Superfícies Invariantes no Espaço Homogêneo Sol com Curvatura Constante.

❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

❙✉♣❡r❢í❝✐❡s ■♥✈❛r✐❛♥t❡s ♥♦ ❊s♣❛ç♦ ❍♦♠♦❣ê♥❡♦

S

_ol

❝♦♠ ❈✉r✈❛t✉r❛ ❈♦♥st❛♥t❡✳

P♦r

●✉✐❧❤❡r♠❡ ▲✉✐③ ❞❡ ❖❧✐✈❡✐r❛ ◆❡t♦

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ P❡❞r♦ ❆♥t♦♥✐♦ ❍✐♥♦❥♦s❛ ❱❡r❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

por

Guilherme Luiz de Oliveira Neto

Dissertação apresentada ao Corpo Docente do Programa de Pós-Graduação em Matemática-CCEN-UFPB, como requisito parcial para

obtenção do título de Mestre em Matemática.

Área de Concentração: Geometria Diferencial

ntonio Hinojosa Vera rientador

Aprovada por:

r. Jobson de Queiroz Oliveira Ex~or

~

Prof. Dr. Jorg Antonio Hinojosa Vera Examinador

Universidade Federal da Paraíba Centro de Ciências Exatas e da Natureza Programa de Pós-Graduação em Matemática

Curso de Mestrado em Matemática

Julho - 2012

O48s Oliveira Neto, Guilherme Luiz de.

Superfícies invariantes no espaço homogêneo ol com curvatura constante / Guilherme Luiz de Oliveira Neto.-- João Pessoa, 2012.

68f.

Orientador: Pedro Antonio Hinojosa Vera

Dissertação (Mestrado) – UFPB/CCEN 1. Matemática. 2. Grupos de Lie. 3. Espaço ol. 4. Superfícies Invariantes. 5. Superfícies Mínimas. 6. Curvatura Média. 7. Curvatura Gaussiana. 8. Superfícies de Weingarten Linear.

Autor: Guilherme Luiz de Oliveira Neto

TItulo:

Superfícies Invariantes no Espaço

Homogêneo

Sol

com Curvatura

Constante.

Depto.: Matemática

Grau: M.Sc. Convocação: Julho Ano: 2012

Permissão está juntamente concedida pela Universidade Federal da Paraíba à circular e ser copiado para propósitos não comerciais, em sua

descrição, o título acima sob a requisição de indivíduos ou instituições.zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

❣✉r♦ ❡ ❛ ♣r✐♥❝✐♣❛❧ r❡s♣♦♥sá✈❡❧ ♣♦r ❡st❛

❝♦♥q✉✐st❛✳

P♦r ❞❡trás ❞❛s ♥♦ss❛s r❡❛❧✐③❛çõ❡s ♣❡ss♦❛✐s✱ ❛❧é♠ ❞❡ ✉♠ ❝♦♥s✐❞❡rá✈❡❧ ❡s❢♦rç♦ ♣ró✲ ♣r✐♦✱ ❡s❝♦♥❞❡✲s❡ ♥♦r♠❛❧♠❡♥t❡ ✉♠ ♥ú♠❡r♦ ♠✉✐t♦ ❣r❛♥❞❡ ❞❡ ❝♦♥tr✐❜✉✐çõ❡s✱ ❛♣♦✐♦s✱ s✉✲ ❣❡stõ❡s✱ ❝♦♠❡♥tár✐♦s ♦✉ ❝rít✐❝❛s ✈✐♥❞♦s ❞❡ ♠✉✐t❛s ♣❡ss♦❛s✳ ❆ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ❛ss✉♠❡✱ ♥♦ ❝❛s♦ ♣r❡s❡♥t❡✱ ✉♠❛ ✈❛❧✐❛ tã♦ ♣r❡❝✐♦s❛ q✉❡✱ s❡♠ ❡❧❛s✱ ❝♦♠ t♦❞❛ ❛ ❝❡rt❡③❛✱ t❡r✐❛ s✐❞♦ ♠✉✐t♦ ❞✐❢í❝✐❧ ❝❤❡❣❛r ❛ q✉❛❧q✉❡r r❡s✉❧t❛❞♦ ❞✐❣♥♦ ❞❡ ♠❡♥çã♦✳ ▼❡♥❝✐♦♥❛r ❛q✉✐ ♦ ♥♦♠❡ ❞❡ss❛s ♣❡ss♦❛s ❝♦♥st✐t✉✐ ✉♠ ♣r❡✐t♦ ❞❡ ❥✉st✐ç❛ ❡ ❞❡ ❤♦♠❡♥❛❣❡♠ s❡♥t✐❞❛ ♣♦r ♠✐♥❤❛ ♣❛rt❡✿ ❆ ❉❡✉s✱ ♣❡❧❛s ♦♣♦rt✉♥✐❞❛❞❡s q✉❡ ♠❡ ❢♦r❛♠ ❞❛❞❛s ♥❛ ✈✐❞❛✱ ♣♦r ♠❡ ❛♠♣❛r❛r ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✱ ♠❡ ❞❛r ❢♦rç❛ ✐♥t❡r✐♦r ♣❛r❛ s✉♣❡r❛r ❛s ❞✐✜❝✉❧❞❛❞❡s✱ ♠♦str❛r ♦s ❝❛♠✐♥❤♦s ♥❛s ❤♦r❛s ✐♥❝❡rt❛s ❡ ♠❡ s✉♣r✐r ❡♠ t♦❞❛s ❛s ♠✐♥❤❛s ♥❡❝❡ss✐❞❛❞❡s✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❛ q✉❛❧ ❛♠♦ ♠✉✐t♦✱ ♣❡❧♦ ❝❛r✐♥❤♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ✐♥❝❡♥t✐✈♦✳ ❊♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ♣❛✐s✱ ▲✉✐③ ❘❡❣✐♥❛❧❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❡ ▼❛r✐❧✐ ❋❛r✐❛s ❞❡ ❖❧✐✈❡✐r❛✱ s❡♠ ♦s q✉❛✐s ♥ã♦ ❡st❛r✐❛ ❛q✉✐✱ ❡ ♣♦r t❡r❡♠ ♠❡ ❢♦r♥❡❝✐❞♦ ❝♦♥❞✐çõ❡s ♣❛r❛ ♠❡ t♦r♥❛r ♦ ♣r♦✜ss✐♦♥❛❧ ❡ ❍♦♠❡♠ q✉❡ s♦✉✳

❆♦s ♠❡✉s ✐r♠ã♦s✱ ●❡r♠❛♥❛ ▲✉✐③❛ ❋❛r✐❛s ❖❧✐✈❡✐r❛ ❞❡ ▼❡✐r❛ ❡ ▲✉✐③ ●✉st❛✈♦ ❋❛r✐❛s ❞❡ ❖❧✐✈❡✐r❛✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♠♣r❡❡♥sã♦ ♥♦s ♣❡rí♦❞♦s ❞❡ ♠❡♥♦s ❛t❡♥çã♦✳

➚ ♠✐♥❤❛ ❛♠❛❞❛ ♥♦✐✈❛✱ ◆í✈❡❛ ●♦♠❡s ◆❛s❝✐♠❡♥t♦✱ ♥ã♦ ❛♣❡♥❛s ♣♦r s❡r ♦ ❛♠♦r ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ ♠❛s t❛♠❜é♠ ♣❡❧❛s ✈ár✐❛s ❞♦s❡s ❞❡ ❛♣♦✐♦ ♠♦r❛❧ q✉❡ ♠❡ t❡♠ ❞❛❞♦ ♣❛r❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ P❡❧♦ s❡✉ ✐♥❝❡♥t✐✈♦ ❡ ❡①❡♠♣❧♦ ❞❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦✳ P♦r s♦♥❤❛r ❡ ❧✉t❛r ❥✉♥t♦ ❝♦♠✐❣♦✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ P❡❞r♦ ❆✳ ❍✐♥♦❥♦s❛ ❱❡r❛✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ❡ss❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❛♣r❡♥❞❡r ❝♦♠ s❡✉s sá❜✐♦s ❝♦♥s❡❧❤♦s ❡ ❛❞✈❡rtê♥❝✐❛ s❡♠♣r❡ ❢♦r♥❡❝✐❞❛s ♥♦ ♠♦♠❡♥t♦ ❝❡rt♦✳ P♦r s✉❛ ♦r✐❡♥t❛çã♦ sér✐❛ ❡ ♠❡t✐❝✉❧♦s❛✱ ♣❡❧❛s ❝rít✐❝❛s ❝♦♥str✉t✐✈❛s✱ ❡ ❛ s✉❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❞❡ t♦❞♦s ♦s ♠♦♠❡♥t♦s✳

❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋P❇ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s q✉❡ ♠❡ ✜③❡r❛♠ s❡r ♠❡❧❤♦r ❛♦ ♥í✈❡❧ ♣❡s✲ s♦❛❧ ❡ ♣r♦✜ss✐♦♥❛❧✱ ❡ ❛ q✉❡ t✐✈❡ ♦ ♣r❛③❡r ❞❡ ❝♦♥✈✐✈❡r ❞✉r❛♥t❡ ❡ss❡ t❡♠♣♦✳ ❙♦✉ ♣r♦❢✉♥✲ ❞❛♠❡♥t❡ ❣r❛t♦ ❛♦s ♣r♦❢❡ss♦r❡s ❉r✳ ❆♥t♦♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛✱ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦✱ ❉r✳ ❉❛♥✐❡❧ ▼❛r✐♥❤♦ P❡❧❧❡❣r✐♥♦✱ ❉r❛✳ ❊❧✐s❛♥❞r❛ ❞❡ ❋át✐♠❛ ●❧♦ss ❞❡ ▼♦r❛❡s ❡ ❉r✳ ▲✐③❛♥❞r♦ ❙❛♥❝❤❡③ ❈❤❛❧❧❛♣❛ ♣❡❧❛ ❛❥✉❞❛✱ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦✳

◗✉❡r♦ ❛❣r❛❞❡❝❡r t❛♠❜é♠ ♣❡❧❛s ❣r❛♥❞❡s ❛♠✐③❛❞❡s q✉❡ ♣✉❞❡ ❝♦♥str✉✐r ❞✉r❛♥t❡ ❡ss❛ ❝♦♥q✉✐st❛✱ ❛♠✐❣♦s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡ss❡s ♠♦♠❡♥t♦s s❡♠♣r❡ ♠❡ ❛❥✉❞❛♥❞♦ ❡ ✐♥❝❡♥t✐✲ ✈❛♥❞♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❛♦ ❆✐❧t♦♥ ❘✳ ❞❡ ❆ss✐s✱ ❇r✉♥❛ ❉✳ ❙❛♥❞❡s✱ ❉❛②✈✐❞ ●❡✈❡rs♦♥ ▲✳ ▼❛rq✉❡s✱ ❉✐❡❣♦ ❋✳ ❞❡ ❙♦✉③❛✱ ❊❧✐sâ♥✐❛ ❙✳ ❞❡ ❖❧✐✈❡✐r❛✱ ❋r❛♥❝✐s❝♦ ❱✳ ❞❡ ❖❧✐✈❡✐r❛✱ ●❛❜r✐✲ ❡❧❛ ❲✳ ❙✳ ❞❛s ◆❡✈❡s✱ ●✐❧s♦♥ ▼✳ ❞❡ ❈❛r✈❛❧❤♦✱ ❏♦s❡♥✐❧❞♦ ❇✳ ❙❛♥t♦s✱ P❛♠♠❡❧❧❛ ◗✳ ❞❡ ❙♦✉③❛✱ P❛✉❧♦ ❞♦ ◆✳ ❙✐❧✈❛✱ P❡❞r♦ ❆✳ ❊✉❣ê♥✐♦✱ ❘❡❣✐♥❛❧❞♦ ❆✳ ❈✳ ❏✉♥✐♦r✱ ❘♦s✐♥â♥❣❡❧❛ ❈✳ ❞❛ ❙✐❧✈❛✱ ❨❛♥❡ ▲✐s❧❡② ❘✳ ❆r❛ú❥♦✱ ❡♥tr❡ ♦✉tr♦s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡st❛ ❝❛♠✐♥❤❛❞❛✳

