Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛
❙✉♣❡r❢í❝✐❡s ❡♠
R
4❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ t❡♦r✐❛ ❞❛s
s✐♥❣✉❧❛r✐❞❛❞❡s
P♦r
P❛✉❧♦ ❞♦ ◆❛s❝✐♠❡♥t♦ ❙✐❧✈❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ▲✐③❛♥❞r♦ ❙❛♥❝❤❡③ ❈❤❛❧❧❛♣❛
❉✐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✐❝❛✳
♣♦r
P❛✉❧♦ ❞♦ ◆❛s❝✐♠❡♥t♦ ❙✐❧✈❛
❉✐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✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❙✐♥❣✉❧❛r✐❞❛❞❡s ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ▲✐③❛♥❞r♦ ❙❛♥❝❤❡③ ❈❤❛❧❧❛♣❛ ❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❆❧❡①❛♥❞r❡ ❈és❛r ●✳ ❋❡r♥❛♥❞❡s ❊①❛♠✐♥❛❞♦r
Pr♦❢✳ ❉r✳ P❡❞r♦ ❆♥t♦♥✐♦ ●♦♠❡③ ❱❡♥❡❣❛s ❊①❛♠✐♥❛❞♦r
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛
♠❛✐♦ ✲ ✷✵✶✸
❉❛t❛✿ ♠❛✐♦ ✲ ✷✵✶✸ ❆✉t♦r✿ P❛✉❧♦ ❞♦ ◆❛s❝✐♠❡♥t♦ ❙✐❧✈❛
❚ìt✉❧♦✿
❙✉♣❡r❢í❝✐❡s ❡♠
R
4❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛
❞❛ t❡♦r✐❛ ❞❛s s✐♥❣✉❧❛r✐❞❛❞❡s
❉❡♣t♦✳✿ ▼❛t❡♠át✐❝❛
●r❛✉✿ ▼✳❙❝✳ ❈♦♥✈♦❝❛çã♦✿ ♠❛✐♦ ❆♥♦✿ ✷✵✶✸
P❡r♠✐ssã♦ ❡stá ❥✉♥t❛♠❡♥t❡ ❝♦♥❝❡❞✐❞❛ ♣❡❧❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ à ❝✐r❝✉❧❛r ❡ s❡r ❝♦♣✐❛❞♦ ♣❛r❛ ♣r♦♣ós✐t♦s ♥ã♦ ❝♦♠❡r❝✐❛✐s✱ ❡♠ s✉❛ ❞❡s❝r✐çã♦✱ ♦ tít✉❧♦ ❛❝✐♠❛ s♦❜ ❛ r❡q✉✐s✐çã♦ ❞❡ ✐♥❞✐✈í❞✉♦s ♦✉ ✐♥st✐t✉✐çõ❡s✳
❆ss✐♥❛t✉r❛ ❞♦ ❆✉t♦r
♠ã❡✱ ❛♦ ♠❡✉ ✐r♠ã♦ ❡ à ♠✐♥❤❛ ♥♦✐✈❛✳
P❛r❛ ❝♦♥s❡❣✉✐r ♦❜t❡r ♦ ❞✐♣❧♦♠❛ ❞❡ ♠❡str❡ ❢♦r❛♠ ♥❡❝❡ssár✐♦s ♠✉✐t♦s ❞✐❛s ❡ ♥♦✐t❡s ❞❡ ❡st✉❞♦✱ ❡ ♠✉✐t❛s ✈❡③❡s ❛❜❞✐❝❛r ❞❡ ♠♦♠❡♥t♦s ❝♦♠ ❛ ❢❛♠í❧✐❛✱ ♥♦✐✈❛ ❡ ❛♠✐❣♦s✱
s❡♠ ❡sq✉❡❝❡r ❞❛s ♠✉✐t❛s ♦r❛çõ❡s q✉❡ ✜③ ❡ q✉❡ ✜③❡r❛♠ ♣♦r ♠✐♠ ❞✉r❛♥t❡ ❡ss❡ t❡♠♣♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ❛❣r❛❞❡ç♦ à ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ ❢♦rç❛s✱ ♣❛③ ✐♥t❡r✐♦r ❡
s❛❜❡❞♦r✐❛ ❞✉r❛♥t❡ ❡st❡ ❝✉rs♦✳
➚ ♠✐♥❤❛ ♠ã❡ q✉❡ s❡♠♣r❡ ❝✉✐❞♦✉ ❜❡♠ ❞❡ ♠✐♠✱ ❡♥s✐♥❛♥❞♦✲♠❡ ✈❛❧♦r❡s ❡ ❞❛♥❞♦ ✉♠❛ ❜♦❛ ❡❞✉❝❛çã♦✱ ❛❧é♠ ❞❡ s❡♠♣r❡ ❛❝r❡❞✐t❛r ❡♠ ♠✐♠ q✉❛♥❞♦ ♥❡♠ ♠❡s♠♦ ❡✉ ❛❝r❡❞✐t❛✈❛✳
