,

♦✉ s❡❥❛✱ ♦s ✈❡t♦r❡s ❞❡ U sã♦ ▲♦r❡♥t③ ♦rt♦❣♦♥❛✐s ❛ w✳

■st♦ s✉❣❡r❡ ✉♠❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦ s✉❜❡s♣❛ç♦ U ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ❝❛rát❡r ❝❛✉s❛❧ ❞♦ ✈❡t♦r w✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ w é ✈❡t♦r t✐♣♦✲t❡♠♣♦✱ ❡♥tã♦ t♦❞♦s ♦s ✈❡t♦r❡s ❞❡ U sã♦ t✐♣♦✲❡s♣❛ç♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❝♦r♦❧ár✐♦ ✶✳✸✳ ❆ss✐♠✱U é t✐♣♦✲❡s♣❛ç♦✳ ❙❡wé ✉♠ ✈❡t♦r

t✐♣♦✲❡s♣❛ç♦ ✭♣♦❞❡♠♦s s✉♣♦r w✉♥✐tár✐♦✮✱ ❡①✐st❡ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③ ❞❡

En1 q✉❡ ❝♦♥t❡♠ w✱ ❧♦❣♦✱ ♦ ✈❡t♦r❡s r❡st❛♥t❡ ♥❡st❛ ❜❛s❡ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ❞❡ U ❝♦♠

✉♠ ✈❡t♦r t✐♣♦✲t❡♠♣♦✱ ♣♦rt❛♥t♦✱U é t✐♣♦✲t❡♠♣♦✳ ◗✉❛♥❞♦wé t✐♣♦✲❧✉③ t❡♠♦sw_∈U✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ U é t✐♣♦✲❧✉③✱ ♣♦✐s✱ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡❣❡♥❡r❛❞❛ q✉❛♥❞♦ r❡str✐t❛

❛♦ s✉❜❡s♣❛ç♦ U✳

✶✳✷ ❚r❛♥s❢♦r♠❛çõ❡s ❞❡ ▲♦r❡♥t③✳

◆❡st❛ s❡çã♦✱ tr❛t❛♠♦s ❞❛s ❛♣❧✐❝❛çõ❡s ❞❡ En1 ♣❛r❛ En1✱ ❡q✉✐✈❛❧❡♥t❡s às ■s♦♠❡tr✐❛s

❧✐♥❡❛r❡s ❞❡ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✳ ❚❛✐s ❛♣❧✐❝❛çõ❡s✱ sã♦ ✉té✐s ♥♦ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ❝✉r✈❛s ♥♦ ❡s♣❛ç♦ ❞❡ ▲♦r❡♥t③✲▼✐♥❦♦✇s❦✐ ❞❡ ❞✐♠❡♥sã♦ n ❡ ❝♦♠♦ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐✲ ❛♥♦✱ t❡♠ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ❣❡♦♠❡tr✐❛ ❛❣r❡❣❛❞❛ ❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ ❢✉♥çõ❡s ❝♦♠♦ ❛ ❝✉r✈❛t✉r❛ ❡ t♦rçã♦ ♣♦r ♠♦✈✐♠❡♥t♦ rí❣✐❞♦✱ ❡♥tr❡ ♦✉tr♦s✳

❉❡✜♥✐çã♦ ✶✳✹✳ ❯♠❛ ❛♣❧✐❝❛çã♦ T : En1 →En1 é ❝❤❛♠❛❞❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③

s❡

T xT y=xy, ♣❛r❛ t♦❞♦x, y ❡♠ _En

1.

◆♦ ♣ró①✐♠♦ t❡♦r❡♠❛✱ ♠♦str❛♠♦s q✉❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③ é ✉♠❛ tr❛♥s✲ ❢♦r♠❛çã♦ ❧✐♥❡❛r q✉❡ ♣r❡s❡r✈❛ ❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❞❡ En1 ❝♦♠♦ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡

▲♦r❡♥t③✳

❚❡♦r❡♠❛ ✶✳✹✳ ❯♠❛ ❢✉♥çã♦ T : _Rn _{→ R}n é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ T é ❧✐♥❡❛r ❡ _{T e1, ..., T en} é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③ ❞❡En1.

Pr♦✈❛ ✿ ❙❡ T é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③✱ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ T ei

T ej =eiej✱ ❧♦❣♦✱ {T e1, ..., T en} é ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③ ❞❡E

n

1✳ ❊♥tã♦✱ ♣❛r❛

t♦❞♦v _∈En1✱ v =

n

X

i=1

aiei✱ s✉♣♦♥❞♦T v = n

X

i=1

biT ei t❡♠♦saj =vej =T vT ej =bj✱

▲♦r❡♥t③ ❡ T ❧✐♥❡❛r ♣❛r❛ v =

n

X

i=1

aiei✱ u= n

X

j=1

bjej ❡♠ En1t❡♠♦s

T vT u = T

n

X

i=1

aieiT

n

X

j=1

bjej

=

n

X

i=1

aiT ei

n

X

j=1

bjT ej

=

n

X

i,j=1

aibj(T eiT ej)

=

n

X

i,j=1

aibj(eiej)

=

n

X

i=1

aiei

n

X

j=1

bjej

= vu.