❆♦ ■♥st✐t✉t♦ ❋❡❞❡r❛❧ ❞❡ ❊❞✉❝❛çã♦✱ ❈✐ê♥❝✐❛ ❡ ❚❡❝♥♦❧♦❣✐❛ ❞♦ P✐❛✉í ✲ ■❋P■✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛♦ ❘❡✐t♦r ❋r❛♥❝✐s❝♦ ❞❛s ❈❤❛❣❛s ❙❛♥t❛♥❛ ❡ ❛♦ ❉✐r❡t♦r ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❘❡❝✉rs♦s ❍✉♠❛♥♦s ❆♥tô♥✐♦ ❏♦ã♦ ❘♦❞r✐❣✉❡s✱ ♣❡❧❛ ❧✐❜❡r❛çã♦ ❝♦♥❝❡❞✐❞❛ ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❡st❛ ♣ós✲❣r❛❞✉❛çã♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛♦ ❉✐r❡t♦r ●❡r❛❧✱ Pr♦❢✳ ❉r✳ ❉❛r❧❡② ❋✐á❝r✐♦ ❞❡ ❆rr✉❞❛ ❙❛♥t✐❛❣♦✱ ❞♦ ❈❛♠♣✉s ❞❡ ❋❧♦r✐❛♥♦✱ ♣♦r t❡r ❛❝❡✐t❛❞♦ ♦ ❛❢❛st❛♠❡♥t♦ ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s✳

❆♦s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ❞♦ ■♥st✐t✉t♦ ❋❡❞❡r❛❧ ❞❡ ❊❞✉❝❛çã♦✱ ❈✐ê♥❝✐❛ ❡ ❚❡❝♥♦❧♦❣✐❛ ❞♦ P✐❛✉í✱ ❈❛♠♣✉s ❋❧♦r✐❛♥♦✿ Pr♦❢✳ ❊s♣✳ ❆♥❞ré ▲✉✐③ ❋❡rr❡✐r❛ ▼❡❧♦✱ Pr♦❢✳ ❊s♣✳ ●✐❧❞♦♥ ❈és❛r ❞❡ ❖❧✐✈❡✐r❛✱ Pr♦❢✳ ❊s♣✳ ▼❛r❝❡❧♦ ❚❡✐①❡✐r❛ ❈❛r♥❡✐r♦ ❡ ▼s✳ ❖❞✐♠ó❣❡♥❡s ❙♦❛r❡s ▲♦♣❡s ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦ ♥❛ ♠✐♥❤❛ ❛✉sê♥❝✐❛✳

❆♦s ♠❡✉s ❛♥t✐❣♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❈❛♠♣✐♥❛ ●r❛♥❞❡✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❣r❛♥❞❡ ♣r♦❢❡ss♦r ❡ ❛♠✐❣♦ ❉r✳ ❉❛♥✐❡❧ ❈♦r❞❡✐r♦ ❞❡ ▼♦r❛✐s ❋✐❧❤♦ ♣❡❧❛ ❛❥✉❞❛ ❡ ♣❛❧❛✈r❛s ❞❡ ✐♥❝❡♥t✐✈♦ ❞❛❞❛s ❞❡s❞❡ ❛ ❣r❛❞✉❛çã♦ ❛♦ ♠❡str❛❞♦✳

❊♥✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞❡ ♠❛♥❡✐r❛ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ q✉❡ ❡st❡ tr❛❜❛❧❤♦ s❡ ❝♦♥❝r❡t✐③❛ss❡✳

❆❣r❛❞❡❝✐♠❡♥t♦s ✈

❘❡s✉♠♦ ✈✐✐✐

❆❜str❛❝t ✐①

■♥tr♦❞✉çã♦ ①

✶ Pr❡❧✐♠✐♥❛r❡s ✶

✶✳✶ ●r✉♣♦s ❞❡ ▲✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✷ ❖ ❊s♣❛ç♦ ❍♦♠♦❣ê♥❡♦ S_ol ✶✻

✷✳✶ ●❡♦♠❡tr✐❛ ❞♦ ❊s♣❛ç♦ S_ol ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❙✉♣❡r❢í❝✐❡s ■♥✈❛r✐❛♥t❡s ♥♦ ❊s♣❛ç♦ S_ol ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸ ❙✉♣❡r❢í❝✐❡s ■♥✈❛r✐❛♥t❡s ♥♦ ❊s♣❛ç♦ S_ol ❝♦♠ ❈✉r✈❛t✉r❛ ❈♦♥st❛♥t❡ ✷✾ ✸✳✶ ❙✉♣❡r❢í❝✐❡s ❝♦♠ ❈✉r✈❛t✉r❛ ▼é❞✐❛ ❈♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✷ ❙✉♣❡r❢í❝✐❡s ❝♦♠ ❈✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❈♦♥st❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✹ ❙✉♣❡r❢í❝✐❡s ❞❡ ❲❡✐♥❣❛rt❡♥ ▲✐♥❡❛r ✺✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✻

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛ ✉♠ ❡st✉❞♦ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥s✲ t❛♥t❡ ❡ ❞❛s s✉♣❡r❢í❝✐❡s ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛ ❝♦♥st❛♥t❡ ♥♦ ❡s♣❛ç♦ S_ol q✉❡ sã♦ ✐♥✈❛r✐❛♥t❡s s♦❜ ❛ ❛çã♦ ❞❡ ❞♦✐s ❣r✉♣♦s ❛ ✶✲♣❛râ♠❡tr♦ ❞❡ ✐s♦♠❡tr✐❛s ❞♦ ❡s♣❛ç♦ ❛♠✲ ❜✐❡♥t❡✳ ❆❧é♠ ❞✐ss♦✱ ❝❧❛ss✐✜❝❛♠♦s ❛s s✉♣❡r❢í❝✐❡s q✉❡ s❛t✐s❢❛③❡♠ ✉♠❛ r❡❧❛çã♦ ❞♦ t✐♣♦

k1 =mk2✱ ♦♥❞❡ k1 ❡ k2 sã♦ ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❞❛ s✉♣❡r❢í❝✐❡ ❡ m ∈R✳

P❛❧❛✈r❛s✲❈❤❛✈❡✿

●r✉♣♦s ❞❡ ▲✐❡✱ ❊s♣❛ç♦ S_ol✱ ❙✉♣❡r❢í❝✐❡s ■♥✈❛r✐❛♥t❡s✱ ❙✉♣❡r❢í❝✐❡s ▼í♥✐♠❛s✱ ❈✉r✈❛✲ t✉r❛ ▼é❞✐❛✱ ❈✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛✱ ❙✉♣❡r❢í❝✐❡s ❞❡ ❲❡✐♥❣❛rt❡♥ ▲✐♥❡❛r✳

■♥ t❤✐s ♣❛♣❡r ✇❡ st✉❞✐❡❞ s✉r❢❛❝❡s ✇✐t❤ ❝♦♥st❛♥t ♠❡❛♥ ❝✉r✈❛t✉r❡ ❛♥❞ s✉r❢❛❝❡s ✇✐t❤ ❝♦♥st❛♥t ●❛✉ss✐❛♥ ❝✉r✈❛t✉r❡ ✐♥ t❤❡ S_ols♣❛❝❡ ✇❤✐❝❤ ❛r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t✇♦ ♦♥❡✲♣❛r❛♠❡t❡r s✉❜❣r♦✉♣s ♦❢ ✐s♦♠❡tr✐❡s ♦❢ t❤❡ ❛♠❜✐❡♥t s♣❛❝❡✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❝❧❛ss✐❢② t❤❡ s✉r❢❛❝❡s t❤❛t s❛t✐s❢② ❛ r❡❧❛t✐♦♥s❤✐♣ ♦❢ t②♣❡ k1 =mk2✱ ✇❤❡r❡ k1 ❛♥❞ k2 ❛r❡ t❤❡ ♣r✐♥❝✐♣❛❧ ❝✉r✈❛t✉r❡s ♦❢ t❤❡ s✉r❢❛❝❡ ❛♥❞ m _∈R✳

❑❡②✇♦r❞s✿

▲✐❡ ●r♦✉♣s✱ S_ol ❙♣❛❝❡✱ ■♥✈❛r✐❛♥t ❙✉r❢❛❝❡✱ ▼✐♥✐♠❛❧ ❙✉r❢❛❝❡✱ ▼❡❛♥ ❈✉r✈❛t✉r❡✱ ●❛✉ss✐❛♥ ❈✉r✈❛t✉r❡✱ ▲✐♥❡❛r ❲❡✐♥❣❛rt❡♥ ❙✉r❢❛❝❡✳

❆ ❣❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧ t❡♠ ❡st✉❞❛❞♦ ❝♦♥st❛♥t❡♠❡♥t❡ s✉♣❡r❢í❝✐❡s ❡♠ ❡s♣❛ç♦s ❤♦✲ ♠♦❣ê♥❡♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ s❡rã♦ ❡st✉❞❛❞❛s s✉♣❡r❢í❝✐❡s ♥♦ ❡s♣❛ç♦ ❤♦♠♦❣ê♥❡♦ S_ol ❞❡ ❞✐♠❡♥sã♦ três✳ ❖ ❡s♣❛ç♦ S_ol é ✉♠❛ ₃✲✈❛r✐❡❞❛❞❡ ❤♦♠♦❣ê♥❡❛ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛✱ ❝✉❥♦ ❣r✉♣♦ ❞❡ ✐s♦♠❡tr✐❛s t❡♠ ❞✐♠❡♥sã♦ 3 ❡ é ✉♠ ❞♦s ♦✐t♦s ♠♦❞❡❧♦s ❞❡ 3✲❣❡♦♠❡tr✐❛s ❞❡ ❚❤✉rst♦♥ ❬✶✾❪✳

❊st❡ tr❛❜❛❧❤♦ ❜❛s❡✐❛✲s❡ ♥♦ ❛rt✐❣♦ ✏■♥✈❛r✐❛♥t s✉r❢❛❝❡s ✐♥ t❤❡ ❤♦♠♦❣❡♥❡♦✉s s♣❛❝❡ S_ol ✇✐t❤ ❝♦♥st❛♥t ❝✉r✈❛t✉r❡✑ ❞❡ ❘❛❢❛❡❧ ▲ó♣❡s ❡ ▼❛r✐❛♥ ■✳ ▼✉♥t❡❛♥✉ ✭✈❡❥❛ ❬✶✷❪✮ ❡ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✳

◆♦ ❝❛♣ít✉❧♦ 1✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ❞❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s ❞❡ ▲✐❡ ❡ s✉❛s r❡❧❛çõ❡s ❝♦♠ ❛s á❧❣❡❜r❛s ❞❡ ▲✐❡ q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛✲ ❜❛❧❤♦✳ ❋✐♥❛❧✐③❛♠♦s✱ ❡st❡ ❝❛♣ít✉❧♦ ❝♦♠ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✳

◆♦ ❝❛♣ít✉❧♦ 2✱ ❡st✉❞❛♠♦s ♦ ❡s♣❛ç♦ S_ol✱ ♦ ♥♦ss♦ ❡s♣❛ç♦ ❛♠❜✐❡♥t❡✱ ❡ s✉❛ ❣❡♦♠❡tr✐❛✳ ❉❡t❡r♠✐♥❛♠♦s ♥❡st❡ ❡s♣❛ç♦✱ ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛✱ ♦s ❝♦❧❝❤❡t❡s ❞❡ ▲✐❡ ❞♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛✱ ♦s ❝❛♠♣♦s ❞❡ ❑✐❧❧✐♥❣ ❡ ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐♥❛ ∇✳

❆❧é♠ ❞✐ss♦✱ ❝❛❧❝✉❧❛♠♦s ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛ ❡ ●❛✉ss✐❛♥❛ ✭✐♥trí♥s❡❝❛ ❡ ❡①trí♥s❡❝❛✮ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ✐♥✈❛r✐❛♥t❡ ✐♠❡rs❛ ♥❡st❡ ❡s♣❛ç♦✳

◆♦ ❝❛♣ít✉❧♦ 3✱ ❡st✉❞❛♠♦s ❛s s✉♣❡r❢í❝✐❡s ✐♥✈❛r✐❛♥t❡s ♥♦ S_ol q✉❡ s❛t✐s❢❛③❡♠ ❝❡rt❛s ❝♦♥❞✐çõ❡s s♦❜r❡ s✉❛s ❝✉r✈❛t✉r❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❝✉r✈❛t✉r❛ ♠é❞✐❛ H ❡ ❝✉r✈❛t✉r❛ ●❛✉s✲

s✐❛♥❛ (Kint ❡ Kext) ❝♦♥st❛♥t❡s✳ ❈❧❛ss✐✜❝❛♠♦s t♦❞❛s ❛s s✉♣❡r❢í❝✐❡s ✐♥✈❛r✐❛♥t❡s ♥♦ Sol ❝♦♠ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ H✱ ✐♥❝❧✉✐♥❞♦ s✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ✭❛❧❣✉♠❛s ✜❣✉r❛s ❞❡

s✉♣❡r❢í❝✐❡s ❝♦♠ H ₆= 0 ❡stã♦ ❡♠ ❬✼❪✮ ❡ t♦❞❛s ❝♦♠ ❝✉r✈❛t✉r❛ ●❛✉ss✐❛♥❛✱ ✐♥trí♥s❡❝❛ ❡

s✐❞♦ r❡❝❡♥t❡♠❡♥t❡ ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ ♠✉✐t♦s ❣❡ô♠❡tr❛s✳ ❱ár✐♦s r❡s✉❧t❛❞♦s ❢♦✲ r❛♠ ♦❜t✐❞♦s ♥♦ ❣r✉♣♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ✭✈❡❥❛ ❬✶❪✱ ❬✽❪✱ ❬✾❪✱ ❬✶✹❪✱ ❬✷✵❪✮ ❡ ♥♦ ❡s♣❛ç♦ ♣r♦❞✉t♦ H2_{×R ✭✈❡❥❛ ❬✶✸❪✱ ❬✶✺❪✱ ❬✶✻❪✮✳}

❋✐♥❛❧♠❡♥t❡✱ ♥♦ ❝❛♣ít✉❧♦ 4✱ ❡st✉❞❛♠♦s ❡ ❝❧❛ss✐✜❝❛♠♦s ❛s s✉♣❡r❢í❝✐❡s ✐♥✈❛r✐❛♥t❡s ❞❡ ❲❡✐♥❣❛rt❡♥ ▲✐♥❡❛r ♥♦ S_ol✱ q✉❡ s❛t✐s❢❛③ ✉♠❛ r❡❧❛çã♦ ❞♦ t✐♣♦ _{k1 =mk}2✱ ♦♥❞❡ _k1 ❡ _k2 sã♦ ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❞❛ s✉♣❡r❢í❝✐❡ ❡ m_∈R✳

Pr❡❧✐♠✐♥❛r❡s

❆s á❧❣❡❜r❛s ❞❡ ▲✐❡ sã♦ ♦❜❥❡t♦s ❛❧❣é❜r✐❝♦s ♣♦r ❡①❝❡❧ê♥❝✐❛✱ ❡♥q✉❛♥t♦ q✉❡ ♦s ❣r✉♣♦s ❞❡ ▲✐❡ tê♠ ✉♠❛ ♥❛t✉r❡③❛ ❣❡♦♠étr✐❝❛ ❡ ❝♦♥st✐t✉❡♠ ✉♠ ❛ss✉♥t♦ ♣❛rt✐❝✉❧❛r♠❡♥t❡ r✐❝♦ ❡ ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♥❛ ▼❛t❡♠át✐❝❛ ❝♦♥t❡♠♣♦râ♥❡❛✳

■♥✐❝✐❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ ❞❡ ❣r✉✲ ♣♦s ❞❡ ▲✐❡ ❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ❛s á❧❣❡❜r❛s ❞❡ ▲✐❡✱ ❛ ✜♠ ❞❡ ❞❛r ❛♦ ❧❡✐t♦r ❛s ❢❡rr❛♠❡♥t❛s ❜ás✐❝❛s ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞♦ t❡①t♦✳ ❆ t❡♦r✐❛ ❝❧áss✐❝❛ ❞❡ ❣r✉♣♦s ❞❡ ▲✐❡ é ❡①♣♦st❛ ❝♦♠ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ♥❛s r❡❢❡rê♥❝✐❛s ❬✸❪✱ ❬✶✶❪ ❡ ❬✶✽❪✳

❋✐♥❛❧✐③❛♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❝♦♠ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ ●❡♦♠❡✲ tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ q✉❡ s❡rã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡st✉❞♦ q✉❡ s❡rá ❢❡✐t♦ ♥♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥t❡s ✭✈❡❥❛ ❬✻❪✮✳

❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ❛s ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♥s✐❞❡r❛❞❛s s❡rã♦ ❞❡ ❍❛✉s❞♦r✛ ❝♦♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧✳

✶✳✶ ●r✉♣♦s ❞❡ ▲✐❡

❉❡✜♥✐çã♦ ✶✳✶✳ ❯♠ ❣r✉♣♦ ❞❡ ▲✐❡ G é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞♦t❛❞❛ ❞❡ ✉♠❛

❡str✉t✉r❛ ❞❡ ❣r✉♣♦✱ ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ♦♣❡r❛çã♦ ∗✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ❛♣❧✐❝❛çã♦

µ:G_×G _−→ G

(x, y) _7−→ µ(x, y) = x_∗y−1

é ❞✐❢❡r❡♥❝✐á✈❡❧✱ ♦♥❞❡ y−1 ❞❡♥♦t❛ ♦ ❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❞❡ y✳

❉❡❝♦rr❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡✱ ♥✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ G✱ ♣❛r❛ ❝❛❞❛ x _∈ G

❛s ❛♣❧✐❝❛çõ❡s

Lx :G −→ G

y _7−→ x_∗y ❡

Rx :G −→ G

y _7−→ y_∗x

sã♦ ❞✐❢❡♦♠♦r✜s♠♦s ❞❡ G✳ ❊st❛s ❛♣❧✐❝❛çõ❡s sã♦ ❝❤❛♠❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ tr❛♥s❧❛✲

çã♦ à ❡sq✉❡r❞❛ ♣♦rx ❡ tr❛♥s❧❛çã♦ à ❞✐r❡✐t❛ ♣♦rx✳ ■♥❞✐❝❛r❡♠♦s ♣♦r ❡ ♦ ❡❧❡♠❡♥t♦

✐❞❡♥t✐❞❛❞❡ ❞❡ G✳

❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❣r✉♣♦s ❞❡ ▲✐❡✳

❊①❡♠♣❧♦ ✶✳✷✳ ❖ ❝♦♥❥✉♥t♦ R ❞♦s ♥ú♠❡r♦s r❡❛✐s ❝♦♠ ❛ ♦♣❡r❛çã♦ s♦♠❛ ❡ ❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ✉s✉❛❧✳

❊①❡♠♣❧♦ ✶✳✸✳ ❙❡❥❛ S1 _{= {z ∈ C; |z| = 1}✱ ♦♥❞❡ C é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s} ❝♦♠♣❧❡①♦s✳ ❈♦♥s✐❞❡r❡♠♦s ❡♠ S1 ❛ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✿ s❡ α, β _∈ S1✱

❡♥tã♦ α.β é ♦ ♣r♦❞✉t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s α ❡ β✳ ❈♦♠♦ ❛s ❛♣❧✐❝❛çõ❡s

C_×C _−→ C

(x, y) _7−→ x_·y

C_{− {0} −→} C_{− {0}}

x _7−→ x−1

sã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ s✉❛s r❡str✐çõ❡s ❛ S1 tê♠ ✐♠❛❣❡♥s ❡♠ S1✱ S1 é ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡✳

❊①❡♠♣❧♦ ✶✳✹✳ ❖ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❣r✉♣♦s ❞❡ ▲✐❡ G ❡ H é ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ G_×H✱

❝♦♠ ❛ ❡str✉t✉r❛ ❞❡ ✈❛r✐❡❞❛❞❡ ♣r♦❞✉t♦ ❡ ♣r♦❞✉t♦ ❞✐r❡t♦ ❞❡ ❣r✉♣♦s✿

(g1, h1)⊙(g2, h2) = (g1·g2, h1·h2),

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ g1✱ g2 ❡♠ G❡ h1✱ h2 ❡♠ H✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ♣❛rt✐r ❞♦s ❡①❡♠♣❧♦s ✭✶✳✷✮ ❡ ✭✶✳✸✮ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ Rn ₌ R _{× · · · ×} R ❡ ♦ t♦r♦ _n✲ ❞✐♠❡♥s✐♦♥❛❧ Tn =S1×S1×...×S1 sã♦ ❣r✉♣♦s ❞❡ ▲✐❡✳

❊①❡♠♣❧♦ ✶✳✺✳ ❆ ✈❛r✐❡❞❛❞❡GL(n,R₎❞❛s ♠❛tr✐③❡s r❡❛✐s _n×n✐♥✈❡rtí✈❡✐s✱ ♠✉♥✐❞♦ ❝♦♠ ❛ ♦♣❡r❛çã♦ ✉s✉❛❧ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s ❡ ❝♦♠ ❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ✉s✉❛❧ ❞♦ Rn2 é ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡✳ ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ ❛s ❢✉♥çõ❡s ❛❜❛✐①♦ sã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s

f : GL(n,R_)×GL(n,R_)−→GL(n,R₎ ❞❛❞❛ ♣♦r _{f(A, B) = AB} ❡

g : GL(n,R_)−→GL(n,R₎ ❞❛❞❛ ♣♦r _{g(A) =A}−1_.