P♦rt❛♥t♦✱ T é ✉♠❛ tr❛♥❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③✳

❯♠❛ ♠❛tr✐③A❞❡ ♦r❞❡♠n é ❧♦r❡♥t③✐❛♥❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❛ss♦❝✐❛❞❛A :_En

1 →En1✱ ❞❡✜♥✐❞❛ ♣♦rA(x) = Axé ❧♦r❡♥t③✐❛♥❛✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s

❛s ♠❛tr✐③❡s ❧♦r❡♥t③✐❛♥❛s ❞❡ ♦r❞❡♠ n ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ♠❛tr✐③❡s ❢♦r♠❛♠ ♦ ❣r✉♣♦ O(n₋1,1)✱ ❝♦♥❤❡❝✐❞♦ ♣♦r ❣r✉♣♦ ❞❡ ▲♦r❡♥t③ ❞❛s ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ n✳

❚❡♦r❡♠❛ ✶✳✺✳ ❙❡❥❛ A ✉♠❛ ♠❛tr✐③ r❡❛❧n_×n✱ ❡ s❡❥❛ J ✉♠❛ ♠❛tr✐③ n_×n ❞✐❛❣♦♥❛❧ ❞❡✜♥✐❞❛ ♣♦r J =         

1 0 . . . 0 0 0 1 . . . 0 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

0 0 . . . 1 0 0 0 . . . 0 ₋1

         .

❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿ ✭✶✮❆ ♠❛tr✐③ A é ❧♦r❡♥t③✐❛♥❛✳

✭✷✮❆s ❝♦❧✉♥❛s ❞❡ A ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③ ❞❡ Rn✳

✭✸✮❆ ♠❛tr✐③ A s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ AtJA=J✳ ✭✹✮❆ ♠❛tr✐③ A s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ AJAt=J✳

✭✺✮❆s ❧✐♥❤❛s ❞❡ A ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③ ❞❡ Rn✳

Pr♦✈❛✿ (1)_→(2) P♦r ❤✐♣ót❡s❡✱ Aé ❧♦r❡♥t③✐❛♥❛✱ ❧♦❣♦ ❡①✐st❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③ T : Rn → Rn✱ ❞❡✜♥✐❞❛ ♣♦r✱ T v = Av✳ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✹✱ T é ❧✐♥❡❛r✱ ❡ ❛s ❝♦❧✉♥❛s ❞❛ ♠❛tr✐③A sã♦ ❢♦r♠❛❞❛s ♣❡❧❛s ✐♠❛❣❡♥s ❞❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ♣❡❧❛ tr❛♥s❢♦r✲ ♠❛çã♦ T✱ q✉❡ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③✱ ❧♦❣♦ ❛s ❝♦❧✉♥❛s ❞❡ A ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③✳

(2) _→ (3) ❱❛♠♦s r❡♣r❡s❡♥t❛r ❛s ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ ♥ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ A = (aij)✱ At = (bij)✱ J = (cij)✱ JA= (dij) ❡ At(JA) = (eij)✱ ♦♥❞❡ bij = aji✱ cnn = −1 ❡ cij =δij✱ ❝❛s♦ ❝♦♥trár✐♦ ❡ i, j = 1, ..., n✳ P♦r ❤✐♣ót❡s❡✱ ♦s ✈❡t♦r❡s vi = (a1i, ..., ani) i= 1, ..., n ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③✳ ❊♥tã♦✱dij =

n

X

k=1

cikakj ❡✱

eij = n

X

l=1

bildlj = n

X

l=1

ali( n

X

k=1

clkakj) = n

X

l=1

ali(cllalj) = n

X

l=1

cll(alialj) =vivj =cij.