❆ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞❡ f ❞❡❝♦rr❡ ❞❛ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ R✱ ❥á ❛ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞❡ g ❞❡❝♦rr❡ ❞❛ r❡❣r❛ ❞❡ ❈r❛♠❡r ♣❛r❛ ❛ ✐♥✈❡rs❛ ❞❡ ✉♠❛ ♠❛tr✐③✳ ❉❡

❢♦r♠❛ ❛♥á❧♦❣❛ ♣♦❞❡✲s❡ ♠♦str❛r q✉❡ GL(n,C₎ ❛❞♠✐t❡ ❛ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦ ❞❡ ▲✐❡✳ ❖s ❣r✉♣♦s GL(n,R₎ ❡ _GL(n,C₎ sã♦ ❝❤❛♠❛❞♦s ❣r✉♣♦s ❧✐♥❡❛r❡s✳

❖s ❣r✉♣♦s ❧✐♥❡❛r❡s ❝♦♥té♠ ♦s s❡❣✉✐♥t❡s s✉❜❣r✉♣♦s✿

U(n) = _{A _∈GL(n,C_{) : AA}∗ _=I} ✭❣r✉♣♦ ✉♥✐tár✐♦✮

SL(n,C_{) = {A ∈GL(n,}C_{) : detA= 1}} ✭❣r✉♣♦ ❧✐♥❡❛r ❡s♣❡❝✐❛❧✮

O(n,C_{) = {A ∈GL(n,}C_{) : AA}t_=I} ✭❣r✉♣♦ ♦rt♦❣♦♥❛❧ ❝♦♠♣❧❡①♦✮

SU(n) = _{B _∈U(n) : detB = 1_} ✭❣r✉♣♦ ✉♥✐tár✐♦ ❡s♣❡❝✐❛❧✮

SL(n,R_{) = {A ∈GL(n,}R_{) : detA= 1}} ✭❣r✉♣♦ ❧✐♥❡❛r ❡s♣❡❝✐❛❧ r❡❛❧✮

O(n) = {A _∈GL(n,R_{) : AA}t_=I} ✭❣r✉♣♦ ♦rt♦❣♦♥❛❧ r❡❛❧✮

SO(n) = _{B _∈O(n) : detB = 1_} ✭❣r✉♣♦ ♦rt♦❣♦♥❛❧ ❡s♣❡❝✐❛❧✮,

♦♥❞❡ A∗ ❡At ✐♥❞✐❝❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ♠❛tr✐③ ❛❞❥✉♥t❛ ❡ ❛ ♠❛tr✐③ tr❛♥s♣♦st❛ ❞❡ A✳

❱❛♠♦s ♠♦str❛r q✉❡ O(n) é ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡✳ P❛r❛ ✐ss♦ ✈❛♠♦s ♠♦str❛r ♣r✐♠❡✐r♦ q✉❡

O(n) é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ GL(n,R₎✳ ❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦

f : M(n,R_{) −→ s(n,}R_{) := {A∈M(n,}R_{) : A=A}t_}

A _7−→ AAt .

❊st❛ ❛♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣♦✐s✱ ❞❛❞♦ A_∈M(n,R₎✱ t❡♠♦s

(AAt)t= (At)tAt=AAt,

♦✉ s❡❥❛✱ AAt _∈s(n,R₎✳ ❆❧é♠ ❞✐ss♦✱ _f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡

f−1(I) = {A ∈M(n,R_{) : AA}t_=I}=O(n).

❆ss✐♠✱ ♣❛r❛ ✈❡r q✉❡ O(n) é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ GL(n,R₎ ❜❛st❛ ♠♦str❛r q✉❡ _I é ✈❛❧♦r r❡❣✉❧❛r ❞❡ f✳ ❙❡ X, Y _∈M(n,R_)≈Rn2✱ t❡♠♦s q✉❡

dfX(Y) = lim

r→0

f(X+rY)₋f(X)

r

= lim

r→0

f(X+rY)(X+rY)t_−XXt

r

= lim

r→0

rXYt_{+rY X}t_+r2_{Y Y}t

r

= XYt+Y Xt.

❙❡ X _∈f−1_{(I) ❡ S ∈s(n,R)✱ ❡♥tã♦ t♦♠❛♥❞♦ Y =} SX

2 ∈M(n,R)✱ t❡♠♦s q✉❡

dfX(Y) = X

SX 2 t + SX 2

Xt = XX

t_St

2 + SXXt 2 = St 2 + S

2 =S,

♦✉ s❡❥❛✱ dfX é s♦❜r❡❥❡t♦r❛ ♣❛r❛ t♦❞♦ X ∈ f−1(I)✳ ▲♦❣♦✱ I é ✈❛❧♦r r❡❣✉❧❛r ❞❡ f✳

P♦rt❛♥t♦✱ O(n) é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ GL(n,R₎✳ ❆❣♦r❛ t♦♠❡♠♦s ❛s ❛♣❧✐❝❛çõ❡s

ϕ : O(n)_×O(n) _−→ O(n)

(A, B) 7−→ AB

❡

ψ : O(n) _−→ O(n)

A _7−→ A−1 .

❈♦♠♦ ❡ss❛s ❛♣❧✐❝❛çõ❡s sã♦ t❛♠❜é♠ ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♥❝❧✉í♠♦s q✉❡ O(n) é ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡✳

❉❡✜♥✐r❡♠♦s ❛❣♦r❛ ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛ ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ G✳ ▼❛✐s

❛❞✐❛♥t❡ ♠♦str❛r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ss❡s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s é ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❛ss♦❝✐❛❞❛ ❛♦ ❣r✉♣♦ ❞❡ ▲✐❡ G✳

❉❡✜♥✐çã♦ ✶✳✻✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❝❛♠♣♦ X ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ G é ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛ q✉❛♥❞♦ Xxy =dLx(Xy)✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x, y ∈G✳

❆♥❛❧♦❣❛♠❡♥t❡✱ X é ✐♥✈❛r✐❛♥t❡ à ❞✐r❡✐t❛ q✉❛♥❞♦ dRx(Xy) =Xyx✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠

x, y _∈G✳

❖ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛ ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ G s❡rá ❞❡♥♦✲

t❛❞♦ ♣♦r LG✳ ❯♠ ❝❛♠♣♦ X ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ à ❞✐r❡✐t❛✮ ✜❝❛

❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦ q✉❛♥❞♦ s❡ ❝♦♥❤❡❝❡ Xe✱ ♦✉ s❡❥❛✱ ♦ ✈❛❧♦r ❞♦ ❝❛♠♣♦ ♥❛

✐❞❡♥t✐❞❛❞❡ e✱ ♣♦✐s Xx = dLx(Xe)✳ ◆♦t❡ t❛♠❜é♠ q✉❡ LG é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ♣♦✐s

❞❛❞♦s X, Y _∈LG❡ α_∈R✱ t❡♠✲s❡

(X+αY)xy = Xxy +αYxy

= dLx(Xy) +αdLx(Yy)

= dLx(Xy+αYy)

= dLx(X+αY)y.

❉♦♥❞❡ X+αY _∈LG✳

Pr♦♣♦s✐çã♦ ✶✳✼✳ ❆ ❛♣❧✐❝❛çã♦

α:LG _−→ TeG

X _7−→ α(X) = Xe,

♦♥❞❡ TxG ✐♥❞✐❝❛ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ G ♥♦ ♣♦♥t♦ x✱ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❡s♣❛ç♦s

✈❡t♦r✐❛✐s✳

❉❡♠♦♥str❛çã♦✿ ➱ ❝❧❛r♦ q✉❡ α é ❧✐♥❡❛r✳ ❉❡ ❢❛t♦✱ ❞❛❞♦s X, Y _∈LG ❡λ_∈R✱ t❡♠✲s❡

α(X+λY) = (X+λY)e=Xe+λYe =α(X) +λα(Y).

❆❣♦r❛✱ ♠♦str❡♠♦s q✉❡ α é s♦❜r❡❥❡t♦r❛✳ ❉❛❞♦ Z _∈ TeG✱ ❞❡✜♥❛ ✉♠ ❝❛♠♣♦ X ❡♠ G

♣♦r Xx =dLx(Z)✳ ❚❡♠♦s✱

Xxy =dLxy(Z) =dLx◦dLy(Z) =dLx(Xy).

P♦rt❛♥t♦✱ X _∈LG✳ ❆❧é♠ ❞✐ss♦✱

α(X) =Xe=dLe(Z) =I(Z) =Z.

❋✐♥❛❧♠❡♥t❡✱ α é ✐♥❥❡t♦r❛✱ ♣♦✐s s❡

α(X) =α(Y),

t❡♠♦s q✉❡

Xe =Ye.

❊✱ ❞❛❞♦ x_∈G✱ t❡♠♦s

Xx =dLx(Xe) = dLx(Ye) = Yx.

▲♦❣♦✱ X =Y✳

Pr♦♣♦s✐çã♦ ✶✳✽✳ ❙❡ X é ✉♠ ❝❛♠♣♦ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛ ❡♠ G✱ ❡♥tã♦ X é ❞✐❢❡r❡♥✲

❝✐á✈❡❧✳

❉❡♠♦♥str❛çã♦✿ P❛r❛ ♠♦str❛r q✉❡ X é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ x _∈ G✱ ❜❛st❛ ❢❛③❡r ❛ ❞❡✲

♠♦♥str❛çã♦ ♣❛r❛x❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❝♦♦r❞❡♥❛❞❛ ❞❡e✱ ♣♦✐sLx−1 é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦

❞❡ ❝❧❛ss❡C∞✳ ❙❡❥❛θ:U _→Rn✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ _e✱ ❝♦♠_{θ = (x}1_{, ..., x}n₎✱

xi :U _→R❡ _x∈U✳ ❈♦♠♦ ❛s ♦♣❡r❛çõ❡s ❡♠ _G sã♦ ❝♦♥tí♥✉❛s ♣♦❞❡♠♦s t♦♠❛r _{V ⊂U} ✈✐③✐♥❤❛♥ç❛ ❞❡ e_∈Gt❛❧ q✉❡ Lx(V)⊂U✳ ❊♥tã♦✱ t❡♠♦s q✉❡

Xx(xi) = (dLx.Xe)(xi) = Xe.(xi◦Lx).

❆❣♦r❛✱ ❡s❝r❡✈❡♥❞♦

Xe =

X

j

cj

∂ ∂xj (e),

♦♥❞❡ cj sã♦ ❝♦♥st❛♥t❡s✱ t❡♠♦s

Xx xi

=X

j

cj

∂(xi_◦Lx)

∂xj (e).

❙❡❥❛ ❛❣♦r❛ fi : V _×V _→ R ❞❡✜♥✐❞❛ ♣♦r _fi_{(x, y) = x}i_{(x, y)}✱ ♦✉ s❡❥❛✱ _fi_{(x, y)} é ❛

i₋és✐♠❛ ❝♦♦r❞❡♥❛❞❛ ❞♦ ♣r♦❞✉t♦ xy=Lx(y)✳ ❊♥tã♦

Xx(xi) =

X

j

cj

∂(xi_◦Lx) (e)

∂xj =

X

j

cj

∂xi(x, e)

∂xj =

X

j

cj

∂fi(x, e)

∂xj .