(3) _→(4)❆ ❤✐♣ót❡s❡AtJA=J✱ ✐♠♣❧✐❝❛₋1 = det(J) = det(AtJA) = ₋det2(A)✱

❧♦❣♦✱ det(A) ₆= 0 ❡ A é ✐♥✈❡rtí✈❡❧✳ ❉❛ ✐❣✉❛❧❞❛❞❡✱ (JAtJ)A = J(AtJA) = JJ = I✱

t❡♠♦s JAtJ =A−1✱ ❧♦❣♦ AJAtJ =AA−1 =I✱ ♣♦rt❛♥t♦ AJAt =J−1 =J✳

(4) _→ (5) ❈♦♥s✐❞❡r❛♠♦s ❛s r❡♣r❡s❡♥t❛çõ❡s ❞❛s ♠❛tr✐③❡s ❞❡ ♦r❞❡♥ n✿ A = (aij)✱ At = (bij)✱ J = (cij)✱ AJ = (dij) ❡ (AJ)At = (eij)✱ ♦♥❞❡ bij = aji✱ cnn = −1 ❡ cij = δij✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♦♥❞❡✱ i, j = 1, ..., n✳ ❚♦♠❛♥❞♦ ♦s ✈❡t♦r❡s ❧✐♥❤❛s ❞❡ A✱ wi = (ai1, ..., ain) ✈❛❧❡dij =

n

X

k=1

ajkcki ❡✱

cij =eij = n

X

l=1

djlbli = n X l=1 ( n X k=1

ajkckl)ail = n

X

l=1

(cllajl)ail = n

X

l=1

cll(ajlail) =wiwj.

(5) _→(1) ▼♦str❛♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ▲♦r❡♥t③ T :Rn→Rn t❛❧ q✉❡

A s❡❥❛ s✉❛ ♠❛tr✐③ ❞❡ ▲♦r❡♥t③✳ ❚♦♠❛♠♦s ❛ ❛♣❧✐❝❛çã♦ T : Rn → Rn✱ ❞❡✜♥✐❞❛ ♣♦r✱

T v = vA✳ ❙❡ vi é ✉♠❛ ❧✐♥❤❛ ❞❡ A✱ t❡♠♦s T ei = vi✳ ❈♦♠♦ ❛s ❧✐♥❤❛s ❞❡ A ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③✱ ♦ ❝♦♥❥✉♥t♦ {T e1, ..., T en} é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③✳ ❆❧é♠ ❞✐ss♦✱ T é ❧✐♥❡❛r ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✹ é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ▲♦r❡♥t③✳

❙❡A ✉♠❛ ♠❛tr✐③ ❧♦r❡♥t③✐❛♥❛✱ ❡♥tã♦ det(A) = _±1✳ ❉❡♥♦t❛♠♦s ♣♦rSO(n₋1,1)✱

♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♠❛tr✐③❡sA❡♠O(n₋1,1)t❛✐s q✉❡det(A) = 1✳ ❊st❡ s✉❜❣r✉♣♦

SO(n₋1,1)é ❝❤❛♠❛❞♦ ❣r✉♣♦ ❡s♣❡❝✐❛❧ ❞❡ ▲♦r❡♥t③✳

❈♦♠♦ ✈✐♠♦s✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❡♠ Rn t❡♠ ❞✉❛s ❝♦♠✲

♣♦♥❡♥t❡s ❝♦♥❡①❛s✱ ✉♠❛ ❞❛❞❛ ♣❡❧♦s ✈❡t♦r❡s q✉❡ t❡♠ ♣❛r✐❞❛❞❡ ♣♦s✐t✐✈❛ ❡ ❛ ♦✉tr❛ ❞❛❞❛ ♣❡❧♦s ✈❡t♦r❡s q✉❡ t❡♠ ♣❛r✐❞❛❞❡ ♥❡❣❛t✐✈❛✳ ❯♠❛ ♠❛tr✐③A é ❞✐t❛ ♣♦s✐t✐✈❛ ✭r❡s♣✳ ♥❡❣❛✲ t✐✈❛✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱A tr❛♥s❢♦r♠❛ ✈❡t♦r❡s ❞♦ t✐♣♦✲t❡♠♣♦ ❞❡ ♣❛r✐❞❛❞❡ ♣♦s✐t✐✈❛ ❡♠ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❞❡ ♣❛r✐❞❛❞❡ ♣♦s✐t✐✈❛ ✭r❡s♣✳ ♥❡❣❛t✐✈❛✮✳ P♦r ❡①❡♠♣❧♦✱ ❛ ♠❛tr✐③ J ❝✐t❛❞❛ ♥♦ ❚❡♦r❡♠❛ ✶✳✺ é ✉♠❛ ♠❛tr✐③ ♥❡❣❛t✐✈❛✳

❙❡❥❛ P O(n ₋1,1) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♠❛tr✐③❡s ♣♦s✐t✐✈❛s ❡♠ O(n ₋ 1,1)✳

❊♥tã♦ P O(n₋1,1) é ✉♠ s✉❜❣r✉♣♦ ❞❡ í♥❞✐❝❡ ❞♦✐s ❡♠ O(n ₋1,1)✳ ❉❡ ❢❛t♦✱ ❝♦♠♦

A _∈ O(n₋1,1)✱ ♣❛r❛ t♦❞♦ v _{∈ E}n

1✱ AvAv =v v✳ ❉❡st❛ ❢♦r♠❛✱ A é ♣♦s✐t✐✈❛ ♦✉

♥❡❣❛t✐✈❛✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ♦✉A_∈P O(n₋1,1)♦✉ ❡st❛ ♥♦ s❡✉ ❝♦♠♣❧❡♠❡♥t❛r✱ ♦✉ s❡❥❛✱