❈♦♠♦ ❛s fi sã♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ x✱ X(xi) é ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ x✳

P♦rt❛♥t♦✱ X é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ x_∈V✳

❉❡✜♥✐çã♦ ✶✳✾✳ ❯♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ s♦❜r❡ R é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ _g✱ ♠✉♥✐❞♦ ❞❡ ✉♠❛ ♦♣❡r❛çã♦ ❜✐❧✐♥❡❛r [·,_·] :g_×g_−→g✱ ❞❡♥♦♠✐♥❛❞❛ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✱ s❛t✐s❢❛③❡♥❞♦ ❛s

s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✳ [X, Y] =−[Y, X] ✭❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡✮

✷✳ [[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0 ✭✐❞❡♥t✐❞❛❞❡ ❞❡ ❏❛❝♦❜✐✮

♣❛r❛ t♦❞♦ X, Y ❡ Z ♣❡rt❡♥❝❡♥t❡s ❛ g✳

❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ á❧❣❡❜r❛s ❞❡ ▲✐❡✳

❊①❡♠♣❧♦ ✶✳✶✵✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ M(n,R₎ ❞❛s ♠❛tr✐③❡s q✉❛❞r❛❞❛s r❡❛✐s ❝♦♠ ♦ ❝♦❧✲ ❝❤❡t❡ ❞❡✜♥✐❞♦ ♣♦r

[A, B] =AB₋BA,

♦♥❞❡ AB ✐♥❞✐❝❛ ♦ ♣r♦❞✉t♦ ✉s✉❛❧ ❞❡ ♠❛tr✐③❡s✳

❊①❡♠♣❧♦ ✶✳✶✶✳ R3 ❝♦♠ ♦ ❝♦❧❝❤❡t❡ ❞❛❞♦ ♣♦r [x, y] =x_∧y,

♦♥❞❡ ∧ ✐♥❞✐❝❛ ♦ ♣r♦❞✉t♦ ✈❡t♦r✐❛❧ ✉s✉❛❧ ❞❡ R3✳

❊①❡♠♣❧♦ ✶✳✶✷✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❉❡♥♦t❡♠♦s ♣♦r X(M) ♦ ❝♦♥✲ ❥✉♥t♦ ❞♦s ❝❛♠♣♦s C∞ _{t❛♥❣❡♥t❡s ❛ M✳ X(M) é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠ ❛s ♦♣❡r❛çõ❡s}

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

(X+Y)x :=Xx+Yx ❡ (λX)x :=λXx.

P❛r❛ X, Y _∈X(M)✱ f :M _−→R ❞❡ ❝❧❛ss❡ _C∞ ❡ _x∈M✱ ❞❡✜♥✐♠♦s ♦ ❝♦❧❝❤❡t❡ _{[X, Y]} ❝♦♠♦ ♦ ❝❛♠♣♦ ❡♠ X(M) t❛❧ q✉❡

[X, Y]x(f) =Xx(Y f)−Yx(Xf),

♦♥❞❡ Xf é ❛ ❛♣❧✐❝❛çã♦ ❞❛❞❛ ♣♦r✿

Xf :M _−→ R

x 7−→ (Xf)(x) = df(x)Xx.

❆❧é♠ ❞✐ss♦✱ s❡ X _∈ X(M) ❡ f _∈ C∞(M)✱ ❡♥tã♦ f X _∈ X(M)✳ ❈♦♠ ❡st❛ ♦♣❡r❛çã♦

X(M) é ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡✳ ▼❛✐s ❞❡t❛❧❤❡s ❡♥❝♦♥tr❛♠✲s❡ ❡♠ [✷]✳

❉❡✜♥✐çã♦ ✶✳✶✸✳ ❙❡❥❛♠M ❡N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ ϕ:M _→N ✉♠❛ ❛♣❧✐❝❛çã♦

❞❡ ❝❧❛ss❡ C∞_{✳ ❉✐③❡♠♦s q✉❡ ♦s ❝❛♠♣♦s X ∈ X(M✮ ❡ Y ∈X(N) sã♦ ϕ✲r❡❧❛❝✐♦♥❛❞♦s✱}

s❡ dϕ_◦X =Y _◦ϕ✱ ♦✉ s❡❥❛✱ s❡ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❝♦♠✉t❛

M ϕ //

X

N

Y

T M

dϕ

/

/_{T N.}

Pr♦♣♦s✐çã♦ ✶✳✶✹✳ ❙❡❥❛ ϕ : M _→ N ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❝❧❛ss❡ C∞_{✱ ♦♥❞❡ M, N sã♦}

✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❙❡ X, X1 ∈ X(M✮ sã♦ ϕ✲r❡❧❛❝✐♦♥❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♠ Y, Y1 ∈X(N)✱ ❡♥tã♦ [X, X1] é ϕ✲r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ [Y, Y1]✳

❉❡♠♦♥str❛çã♦✿ ▼♦str❡♠♦s q✉❡ ♣❛r❛ ❝❛❞❛ p _∈M ❡ ♣❛r❛ ❝❛❞❛ f _∈C∞(M) ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡

dϕ[X, X1]p(f) = [Y, Y1]ϕ(p)(f).

❉❡ ❢❛t♦✱

dϕ[X, X1]p(f) = [X, X1]p(f◦ϕ)

= Xp(X1(f◦ϕ))−(X1)_p(X(f ◦ϕ))

= Xp(dϕ◦X1) (f)−(X1)_p(dϕ◦X) (f)

= Xp(Y1◦ϕ) (f)−(X1)_p(Y ◦ϕ) (f)

= Xp(Y1(f)◦ϕ)−(X1)p(Y (f)◦ϕ)

= dϕ(Xp) (Y1(f))−dϕ(X1)p(Y (f))

= Yϕ(p)(Y1(f))−(Y1)ϕ(p)(Y (f))

= [Y, Y1]ϕ(p)(f).

❈♦r♦❧ár✐♦ ✶✳✶✺✳ ❙❡ X, Y _∈LG✱ ❡♥tã♦ [X, Y]_∈LG✳

❉❡♠♦♥str❛çã♦✿ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ dLx[X, Y]y = [X, Y]xy✳ ❙❡X ∈LG ❡x∈G✱

❡♥tã♦ X é ϕ✲r❡❧❛❝✐♦♥❞❛❞♦ ❝♦♥s✐❣♦ ♠❡s♠♦✳ ❉❡ ❢❛t♦✱

(dLx ◦X) (y) = dLx(X (y)) =dLx(Xy) =X_xy

(X _◦Lx) (y) = X (Lx(y)) = X(xy) = Xxy

▲♦❣♦✱ t❡♠♦s q✉❡

dLx◦X =X◦Lx.

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ t❡♠♦s q✉❡

dLx◦Y =Y ◦Lx

❡ ♣♦rt❛♥t♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✭✶✳✶✹✮✱ t❡♠♦s q✉❡[X, Y]éLx✲r❡❧❛❝✐♦♥❛♥❞♦ ❝♦♥s✐❣♦ ♠❡s♠♦✱

♦✉ s❡❥❛✱

dLx◦[X, Y](y) = [X, Y]◦Lx(y).

■st♦ ✐♠♣❧✐❝❛ q✉❡

dLx[X, Y]y = [X, Y](xy) = [X, Y]xy.

P♦rt❛♥t♦✱ [X, Y]_∈LG✳

❊①❡♠♣❧♦ ✶✳✶✻✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ ❡ LG ♦ ❡s♣❛ç♦ ❞♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à

❡sq✉❡r❞❛✳ LG é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✭✶✳✶✺✮ é ❢❡❝❤❛❞♦ ❡♠ r❡❧❛çã♦ ❛

♦♣❡r❛çã♦ ❝♦❧❝❤❡t❡ ❞❡ ❝❛♠♣♦s ❞❡✜♥✐❞❛ ♥♦ ❡①❡♠♣❧♦ ✭✶✳✶✷✮✳ ❆ss✐♠✱ LG é ✉♠❛ á❧❣❡❜r❛

❞❡ ▲✐❡✳

❏á ✈✐♠♦s ♥❛ Pr♦♣♦s✐çã♦ ✭✶✳✼✮ q✉❡ LG❡TeGsã♦ ✐s♦♠♦r❢♦s ❝♦♠♦ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳

❆ss✐♠✱ ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r ❡♠ TeG ✉♠❛ ❡str✉t✉r❛ ❞❡ á❧❣❡❜r❛ ❞❡ ▲✐❡ ♣❛ss❛♥❞♦ ♦

❝♦❧❝❤❡t❡ ❞❡ ❝❛♠♣♦s ❡♠ LG ♣❛r❛ TeG✳

❉❡✜♥✐çã♦ ✶✳✶✼✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡✳ ❉❡✜♥✐♠♦s ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❞❡ G ❝♦♠♦

s❡♥❞♦ ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t❛♥❣❡♥t❡ ❛ G♥♦ ♣♦♥t♦ e✱ TeG✱ ♦♥❞❡ ❡ é ♦ ❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡

❞❡ G✳

❆ss✐♠✱ ♣❛r❛ Vb✱ cW _∈TeG✱ ❞❡✜♥✐♠♦s [V ,b cW] := [V, W]e✱ ♦♥❞❡ V✱W ∈LG sã♦ t❛✐s

q✉❡

Vx =dLxVb e Wx =dLxW .c

❉❡♥♦t❛r❡♠♦s ♣♦r Gb ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❞♦ ❣r✉♣♦ ❞❡ ▲✐❡ G✳

❉❡✜♥✐çã♦ ✶✳✶✽✳ ❯♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ♥✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ G é ✐♥✈❛r✐❛♥t❡ à

❡sq✉❡r❞❛ s❡ ❛s tr❛♥s❧❛çõ❡s à ❡sq✉❡r❞❛ sã♦ ✐s♦♠❡tr✐❛s✱ ♦✉ s❡❥❛✱

hu, v_iy =hd(Lx)yu, d(Lx)yviLx(y), ∀x, y ∈G, ∀u, v ∈TyG.

❆♥❛❧♦❣❛♠❡♥t❡ ❞❡✜♥❡✲s❡ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ à ❞✐r❡✐t❛✳

❯♠❛ ♠étr✐❝❛ q✉❡ é ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛ ❡ à ❞✐r❡✐t❛ ❞✐③✲s❡ ❜✐✲✐♥✈❛r✐❛♥t❡✳

P❛r❛ ✐♥tr♦❞✉③✐r ✉♠❛ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛ ❡♠ G ♣♦❞❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱

t♦♠❛r ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ q✉❛❧q✉❡r ❡♠ TeG:=Gb ❡ ❞❡✜♥✐r

hu, v_ix =hd(Lx−1)_yu, d(L_x−1)_yvi_Lx(y), ∀x∈G, ∀u, v ∈T_xG.

■st♦ ❞❡✜♥❡✱ ❞❡ ❢❛t♦✱ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ G✱ ♣♦✐s Lx ❞❡♣❡♥❞❡ ❞✐❢❡r❡♥❝✐❛✲

✈❡❧♠❡♥t❡ ❞❡ x✱ ❡✱ é ❝❧❛r♦ q✉❡✱ t❛❧ ♠étr✐❝❛ s❡rá ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛✳

❯♠❛ ♠étr✐❝❛ ❤♦♠♦❣ê♥❡❛ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ M é ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ t❛❧

q✉❡ ❞❛❞♦s ❞♦✐s ♣♦♥t♦s x, y _∈ M ❡①✐st❡ ✉♠❛ ✐s♦♠❡tr✐❛ ❞❡ M q✉❡ ❧❡✈❛ x ❡♠ y✳ ❈♦♠

t❛❧ ♠étr✐❝❛✱ M é ❞✐t❛ ❤♦♠♦❣ê♥❡❛✳

❯♠ ❣r✉♣♦ ❞❡ ▲✐❡ ❝♦♠ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛ é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❤♦♠♦❣ê♥❡❛✱ ♥♦ s❡♥t✐❞♦ q✉❡✿ ❞❛❞♦s x, y _∈G❡①✐st❡ ✉♠❛ ✐s♦♠❡tr✐❛ ❞❡ G q✉❡ ❧❡✈❛ x ❡♠ y✱ ❛ s❛❜❡r

Lyx−1 : G → G

x _7→ Lyx−1(x) =yx−1x=y

.

❆ss✐♠ Gé t❛♠❜é♠ ❝♦♠♣❧❡t♦ ❝♦♠♦ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ♣♦✐s q✉❛❧q✉❡r ✈❛r✐❡❞❛❞❡

❤♦♠♦❣ê♥❡❛ é ❝♦♠♣❧❡t❛✳

❙❡❥❛ G✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ ❝♦♠ ♠étr✐❝❛ _h·,_·i ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛✳ ▲❡♠❜r❡♠♦s q✉❡

❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ❛ss♦❝✐❛❞❛ ∇ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ❝♦♥❞✐çã♦ ❞❡ s✐♠❡tr✐❛

∇XY − ∇YX = [X, Y] ✭✶✳✶✮

❡ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ ✭❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❝♦♠ ❛ ♠étr✐❝❛✮

X_hY, Z_i=h∇XY, Zi+hY,∇XZi, ✭✶✳✷✮

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ❝❛♠♣♦sX, Y, Z✳ ❆ ♣❛rt✐r ❞❡ ✭✶✳✶✮ ❡ ✭✶✳✷✮✱ ♣❡r♠✉t❛♥❞♦ X, Y, Z

♦❜té♠✲s❡ ❛ ❢ór♠✉❧❛✿

2_h∇XY, Zi=h[X, Y], Zi − h[Y, Z], Xi+h[Z, X], Yi+XhY, Zi+YhZ, Xi −ZhX, Yi

❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢ór♠✉❧❛ ❞❡ ❑♦s③✉❧✳ ❆❣♦r❛✱ s❡ X, Y _∈ LG ❡♥tã♦ _hX, Y_i é ❝♦♥st❛♥t❡✱

♣♦✐s

hXx, Yxi=h(dLx)_e,(dLx)_eYei=hXe, Yei. ✭✶✳✸✮

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♥♦ ❝❛s♦ ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡✱ ❛ ❢ór♠✉❧❛ ❞❡ ❑♦s③✉❧ r❡❞✉③✲s❡ ❛ 2_h∇XY, Zi=h[X, Y], Zi − h[Y, Z], Xi+h[Z, X], Yi. ✭✶✳✹✮

❆ss✐♠✱ s❡ x_∈Gé ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❡ G✱ ❡♥tã♦ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✭✶✳✶✺✮✱ t❡♠♦s q✉❡

2_h∇XY, Zi(x) = h[X, Y], Zi(x)− h[Y, Z], Xi(x) +h[Z, X], Yi(x)

= _h[X, Y]_e, Zei − h[Y, Z]_e, Xei+h[Z, X]_e, Yei

= 2_h(_∇XY)e, Zei.

P♦r ♦✉tr♦ ❧❛❞♦✱

hd(Lx)e(∇XY)e, Zxi = hd(Lx)e(∇XY)e, d(Lx)eZxi

= _h(_∇XY)e, Zei.

P♦rt❛♥t♦✱

h(_∇XY)x, Zxi=hd(Lx)e(∇XY)e, Zxi, ∀Z ∈LG,

♦✉ s❡❥❛✱ s❡ X, Y _∈LG✱ ❡♥tã♦_∇XY ∈LG✳ ❆ss✐♠✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ X ∈LG❞❡✜♥❡ ✉♠❛

tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❛♥t✐ss✐♠étr✐❝❛

∇XY : LG −→ LG

Y 7−→ ∇XY

.

❆ ❛♥t✐ss✐♠❡tr✐❛ é ❝♦♥s❡q✉ê♥❝✐❛ ❞✐r❡t❛ ❞❛ s✐♠❡tr✐❛ ❞❛ ❝♦♥❡①ã♦ ❡ ❞♦ ❢❛t♦ ❞❡ hX, Y_i

s❡r ❝♦♥st❛♥t❡✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ hY, Z_i é ❝♦♥st❛♥t❡✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ X ❞❡ LG t❡♠♦s X_hY, Z_i= 0 ❡ ❧♦❣♦_h∇XY, Zi=−hY,∇XZi✳

◆♦ q✉❡ s❡❣✉❡✱ M ❞❡♥♦t❛rá ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❡ ✏s✉❛✈❡✑ ✐♥❞✐❝❛rá ❛ ❝❧❛ss❡

❞❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ C∞_✳

❉❡✜♥✐çã♦ ✶✳✶✾✳ ❙❡❥❛ V ✉♠ ❝❛♠♣♦ s✉❛✈❡ ❡♠ M✳ ❉✐③❡♠♦s q✉❡ V é ✉♠ ❝❛♠♣♦ ❞❡

❑✐❧❧✐♥❣ ❡♠ M s❡ V s❛t✐s❢❛③

h∇XV, Yi+hX,∇YVi= 0,

♣❛r❛ t♦❞♦ X, Y ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s s✉❛✈❡s ❡♠ M✳

❉❡s❡❥❛♠♦s r❡❧❛❝✐♦♥❛r ♦s ❝❛♠♣♦s ❞❡ ❑✐❧❧✐♥❣ ❝♦♠ ♦ ❣r✉♣♦ ❞❛s ✐s♦♠❡tr✐❛s ❞❛ ✈❛r✐❡✲ ❞❛❞❡✳ ❯♠ r❡s✉❧t❛❞♦ q✉❡ ♥❡❝❡ss✐t❛♠♦s ♥❡st❛ ❞✐r❡çã♦✱ ❝✉❥❛ ♣r♦✈❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✹❪✱ ♣✳ 63✱ é ♦ q✉❡ s❡❣✉❡✿

❚❡♦r❡♠❛ ✶✳✷✵✳ ❙❡❥❛ X ✉♠ ❝❛♠♣♦ s✉❛✈❡ ❡♠ ✉♠ ❛❜❡rt♦ W ❞❡ M ❡ s❡❥❛ p ∈ W✳

❊♥tã♦ ❡①✐st❡♠ ✉♠ ❛❜❡rt♦ U _⊂ W✱ p _∈ U✱ ✉♠ ♥ú♠❡r♦ ε > 0 ❡ ✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡

ϕ: (₋ε, ε)_×U _−→W t❛✐s q✉❡ ❛ ❝✉r✈❛ t_7−→ϕ(t, q)✱ t_∈(₋ε, ε)✱ é ❛ ú♥✐❝❛ tr❛❥❡tór✐❛ ❞❡ X q✉❡ ♥♦ ✐♥st❛♥t❡ t= 0 ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ q✱ ♣❛r❛ ❝❛❞❛ q_∈U✱ ✐st♦ é✱ ϕ(0, q) =q ❡

d

dtϕ(t, q) = X(ϕ(t, q))

♣❛r❛ t♦❞♦ t_∈(₋ε, ε)✳

❆ ❛♣❧✐❝❛çã♦ ϕt :U −→ W ❞❛❞♦ ♣♦r ϕt(q) =ϕ(t, q) é ❝❤❛♠❛❞❛ ♦ ✢✉①♦ ❞❡ X ❡♠

W✳

❖❜s❡r✈❛♠♦s ♥♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r q✉❡✱ ✜①❛❞♦ t✱ ❝♦♠ _|t_|< ε✱ϕt:U −→ϕt(U)⊂M

❞❛❞♦ ♣♦r ϕt(q) = ϕ(t, q) ❞❡✜♥❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ U ❡♠ ϕt(U) ❡ ϕt◦ϕs = ϕt+s

✈❛❧❡ ♦♥❞❡ ❛♠❜♦s ♦s ❧❛❞♦s ❡stã♦ ❞❡✜♥✐❞♦s✳ ❚❡♥❞♦ ✐st♦ ❡♠ ✈✐st❛✱ ♦❜s❡r✈❛♠♦s q✉❡ ✉♠ ❝❛♠♣♦ X ❣❡r❛ ✉♠ ❣r✉♣♦Ge _={ϕt}❝❤❛♠❛❞♦ ❞❡ s✉❜❣r✉♣♦ ✭❧♦❝❛❧✮ ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s ❧♦❝❛✐s✳

❖ ❝♦♥❥✉♥t♦ ❞❛s ✐s♦♠❡tr✐❛s ❞❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ M ❢♦r♠❛ ✉♠ s✉❜❣r✉♣♦ ❞♦

❣r✉♣♦ ❞♦s ❞✐❢❡♦♠♦r✜s♠♦s ❞❡ M✳ ❆ss✐♠✱ s❡ ✉♠ ❝❛♠♣♦ X ❡♠ M ❣❡r❛ ✉♠❛ ❢❛♠í❧✐❛

❛ ✉♠ ♣❛râ♠❡tr♦ ❝♦♥st✐t✉í❞❛ ❞❡ ✐s♦♠❡tr✐❛s✱ ❞✐③❡♠♦s q✉❡ ❡❧❛ ❣❡r❛ ✉♠ s✉❜❣r✉♣♦ ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ✐s♦♠❡tr✐❛s✳

Pr♦♣♦s✐çã♦ ✶✳✷✶✳ ❙❡❥❛ X ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s s✉❛✈❡ ❞❡ M✳ ❊♥tã♦ X é ✉♠ ❝❛♠♣♦

❞❡ ❑✐❧❧✐♥❣✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ X ❣❡r❛ ✉♠ s✉❜❣r✉♣♦ ✭❧♦❝❛❧✮ ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ✐s♦♠❡tr✐❛s

❧♦❝❛✐s ❞❡ M✳

❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✶✵❪✱ ♣á❣✳ 48✳

❈♦♥s✐❞❡r❛♥❞♦ G_={φr}r∈R ✉♠ s✉❜❣r✉♣♦ ✭❧♦❝❛❧✮ ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ✐s♦♠❡tr✐❛s ❞❡

M✱ s❡❣✉❡ ❞♦ ❡①♣♦st♦ ❛❝✐♠❛ q✉❡✱ ♣❛r❛ t♦❞♦ r _∈ R✱ _{φr :M −→M} é ✉♠❛ ✐s♦♠❡tr✐❛ ❡ q✉❡

X(p) = d

drφr(p)

r=0 é ✉♠ ❝❛♠♣♦ ❞❡ ❑✐❧❧✐♥❣ ❡♠ M✳

❊♠ ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ ❝♦♠ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛✱ t♦❞♦ ❝❛♠♣♦ ✐♥✈❛r✐❛♥t❡ à ❞✐r❡✐t❛ Y é ❞❡ ❦✐❧❧✐♥❣✳ ❉❡ ❢❛t♦✱ ♦s ❞✐❢❡♦♠♦r✜s♠♦s ❧♦❝❛✐s sã♦ ❞❛ ❢♦r♠❛

Φt:x−→(exp(tY)).x,

❡ ♣♦rt❛♥t♦ sã♦ ✐s♦♠❡tr✐❛s ❣❧♦❜❛✐s✳

P❛r❛ ❝♦♠♣r❡❡♥❞❡r ✉♠ ♣♦✉❝♦ ❛ ❣❡♦♠❡tr✐❛ ❞♦ ❡s♣❛ç♦ S_ol❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❞❡✜♥✐r♠♦s ❢♦❧❤❡❛çõ❡s✳ ■♥t✉✐t✐✈❛♠❡♥t❡ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❞✐♠❡♥sã♦ n s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❞❡

❞✐♠❡♥sã♦ m é ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ M ❡♠ s✉❜✈❛r✐❡❞❛❞❡s ❝♦♥❡①❛s ❞❡ ❞✐♠❡♥sã♦ n✳

❆♥t❡s ❞❡ ❞❛r♠♦s ❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧✱ ❝♦♥s✐❞❡r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ✉♠ ❡①❡♠♣❧♦✱ q✉❡ ❛♣❡s❛r ❞❡ s✐♠♣❧❡s✱ ♥♦s ❞❛rá ✉♠❛ ✈✐sã♦ ❣❡♦♠étr✐❝❛ ❞❡st❡ ❝♦♥❝❡✐t♦✳

❖❜s❡r✈❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ c_∈Rm−n ✜①❛❞♦✱ ♦ ♣❧❛♥♦

Rn_{× {c}} ✭✶✳✺✮

♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ✏❢♦❧❤❛✑ ❡♠ Rm✱ ✭❋✐❣✉r❛ _1.1✮✳ ❱❛r✐❛♥❞♦ _c∈Rm−n ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r Rm ❝♦♠♦

Rm ₌Rn_×Rm−n_.