O(1, n₋1)/P O(n₋1,1) t❡♠ ❛♣❡♥❛s ❞✉❛s ❝❧❛ss❡s✳ ❖ ❣r✉♣♦ ❞❛s ♠❛tr✐③❡s ♣♦s✐t✐✈❛s

P O(n ₋1,1) é ❝❤❛♠❛❞♦ ❣r✉♣♦ ♣♦s✐t✐✈♦ ❞❡ ▲♦r❡♥t③✳ ❙❡❥❛ t❛♠❜é♠ P SO(n₋ 1,1)

♦ ❣r✉♣♦ ❞❛s ♠❛tr✐③❡s ♣♦s✐t✐✈❛s ❡♠ SO(n₋1,1) é ✉♠ s✉❜❣r✉♣♦ ❞❡ í♥❞✐❝❡ ❞♦✐s ❡♠

SO(n₋1,1)✳ ❖ ❣r✉♣♦ SO(n₋1,1) é ❝❤❛♠❛❞♦ ❣r✉♣♦ ❡s♣❡❝✐❛❧ ❞❡ ▲♦r❡♥t③ ♣♦s✐t✐✈♦✳

❆♥t❡s ❞♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✱ ❞❡✜♥✐♠♦s ♦ q✉❡ é ✉♠❛ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦ G s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦S ♥ã♦ ✈❛③✐♦✳

❉❡✜♥✐çã♦ ✶✳✺✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ S ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳ ❉✐③❡♠♦s q✉❡ G ❛❣❡ s♦❜r❡ S s❡ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ . : G_×S _→ S✱ ❞❡♥♦t❛❞♦ ♣♦r .(g, s) _7→ g.s s❛t✐s❢❛③❡♥❞♦✿

✭✶✮ e.s =s ♣❛r❛ t♦❞♦ s_∈S✱

✭✷✮ ♣❛r❛ t♦❞♦ g, h_∈G❡ s_∈S ✈❛❧❡ (gh).s=g.(h.s)✳

❖ ❝♦♥❥✉♥t♦ O(s) =_{g.s t❛❧ q✉❡ g _∈G_}é ❝❤❛♠❛❞♦ ❞❡ ór❜✐t❛ ❞❡s✳ ❯♠❛ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦Gs♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ S é ❞✐t❛ tr❛♥s✐t✐✈❛ s❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛✳

❚❡♦r❡♠❛ ✶✳✻✳ ❆ ❛çã♦ ♥❛t✉r❛❧ ❞❡ P O(n ₋1,1) s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s s✉❜❡s♣❛ç♦s

t✐♣♦✲t❡♠♣♦ ❞❡ ❞✐♠❡♥sã♦ m ❡♠ Rn é tr❛♥s✐t✐✈❛✳

Pr♦✈❛ ✿ ❙❡❥❛ V ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ m ❞❡ Rn✳ P♦❞❡♠♦s ✐❞❡♥✲

t✐✜❝❛r Rm ❝♦♠ ♦ s✉❜❡s♣❛ç♦ ❞❡ Rn ❣❡r❛❞♦ ♣❡❧♦s ✈❡t♦r❡s {en−(m+1), ..., en}✳ ❈♦♠♦ en ∈ Rn✱ Rm é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ t✐♣♦✲t❡♠♣♦✳ Pr♦✈❛r❡♠♦s ♦ r❡s✉❧t❛❞♦ ♠♦s✲ tr❛♥❞♦ q✉❡ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ A _∈ P O(n ₋1,1) t❛❧ q✉❡ A(_Rm) = V✱ ✉s❛♥❞♦ ♦ ♣r♦❝❡ss♦ ❞❡ ♦rt♦♥♦r♠❛❧✐③❛çã♦ ❞❡ ●r❛♠✲❙❝❤♠✐❞t✳

❈♦♠♦ V é ✉♠ s✉❜❡s♣❛ç♦ t✐♣♦✲t❡♠♣♦ q✉❛❧q✉❡r✱ t♦❞♦ s✉❜❡s♣❛ç♦ t✐♣♦✲t❡♠♣♦ ❞❡ ❞✐♠❡♥sã♦ m ❞❡ Rn ❡st❛ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ Rm ❡ ♣♦rt❛♥t♦ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ór❜✐t❛✳