P♦rt❛♥t♦ t❛✐s ❢♦❧❤❛s ❞❡✜♥❡♠ ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❞✐♠❡♥sã♦ n ❡♠ Rm✳ ◆♦t❡ q✉❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧ f : U _⊂ Rm _{−→ V ⊂ R}m _{q✉❡ ♣r❡s❡r✈❛ ❛s ❢♦❧❤❛s ✭✶✳✺✮ s❛t✐s❢❛③✱}

♣❛r❛ ❝❛❞❛ c_∈Rm−n✱ ❛ ♣r♦♣r✐❡❞❛❞❡

f(U_∩(Rn_{× {c})) =V ∩(}Rn_{× {c}),}

s❡♥❞♦ c_∈Rm−n_{✱ ✭❋✐❣✉r❛ 1.2✮✳ P♦rt❛♥t♦ ❡st❡ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡✈❡ t❡r ❛ ❢♦r♠❛}

f(x, y) = (f1(x, y), f2(y)), (x, y)∈Rn×Rm−n ✭✶✳✻✮

❋✐❣✉r❛ ✶✳✶✿ ❋♦❧❤❡❛çõ❡s ❡♠ R3✳ ❋✐❣✉r❛ ✶✳✷✿ ❉✐❢❡♦♠♦r✜s♠♦ _f✳

❉❡✜♥✐çã♦ ✶✳✷✷✳ ❯♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❝❧❛ss❡ Cr ❡ ❞✐♠❡♥sã♦ n ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❞❡

❞✐♠❡♥sã♦ m é ✉♠ ❛t❧❛s ♠á①✐♠♦ F✱ ❞❡ ❝❧❛ss❡ _Cr_{✱ q✉❡ s❛t✐s❢❛③ ❛s ❞✉❛s ♣r♦♣r✐❡❞❛❞❡s}

❛❜❛✐①♦✿

✶✳ ❙❡ (U, ϕ)∈F✱ ❡♥tã♦

ϕ(U) = U1×U2 ⊂Rn×Rm−n,

s❡♥❞♦ U1 ⊂Rn ❡ U2 ⊂Rm−n ❞✐s❝♦s ❛❜❡rt♦s❀

✷✳ ❙❡(U, ϕ)✱ (V, ψ)_∈F✱ ❝♦♠_{U∩V 6=∅}✱ ❡♥tã♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s _ψ◦ϕ−1

s❛t✐s❢❛③ ✭✶✳✻✮✱ ♦✉ s❡❥❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

ψ_◦ϕ−1(x, y) = (f1(x, y), f2(y)).

❉✐③❡♠♦s q✉❡M é ❢♦❧❤❡❛❞❛ ♣♦rF✱ ♦✉ q✉❡F é ✉♠❛ ❡str✉t✉r❛ ❢♦❧❤❡❛❞❛ ❞❡ ❞✐♠❡♥sã♦

n ❡ ❝❧❛ss❡Cr _{s♦❜r❡M✳ ■♥❞✐❝❛r❡♠♦s ✉♠❛ ❢♦❧❤❡❛çã♦ ❞❡ ❞✐♠❡♥sã♦ n❡ ❝❧❛ss❡C}r _{❞❡ ✉♠❛}

✈❛r✐❡❞❛❞❡ M ❞❡ ❞✐♠❡♥sã♦ m ♣♦r F✱ s❡♠♣r❡ q✉❡ ♥ã♦ ❤♦✉✈❡r r✐s❝♦ ❞❡ ❝♦♥❢✉sã♦✳

❖ ❊s♣❛ç♦ ❍♦♠♦❣ê♥❡♦

S

_ol

◆❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛♠♦s ♦ ❡s♣❛ç♦ S_ol ❡ s✉❛ ❣❡♦♠❡tr✐❛✳ ❖ S_ol é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ tr✐❞✐♠❡♥s✐♦♥❛❧ ❤♦♠♦❣ê♥❡❛ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛✱ ❝✉❥♦ ❣r✉♣♦ ❞❡ ✐s♦♠❡✲ tr✐❛s t❡♠ ❞✐♠❡♥sã♦ ✸ ❡ é ✉♠ ❞♦s ♦✐t♦ ♠♦❞❡❧♦s ❞❡ ❣❡♦♠❡tr✐❛ ❞❡ ❚❤✉rst♦♥ ✭✈❡❥❛ ❬✶✾❪✮✳ ❈♦♠♦ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✱ ♦ ❡s♣❛ç♦ S_ol ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r R3 ♠✉♥✐❞♦ ❝♦♠ ❛ ♠étr✐❝❛

h·,_·i=ds2 =e2zdx2 +e−2zdy2+dz2,

♦♥❞❡ (x, y, z) sã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛♥ô♥✐❝❛s ❞♦ R3 ❡✱ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡ ❣r✉♣♦ ❞❛❞❛ ♣♦r

(x, y, z)_∗(x′, y′, z′) = (x+e−zx′, y+ezy′, z+z′),

❡st❡ ❡s♣❛ç♦ é ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ ❡ ❛ ♠étr✐❝❛ ds2 ❛❝✐♠❛ é ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛✳

◆❛ ♣r✐♠❡✐r❛ s❡çã♦ ❡st✉❞❛♠♦s ✉♠ ♣♦✉❝♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞♦ ❡s♣❛ç♦ S_ol✱ ♦❜t❡♠♦s ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛✱ ♦s ❝♦❧❝❤❡t❡s ❞❡ ▲✐❡ ❞♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛✱ ♦s ❝❛♠♣♦s ❞❡ ❑✐❧❧✐♥❣✱ ❛s ✐s♦♠❡tr✐❛s ❡ s✉❛s ❢♦❧❤❡❛çõ❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡ ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐♥❛ ∇✳ ❊♥q✉❛♥t♦ q✉❡ ♥❛ s❡❣✉♥❞❛ s❡çã♦✱ ❞❡✜♥✐♠♦s s✉♣❡r❢í❝✐❡s ✐♥✈❛r✐❛♥t❡s

♥♦ S_ol ❡ ♣♦st❡r✐♦r♠❡♥t❡ ❝❛❧❝✉❧❛♠♦s ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛ ❡ ●❛✉ss✐❛♥❛ ✭✐♥trí♥s❡❝❛ ❡ ❡①trí♥s❡❝❛✮ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ✐♥✈❛r✐❛♥t❡ ♥♦ S_ol✳

✷✳✶ ●❡♦♠❡tr✐❛ ❞♦ ❊s♣❛ç♦

S

_ol

◆❡st❛ s❡çã♦ ❡①♣♦♠♦s ❛❧❣✉♠❛s ♥♦çõ❡s ✐♥✐❝✐❛✐s ❞❛ ❣❡♦♠❡tr✐❛ ❞♦ ❡s♣❛ç♦ S_ol✳ ❋❛r❡✲ ♠♦s ✉s♦ ❞♦s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✳ ❖ ❡s♣❛ç♦ S_ol é ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦ q✉❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛s ♠❛tr✐③❡s tr✐❛♥❣✉❧❛r❡s s✉♣❡r✐♦r❡s ❡♠ M(3,R₎✱ ❞❛ ❢♦r♠❛

   

e−z _{0 x}

0 ez y

0 0 1

  

, (x, y, z)∈R3.

❊♠ t❡r♠♦s ❞❛s ❝♦♦r❞❡♥❛❞❛s (x, y, z)✱ ♦ ♣r♦❞✉t♦ ♥♦ ❣r✉♣♦ ❞❡ ▲✐❡ S_ol é ♦❜t✐❞♦ ♣♦r r❡str✐çã♦ ❞♦ ♣r♦❞✉t♦ ✉s✉❛❧ ❞❡ ♠❛tr✐③❡s ❡♠ M(3,R₎✳ ❆ss✐♠✱ ❞❛❞❛s ❛s ♠❛tr✐③❡s _A❡ _B r❡♣r❡s❡♥t❛❞❛s ❡♠ ❝♦♦r❞❡♥❛❞❛s ♣♦r

A_7−→(x, y, z) e B _7−→(x′, y′, z′),

♦ ♣r♦❞✉t♦ ❞❡ ♠❛tr✐③❡s (A, B)_7−→AB é r❡♣r❡s❡♥t❛❞♦ ♥❡st❛s ❝♦♦r❞❡♥❛❞❛s ♣♦r

(x, y, z)_∗(x′, y′, z′) = (x+e−zx′, y+ezy′, z+z′). ✭✷✳✶✮

❱❛♠♦s ❞❡t❡r♠✐♥❛r✱ ❛❣♦r❛✱ ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ♦❜t❡r ♦s ❝♦❧❝❤❡t❡s ❞❡ ▲✐❡ ❞❡st❡s ❝❛♠♣♦s✳ ❆❧é♠ ❞✐ss♦✱ ❝❛❧❝✉❧❛r❡♠♦s ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❞✐r❡✐t❛✳

◆❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ sol3✱ ❞♦ ❣r✉♣♦ ❞❡ ▲✐❡ S_ol✱ ❞❡st❛❝❛♠♦s ♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s

∂x|e :=e1, ∂y|e:=e2 e ∂z|e:=e3,

♦♥❞❡

e1 =    

0 0 1

0 0 0 0 0 0

  

, e2 =    

0 0 0

0 0 1 0 0 0

  

 e e3 =    

−1 0 0

0 1 0

0 0 0

   ,

➱ ❢á❝✐❧ ✈❡r q✉❡✿

[e1, e2] = 0, [e1, e3] =e1, [e2, e3] =−e2.