❚♦♠❛♥❞♦ {u1, ..., um} ✉♠❛ ❜❛s❡ ❞♦ s✉❜❡s♣❛ç♦ ❱✱ ♦♥❞❡ um é ✉♠ ✈❡t♦r t✐♣♦✲ t❡♠♣♦ ❞❡ ♣❛r✐❞❛❞❡ ♣♦s✐t✐✈❛ ♣♦❞❡♠♦s ❝♦♠♣❧❡t❛r ❡st❛ ❜❛s❡ ♣❛r❛ ✉♠❛ ❜❛s❡ B =

{u1, ..., um, p1, ..., pn−m} ❞❡ Rn✳ P♦❞❡♠♦s r❡♦r❞❡♥❛r ❡ r❡❡s❝r❡✈❡r ❛ ❜❛s❡ B ♣♦r B =

{q1, ..., qn}✱ ❞❡ ♠♦❞♦ q✉❡✱pi =qi ♣❛r❛i= 1, ..., n−m ❡qn−i =um−i✱i= 0, ..., m−1✳ ❈♦♥s✐❞❡r❛♥❞♦ wn=

qn

kqnk✱ ♦❜s❡r✈❛♠♦s q✉❡ wnwn =

qn

kqnk qn

kqnk

= 1

kqnk2 qnqn

= ₋kqnk

2

kqnk2

= ₋1.

P♦rt❛♥t♦✱wné ✉♠ ✈❡t♦r t✐♣♦✲t❡♠♣♦✳ ❉❡✜♥✐♥❞♦ ♦ ✈❡t♦rvn−1 =qn−1+(wnqn−1)wn

t❡♠♦s✱

wnvn−1 =wnqn−1+ (wnqn−1)(wnwn) =wnqn−1−wnqn−1 = 0,

❧♦❣♦✱ vn−1 é ✉♠ ✈❡t♦r t✐♣♦✲❡s♣❛ç♦✳ ❚♦♠❛♥❞♦ wn−1 = vn−1/kvn−1k q✉❡ t❡♠ ♥♦r♠❛

❧♦r❡♥t③✐❛♥❛ ✉♥✐tár✐❛✱ ❞❡✜♥✐♠♦s ♦ ✈❡t♦rvn−2 =qn−2+ (wnqn−2)wn−(wnqn−2)wn−1✱

❡ ♥♦t❛♠♦s q✉❡✿

wnvn−2 = wnqn−2+ (wnqn−2)(wnwn)−(wnqn−1)(wnwn−1)

= wnqn−2−wnqn−2 = 0

wn−1vn−2 = wn−1qn−2+ (wn−1qn−2)(wn−1wn)−(wn−1qn−1)(wn−1wn−1)

= wn−1qn−2−wn−1qn−2 = 0,

♦✉ s❡❥❛✱ vn−2 é ▲♦r❡♥t③ ♦rt♦❣♦♥❛❧ ❛ wn ❡ ❛ wn−1✱ ♣♦rt❛♥t♦ é ✉♠ ✈❡t♦r t✐♣♦✲❡s♣❛ç♦✳

❈♦♥s✐❞❡r❡♠♦swn−2 =

vn−2

kvn−2k✳

❘❡❝✉rss✐✈❛♠❡♥t❡✱ ❞❡✜♥✐♠♦s✿

vn−3 = qn−3+ (wnqn−3)wn−(wn−1qn−3)wn−1−(wn−2qn−3)wn−2,

wn−3 =

vn−3

kvn−3k

, ✳✳✳

v1 = q1+ (wnq1)wn−(wn−1q1)wn−1−(wn−2q1)wn−2−...−(w2 q1)w2,

w1 =

v1

kv1k

.

❉❡st❡ ♠♦❞♦✱ {w1, ..., wn} é ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ▲♦r❡♥t③✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞♦s qi✱ ♦s ❡❧❡♠❡♥t♦s ❞❡ {wn−(m+1), ..., wn} ❡stã♦ ❡♠ V✱ ♣♦✐s sã♦ ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ❜❛s❡ {u1, ..., um} ❞❡V✱ ♣♦rt❛♥t♦✱ {wn−m, ..., wn} é ✉♠❛ ❜❛s❡ ❞❡V✳ ❙❡ ❝♦♥s✐❞❡r❛r♠♦s ❛ ♠❛tr✐③Aq✉❡ t❡♠_{w1, ..., wn}❝♦♠♦ ❝♦❧✉♥❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✺ s❡❣✉❡ q✉❡Aé ❧♦r❡♥t③✐❛♥❛ ❡A(Rm) = V✱ ♠❛✐s ❛✐♥❞❛✱ t❡♠♦s q✉❡Aé ♣♦s✐t✐✈❛✱ ❡A(en) = wn é t✐♣♦✲t❡♠♣♦ ♣♦s✐t✐✈♦ ✳

❖❜s❡r✈❛çã♦ ✶✳✷✳ ❖ ❚❡♦r❡♠❛ ✶✳✻ ♣♦❞❡ s❡r ❞❡♠♦♥str❛❞♦ t❛♠❜é♠ ♣❛r❛ ♦ s✉❜❣r✉♣♦ ❞❛s ♠❛tr✐③❡s ♥❡❣❛t✐✈❛s✳ P♦❞❡♠♦s t❛♠❜é♠✱ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ❞❡♠♦♥str❛r q✉❡ ❛ ❛çã♦ ❞♦ ❣r✉♣♦ O(n₋1,1) s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s s✉❜❡s♣❛ç♦s t✐♣♦✲❡s♣❛ç♦s ❞❡ ❞✐♠❡♥sã♦ m ❞❡ Rn é tr❛♥s✐t✐✈❛✳

❚❡♦r❡♠❛ ✶✳✼✳ ❙❡❥❛♠u✱ v ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❝♦♠ ♣❛r✐❞❛❞❡ ♣♦s✐t✐✈❛ ✭r❡s♣✳ ♥❡❣❛t✐✈❛✮ ❡♠ Rn✳ ❊♥tã♦✱ uv ≤ kukkvk ❝♦♠ ✈❛❧✐❞❛❞❡ ❞❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u ❡ v

sã♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s✳

Pr♦✈❛ ✿ ◆❡st❛ ❞❡♠♦♥str❛çã♦ ✉s❛♠♦s k._ke ❡ k.k ♣❛r❛ ❛s ♥♦r♠❛s ❡✉❝❧✐❞✐❛♥❛ ❡ ❧♦r❡♥t③✐❛♥❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✻✱ ❡①✐st❡ ✉♠ A ❡♠ P O(n₋1,1) t❛❧

q✉❡ Au = ten✳ ❈♦♠♦ A ♣r❡s❡r✈❛ ♦ ♣r♦❞✉t♦ ❧♦r❡♥t③✐❛♥♦✱ ♣♦❞❡♠♦s s✉❜st✐t✉✐r u ❡ v ♣♦r Au ❡Av✳ ❙✉♣♦♥❞♦Av= (y1, ..., yn)✉♠ ✈❡t♦r ❞❡ En1 t❡♠♦s✿

ku_k2_kv_k2 = _kAu_k2_kAv_k2

= ₋t2(_ky_k2_e−yn2)

= ₋t2_ky_k2_e+t2yn2

≤ t2_y

n2

= (AuAv)

2

= (uv)

2_{. ✭✶✳✷✮}

❉❡st❛ ❢♦r♠❛✱ ❝♦♠♦ u ❡v sã♦ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ❡A é ♣♦s✐t✐✈❛✱ s❡❣✉❡ q✉❡ Au ❡Av sã♦ ✈❡t♦r❡s ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ❡ ❛ss✐♠

uv =AuAv =₋tyn ≤0.

P♦rt❛♥t♦✱ ❝♦♠♦ ku_kkv_{k ≥} 0✱ s❡❣✉❡ q✉❡ uv ≤ kukkvk✳ ❘❡♣❛r❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡

✈❛❧❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ y = 0✱ ♦✉ s❡❥❛✱ s❡Au ❡ Av sã♦ ✈❡t♦r❡s ❝♦❧✐♥❡❛r❡s✳ ❈♦♠♦ A é ❧✐♥❡❛r ❡ ✐♥✈❡rtí✈❡❧✱ s❡❣✉❡ q✉❡u ❡ v sã♦ ❝♦❧✐♥❡❛r❡s✳

❖✉tr❛ ❞❡♠♦♥str❛çã♦ s✐♠♣❧❡s ❞❡st❡ t❡♦r❡♠❛ é ❢❡✐t❛ ✉s❛♥❞♦ ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ us♦❜r❡ ♦ ✈❡t♦r v✳ ❙✐♠✱ ♣♦✐s s❡λv é ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡us♦❜r❡ ♦ ✈❡t♦r v✱ ❡♥tã♦ λ= uv

vv

✳ ◆♦t❛♠♦s q✉❡vv <0✳ ▲♦❣♦ ♦ ✈❡t♦r z =u−λv q✉❡ é ▲♦r❡♥t③ ♦rt♦❣♦♥❛❧

❛ v é t✐♣♦✲❡s♣❛ç♦✱ ❡♥tã♦

zz ≥0⇒(u−λv)(u−λv) = (u−λv)u=uu−λvu≥0.

■st♦ ✐♠♣❧✐❝❛ q✉❡

−|uv|2

vv

≥ −(uu).

▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ♣♦r vv✱ t❡♠♦s

|uv|

2

≤[₋(uu)].[−(vv)] = (kukkvk)

2

❧♦❣♦✱

|uv| ≤ kukkvk.