❉❡♥♦t❡♠♦s ♣♦rE1✱E2 ❡E3♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛ ❣❡r❛❞♦s ♣❡❧♦s ✈❡t♦r❡s

e1✱ e2 ❡ e3✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖ s✉❜❣r✉♣♦ ❛ ✉♠ ♣❛râ♠❡tr♦ ❣❡r❛❞♦ ♣♦r e1 é ❛ ❝✉r✈❛ ♣❛ss❛♥❞♦ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ e1✳

❊♠ ❝♦♦r❞❡♥❛❞❛s ❡①♣♦♥❡♥❝✐❛✐s✱ ❡st❛ ❝✉r✈❛ ❝♦rr❡s♣♦♥❞❡ à ❝✉r✈❛ t _7−→(t,0,0)✳ ▲♦❣♦✱ ❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞♦ ❝❛♠♣♦ E1 ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ A∈Sol❝♦♠ ❝♦♦r❞❡♥❛❞❛s (x, y, z) é ❞❛❞❛ ♣♦r

LA(exp(te1)) = A(exp(te1))

= (x, y, z)∗(t,0,0)

= (x+e−zt, y, z).

❉❡r✐✈❛♥❞♦ ❡♠ t = 0 ❛ ❝✉r✈❛ (x+e−zt, y, z)✱ t❡♠♦s ♦ ❝❛♠♣♦ E1 ❡♠ A= (x, y, z)✿

E1

(x,y,z) =

d dt

t=0(x+e

−z_{t, y, z))}

= (e−z,0,0)

= e−z(1,0,0)

= e−z∂x.

❆♥❛❧♦❣❛♠❡♥t❡✱ ♦❜t❡♠♦s ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛ E2 ❡ E3 ❣❡r❛❞♦s✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡✱ ♣♦r e2 ❡ e3

E2

(x,y,z) =e

z_∂

y ❡ E3

(x,y,z) =∂z.

❆ss✐♠✱ ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❡sq✉❡r❞❛ ❣❡r❛❞♦s ♣♦r e1 = ∂x✱ e2 =∂y ❡ e3 = ∂z ♥❛

á❧❣❡❜r❛ ❞❡ ▲✐❡ sol3 sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱       

E1 = e−z∂x

E2 = ez∂y

E3 = ∂z.

✭✷✳✷✮

❈❛❧❝✉❧❛♥❞♦ ♦s ❝♦❧❝❤❡t❡s ❞❡ ▲✐❡ ❞❡st❡s ❝❛♠♣♦s✱ t❡♠♦s

[E1, E2](x,y,z) = dL(x,y,z)[E1, E2]e

= dL(x,y,z)[e1, e2]

= 0,

[E1, E3](x,y,z) = dL(x,y,z)[E1, E3]e

= dL(x,y,z)[e1, e3]

= dL(x,y,z)e1

= E1

❡

[E2, E3](x,y,z) = dL(x,y,z)[E2, E3]e

= dL(x,y,z)[e2, e3]

= dL(x,y,z)(−e2)

= ₋E2.

❖✉ s❡❥❛✱

[E1, E2] = 0 ✱ [E1, E3] =E1 ❡ [E2, E3] =−E2. ✭✷✳✸✮

❈❛❧❝✉❧❡♠♦s✱ ❛❣♦r❛✱ ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❞✐r❡✐t❛ ❣❡r❛❞♦s ♣❡❧♦s ✈❡t♦r❡s e1✱e2❡e3✳ ❉❡♥♦t❡♠♦s ♣♦rF1✱F2 ❡ F3 ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❞✐r❡✐t❛ ❣❡r❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r e1✱ e2 ❡e3✳

❖ s✉❜❣r✉♣♦ ❛ ✉♠ ♣❛râ♠❡tr♦ ❣❡r❛❞♦ ♣♦r e1 é ❛ ❝✉r✈❛ ♣❛ss❛♥❞♦ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡e1✱ ❞♦♥❞❡ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❡①♣♦♥❡♥❝✐❛✐s ❡st❛ ❝✉r✈❛ ❝♦rr❡s♣♦♥❞❡ à ❝✉r✈❛ t_7−→(t,0,0)✳ ▲♦❣♦✱ ❛ ❝✉r✈❛ ✐♥t❡❣r❛❧ ❞♦ ❝❛♠♣♦ F1 ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ A∈Sol ❝♦♠ ❝♦♦r❞❡♥❛❞❛s (x, y, z)é ❞❛❞❛ ♣♦r

RA(exp(te1)) = (t,0,0)∗(x, y, z)

= (t+x, y, z).

❉❡r✐✈❛♥❞♦ ❡♠ t = 0 ❛ ❝✉r✈❛ (t+x, y, z)✱ ♦❜t❡♠♦s ♦ ❝❛♠♣♦ F1 ❡♠ A= (x, y, z)✿

F1

(x,y,z) =

d dt

t=0(t+x, y, z)

= (1,0,0)

= ∂x.

❆♥❛❧♦❣❛♠❡♥t❡✱ ♦❜t❡♠♦s ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❞✐r❡✐t❛ F2 ❡ F3 ❣❡r❛❞♦s✱ r❡s♣❡❝t✐✈❛✲ ♠❡♥t❡✱ ♣♦r e2 ❡e3

F2

(x,y,z) =∂y ❡ F3

(x,y,z) =−x∂x+y∂y +∂z.

❆ss✐♠✱ ♦s ❝❛♠♣♦s ✐♥✈❛r✐❛♥t❡s à ❞✐r❡✐t❛ ❣❡r❛❞♦s ♣♦r e1 = ∂x✱ e2 = ∂y ❡ e3 = ∂z ♥❛

á❧❣❡❜r❛ ❞❡ ▲✐❡ sol3 sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱       

F1 = ∂x

F2 = ∂y

F3 = −x∂x+y∂y +∂z.

✭✷✳✹✮

❆ s❡❣✉✐r✱ ❞❡✜♥✐♠♦s ✉♠❛ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛ ♥♦ S_ol ❡ ❡♠ s❡❣✉✐❞❛ ❞❡✲ t❡r♠✐♥❛♠♦s ♦s ❝❛♠♣♦s ❞❡ ❑✐❧❧✐♥❣✱ ❛s ✐s♦♠❡tr✐❛s ❡ s✉❛s ❢♦❧❤❡❛çõ❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❡✱ ❛❧é♠ ❞✐ss♦✱ ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ∇✳

❆ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛✱ ❞❡♥♦t❛❞❛ ♣♦r h·,_·i,❞❡✜♥❡✲s❡ t♦♠❛♥❞♦ ♦s ❝❛♠♣♦s E1✱ E2✱ E3 ❝♦♠♦ ♦rt♦♥♦r♠❛✐s✱ ✐st♦ é✱

hEi, Eji=δij. ✭✷✳✺✮

❉❡ ✭✷✳✹✮ t❡♠♦s✿

      

∂x = ezE1

∂y = e−zE2

∂z = E3.

✭✷✳✻✮

❆ss✐♠✱ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❛ ♠étr✐❝❛ ❡♠ t❡r♠♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡①♣♦♥❡♥❝✐❛✐s sã♦✿

h∂x, ∂xi=e2z h∂x, ∂yi= 0

h∂y, ∂yi=e−2z h∂x, ∂zi= 0

h∂z, ∂zi= 1 h∂y, ∂zi= 0.

❖✉ s❡❥❛✱ ❛ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛ q✉❡ ✜①❛♠♦s ♥♦ ❡s♣❛ç♦ S_ol é ❞❛❞❛ ♣♦r

ds2 =e2zdx2+e−2zdy2+dz2. ✭✷✳✼✮

❆s tr❛♥s❧❛çõ❡s à ❡sq✉❡r❞❛ sã♦ ✐s♦♠❡tr✐❛s✱ ✉♠❛ ✈❡③ q✉❡ ❛ ♠étr✐❝❛ é ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛✳ ❆❧é♠ ❞✐ss♦✱ ❡♠ ✉♠ ❣r✉♣♦ ❞❡ ▲✐❡ ❝♦♠ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ à ❡sq✉❡r❞❛✱ t♦❞♦ ❝❛♠♣♦ ✐♥✈❛r✐❛♥t❡ à ❞✐r❡✐t❛ é ❞❡ ❦✐❧❧✐♥❣✳ ▲♦❣♦✱ ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ❣r✉♣♦ ❞❡ ✐s♦♠❡tr✐❛s ❞♦ ❡s♣❛ç♦ S_ol é ❞❛❞❛ ♣❡❧♦s ❝❛♠♣♦s ❞❡ ❑✐❧❧✐♥❣ ❛❜❛✐①♦✿

      

F1 = ∂x

F2 = ∂y

F3 = −x∂x+y∂y +∂z.

✭✷✳✽✮

❖s s✉❜❣r✉♣♦s ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡ ✐s♦♠❡tr✐❛s ❣❡r❛❞♦ ♣❡❧♦s três ❝❛♠♣♦s ❞❡ ❑✐❧❧✐♥❣ ❞❛❞♦s ❡♠ ✭✷✳✽✮ sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱

      

T1,t(x, y, z) = (x+t, y, z)

T2,t(x, y, z) = (x, y+t, z)

T3,t(x, y, z) = (e−tx, ety, z+t),

✭✷✳✾✮

♦♥❞❡ t_∈R✱ é ✉♠ ♣❛râ♠❡tr♦ r❡❛❧✳

❖ ♣♦♥t♦ ❝❤❛✈❡ ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞♦ ❡s♣❛ç♦ S_ol é ❝♦♥s✐❞❡r❛r ❛s três s❡❣✉✐♥t❡s ❢♦❧❤❡❛çõ❡s✿       

F_{1 : {Pt ={(t, y, z); y, z ∈}R_}}t

∈R; F_{2 : {Qt={(x, t, z); x, z ∈}R_}}t∈R;

F_{3 : {Rt ={(x, y, t); x, y ∈}R_}}t∈R.

✭✷✳✶✵✮

❆s ❞✉❛s ♣r✐♠❡✐r❛s ❢♦❧❤❡❛çõ❡sF₁ ❡F₂ sã♦ ❞❡t❡r♠✐♥❛❞❛s ♣❡❧♦s ❣r✉♣♦s ❞❡ ✐s♦♠❡tr✐❛s

{T1,t}t∈R ❡{T_2,t}_t∈R✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡❧❛s ❞❡s❝r❡✈❡♠ ❛s ú♥✐❝❛s s✉♣❡r❢í❝✐❡s t♦t❛❧♠❡♥t❡ ❣❡♦❞és✐❝❛s ❞♦ ❡s♣❛ç♦ S_ol✱ s❡♥❞♦ ❝❛❞❛ ❢♦❧❤❛ ✐s♦♠étr✐❝❛ ❛ ✉♠ ♣❧❛♥♦ ❤✐♣❡r❜ó❧✐❝♦ ❡✱ ❛ t❡r❝❡✐r❛ ❢♦❧❤❡❛çã♦ F₃ é r❡❛❧✐③❛❞❛ ♣♦r s✉♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❡ t♦❞❛s ❡❧❛s sã♦ ✐s♦♠étr✐❝❛s

❛♦ ♣❧❛♥♦ ❊✉❝❧✐❞✐❛♥♦✳

❆ ♣❛rt✐r ❞❡ ✭✷✳✸✮ ❡ ✉t✐❧✐③❛♥❞♦ ✭✶✳✹✮✱ ❞❡❞✉③✐♠♦s q✉❡ ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ∇❞♦

❡s♣❛ç♦ S_ol s❛t✐s❢❛③✿

∇E1E1 =−E3 ∇E1E2 = 0 ∇E1E3 =E1

∇E2E1 = 0 ∇E2E2 =E3 ∇E2E3 =−E2

∇E3E1 = 0 ∇E3E2 = 0 ∇E3E3 = 0,