❈♦♠♦ u❡ v sã♦ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❞❡ ♠❡s♠❛ ♣❛r✐❞❛❞❡ ♦ ♣r♦❞✉t♦ ❧♦r❡♥t③✐❛♥♦ ❡♥tr❡ ❡❧❡s é ♥❡❣❛t✐✈♦✱ ❧♦❣♦✱

uv ≤ kukkvk.

◆♦t❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ✈❡t♦r z é ♦ ✈❡t♦r ♥✉❧♦✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ u=λv✳

✶✳✸ Pr♦❞✉t♦ ❱❡t♦r✐❛❧ ❡♠

E

✳

◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛♠♦s ♦ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ❡♠E31 ❡ ❞❡♠♦♥str❛♠♦s ❛❧❣✉♠❛s ❞❡

s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❙❡❥❛♠ u✱v ✈❡t♦r❡s ❞❡ _E3 1 ❡

J =



 

1 0 0 0 1 0 0 0 ₋1



 .

❖ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ❧♦r❡♥t③✐❛♥♦ ❞❡ u ❡ v é ❞❡✜♥✐❞♦ ♣♦r✱

u_∧v =J(u_×v).

❖❜s❡r✈❛♠♦s q✉❡ J(u_×v) é ❡①❛t❛♠❡♥t❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ú❧t✐♠❛ ❝♦♦r❞❡♥❛❞❛ ❞❡

u_×v ♣♦r ₋1✳ ❙❡♥❞♦ ❛ss✐♠✱ ♣❛r❛ q✉❛❧q✉❡r ✈❡t♦r w ❞❡R3✱ wJ(u×v)é ♦ ♣r♦❞✉t♦

✐♥t❡r♥♦ ✉s✉❛❧ ❞❡R3 ❡♥tr❡ w❡ J(u_×v)✳ ❉✐st♦ t❡♠♦s✱

u(u∧v) = uJ(u×v) = hu, u×vi = 0,

v (u∧v) = vJ(u×v) = hv, u×vi = 0.

P♦rt❛♥t♦✱ u_∧v é ▲♦r❡♥t③ ♦rt♦❣♦♥❛❧ ❛♦s ✈❡t♦r❡s u ❡ v✳ ◆❛ ♣r♦✈❛ ❞♦ ♣ró①✐♠♦ t❡♦r❡♠❛ ✉s❛♠♦s ❛ s❡❣✉✐♥t❡ ✐❣✉❛❧❞❛❞❡✿

u_∧v =J(u)_×J(v). ✭✶✳✸✮

❱❛♠♦s ♣r♦✈á✲❧❛ ❝♦♥s✐❞❡r❛♥❞♦ u = (u1, u2, u3) ❡ v = (v1, v2, v3) ❡♠ R3 ❡ ❧❡♠❜r❛♥❞♦

q✉❡

ui uj vi vj

=₋

vi vj ui uj

=

vi −vj ui −uj

. ❊♥tã♦✱

u_∧v = J(u_×v)

= J

u2 u3

v2 v3 ,₋

u1 u3

v1 v3 ,

u1 u2

v1 v2 ! =

u2 u3

v2 v3 ,₋

u1 u3

v1 v3 ,₋

u1 u2

v1 v2 ! = ₋

v2 v3

u2 u3 ,

v1 v3

u1 u3 ,

v1 v2

u1 u2 ! =

v2 −v3

u2 −u3 ,₋

v1 −v3

u1 −u3 ,

v1 v2

u1 u2 !

= (v1, v2,−v3)×(u1, u2,−u3)

= J(v)_×J(u).

❚❡♦r❡♠❛ ✶✳✽✳ ❙❡ w, u, v, z sã♦ ✈❡t♦r❡s ❡♠ R3✱ ❡♥tã♦

(1) u_∧v = ₋v_∧u,

(2) (u_∧v)z =

u1 u2 u3

v1 v2 v3

z1 z2 z3 ,

(3) u_∧(v_∧z) = (uv)z−(zu)v,

(4) (u_∧v)(z∧w) =

uw uz

vw vz

.

Pr♦✈❛ ✿ ✭✶✮ ❉❛ ✐♥❞❡♥t✐❞❛❞❡ ✭✷✳✷✮✱

u_∧v =J(v)_×J(u) =₋J(u)_×J(v) = ₋(J(u)_×J(v)) =₋(v_∧u) =₋v_∧u.

✭✷✮

(u_∧v)z = J(u×v)z

= _hu_×v, z_i

= *

u2 u3

v2 v3 , ₋

u1 u3

v1 v3 ,

u1 u2

v1 v2 ! , z + =

u2 u3

v2 v3

z1−

u1 u3

v1 v3

z2+

u1 u2

v1 v2 z3 =

u1 u2 u3

v1 v2 v3

z1 z2 z3 .

t♦r✐❛❧ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✱ ❞❛❞❛ ♣♦ra_×(b_×c) = _ha, c_ib _{− h}a, b_ic, u_∧(v _∧z) = J(v_∧z)_×J(u)

= (J(J(v_×z))_×J(u)) = (v_×z)_×J(u) = ₋(J(u)_×(v_×z))

= ₋(_hJ(u), z_iv_{− h}J(u), v_iz) = (uv)z−(uz)v.

✭✹✮

uw uz

vw vz

= (uw)(vz)−(uz)(v w)

= (u(v z)w)₋(v(uz)w) = (u(v z)−v(uz))w

= ((u_∧v)_∧z)w

=

u_∧v z w = z w u_∧v

= (z_∧w)(u∧v).

❈♦r♦❧ár✐♦ ✶✳✹✳ ❙❡ u ❡ v sã♦ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❝♦♠ ♣❛r✐❞❛❞❡ ♣♦s✐t✐✈❛ ✭♥❡❣❛t✐✈❛✮ ❞❡ R3✱ ❡♥tã♦ u_∧v é t✐♣♦✲❡s♣❛ç♦ ❡

ku_∧v_k=_ku_kkv_ksenh(η(u, v))

✳

Pr♦✈❛ ✿ ❈♦♠♦ u❡ v sã♦ ✈❡t♦r❡s t✐♣♦✲t❡♠♣♦ ❡ u_∧v ▲♦r❡♥t③ ♦rt♦❣♦♥❛❧ ❛u ❡ ❛ v s❡❣✉❡ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✷✳✷ q✉❡ u_∧v é ✉♠ ✈❡t♦r t✐♣♦✲❡s♣❛ç♦✳ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✽ t❡♠♦s✱

ku_∧v_k2 = (u_∧v)(u∧v)

=

uv uu

v v vu

= (uv)

2

− ku_k2_kv_k2

= _ku_k2_kv_k2cosh2(η(u, v))_{− k}u_k2_kv_k2

= _ku_k2_kv_k2senh2(η(u, v)),

η(u, v)>0_⇒senh(η(u, v))>0_{⇒ k}u_kkv_ksenh(η(u, v))>0.

P♦rt❛♥t♦✱ ku_∧v_k=_ku_kkv_ksenh(η(u, v))✳

❈♦r♦❧ár✐♦ ✶✳✺✳ ❙❡ u ❡ v sã♦ ✈❡t♦r❡s t✐♣♦✲❡s♣❛ç♦ ❡♠ R3✱ ❡♥tã♦

✭✶✮ |uv|<kukkvk s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u∧v é t✐♣♦✲t❡♠♣♦✱

✭✷✮ |uv|=kukkvk s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u∧v é t✐♣♦✲❧✉③✱

✭✸✮ |uv|>kukkvk s❡✱ ❡ s♦♠❡♥t❡ s❡✱ u∧v é t✐♣♦✲❡s♣❛ç♦✳

Pr♦✈❛ ✿ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✽ ✐t❡♠ ✭✹✮✱ (u_∧v)(u∧v) = |uv|

2

− ku_k2_kv_k2✳ ❈♦♠ ❡st❛ ✐❣✉❛❧❞❛❞❡✱ ♦❜t❡♠♦s t♦❞♦s ♦s r❡s✉❧t❛❞♦s✳

✶✳✹ ●❡♦♠❡tr✐❛ ❞❡ ❈✉r✈❛s ❡♠

E

❉❡✜♥✐çã♦ ✶✳✻✳ ❯♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛♠❡tr✐③❛❞❛ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✲ ✈❡❧ ✭C∞_✮✱

α : I _→ En1✱ ❞❡ ✉♠ ❛❜❡rt♦ I = (a, b) ❞❛ r❡t❛ r❡❛❧ ❡♠ En1✳ ❖ ❝♦♥❥✉♥t♦

✐♠❛❣❡♠ ❞❡ α✱ α(I) é ❝❤❛♠❛❞♦ ❞❡ tr❛ç♦ ❞❛ ❝✉r✈❛ α✳

❊st❛ ❞❡✜♥✐çã♦ ❞❡ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ é ❡①❛t❛♠❡♥t❡ ❛ ♠❡s♠❛ ❞❛ ❣❡♦♠❡tr✐❛ ❞✐❢❡✲ r❡♥❝✐❛❧ ❞❡ ❝✉r✈❛s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✱ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ ❧❡✈❛ ❝❛❞❛ t _∈ I ❛ ✉♠ ♣♦♥t♦ α(t) = (x1(t), x2(t), ..., xn(t)) ∈ Rn✱ ♦♥❞❡ ❛s ❛♣❧✐❝❛çõ❡s xi : I → R✱ i= 1, ..., n✱ sã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s✳