❆♦ ♠❡✉ ✐r♠ã♦ P❡trô♥✐♦✱ ♣❡❧❛ t♦r❝✐❞❛ ❡ ♣♦r s❡r s❡♠♣r❡ ♣r❡st❛t✐✈♦✳ ➚ ♠✐♥❤❛ ♥♦✐✈❛ ❏✉❧✐❛♥❛✱ ♣♦r s❡✉ ❛♠♦r✱ ❝❛r✐♥❤♦ ❡ ❝♦♠♣r❡❡♥sã♦✳
❆♦ ♠❡✉ ❛♠✐❣♦ ❏❛✐❧s♦♥ ♣♦r s❡r ✉♠ ❞♦s ♣r✐♠❡✐r♦s q✉❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❛ ❝✉rs❛r ♦ ♠❡str❛❞♦✳
❆♦s ❝♦❧❡❣❛s ❞♦ ♠❡str❛❞♦✱ ♣❡❧♦ ♣r❛③❡r ❞❡ s✉❛s ❛♠✐③❛❞❡s✱ ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦ ❡♠ ❣r✉♣♦✱ ♣❡❧❛ tr♦❝❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❧✐st❛s ❞❡ ❡①❡r❝í❝✐♦s ✱ ❝♦♥✈❡rs❛s ✱ ❢✉t❡❜♦❧ ❡ ❡t❝✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❛♦ ❉❛♥✐❧♦✱ ❊❜❡rs♦♥✱ ❊❞♥❛✱ ❊r✐♥❛❧❞♦✱ ❋r❛♥❝✐s❝♦✱ ●✐♥❛❧❞♦✱ ●✉✐❧❤❡r♠❡✱
●✉st❛✈♦✱ ❏♦sé ❈❛r❧♦s✱ ▲✉❛♥✱ ▲✉❛♥❞♦✱ ▼❛r✐❛♥❛✱ ▼❛①✱ ▼ô♥✐❝❛✱ ◆❛❝✐❜✱ P❡❞r♦✱ ❘❡♥❛t♦✱ ❘❡❣✐♥❛❧❞♦✱ ❘✐❝❛r❞♦✱ ❨❛♥❡✱ ❡♥tr❡ ♦✉tr❛s q✉❡ ❝♦♥❤❡❝✐ ❞✉r❛♥t❡ ❡st❛ ❝❛♠✐♥❤❛❞❛✳
❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦ ❋r❛♥❝✐s❝♦ ❱✐❡r❛ ❞❡ ❖❧✐✈❡✐r❛✱ q✉❡ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❝❛♠✐♥❤❛❞❛ s❡ t♦r♥♦✉ ✉♠ ❣r❛♥❞❡ ❛♠✐❣♦✱ s❡♠♣r❡ ❞❛♥❞♦ ❡s♣❡r❛♥ç❛ ❡ ❛♣♦✐♦ ♥♦s
♠♦♠❡♥t♦s ♠❛✐s ♥❡❝❡ssár✐♦s✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦ ❋r❛♥❝✐s❝♦✳
❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦✱ ♣❡❧❛s ❜♦❛s ❛✉❧❛s ♥❛ ❞✐s❝✐♣❧✐♥❛ ■♥tr♦❞✉çã♦ ❛ ➪♥❛❧✐s❡ ❘❡❛❧ ❞✉r❛♥t❡ ♦ ✈❡rã♦ ♣❛r❛ s❡❧❡çã♦ ❞♦ ♠❡str❛❞♦✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ♣r♦❢❡ss♦r❡s ❞♦ ♠❡str❛❞♦✱ ❉r✳ ❆❧❡①❛♥❞r❡ ❞❡ ❇✉st❛♠❛♥t❡ ❙✐♠❛s ✱
♣❛❝✐ê♥❝✐❛✱ ✐♥❝❡♥t✐✈♦s✱ ♣♦r ❛❝r❡❞✐t❛r q✉❡ ❡✉ ❡r❛ ❝❛♣❛③✱ s✉❣❡stõ❡s✱ ❞✐❝❛s✱ ❡♥✜♠ ♣♦r ✉♠❛ ❜♦❛ ♦r✐❡♥t❛çã♦✳
❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❉r✳ ❆❧❡①❛♥❞r❡ ❈és❛r ●✉r❣❡❧ ❋❡r♥❛♥❞❡s ❡ ❉r✳ P❡❞r♦ ❆♥t♦♥✐♦ ●♦♠❡③ ❱❡♥❡❣❛s ♣♦r t❡r❡♠ ❛❝❡✐t❛❞♦ ❢❛③❡r ♣❛rt❡ ❞❛ ❜❛♥❝❛✳
❚❛♠❜é♠ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛♦ ♣r♦❢❡ss♦r ❉r✳ ❘♦❜❡rt♦ ❈❛❧❧❡❥❛s ❇❡❞r❡❣❛❧ ♣♦r t❡r s✐❞♦ ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s r❡s♣♦♥sá✈❡✐s ♣❡❧❛ ♠✐♥❤❛ ✈✐❛❣❡♠ à ❯❙P ❞❡ ❙ã♦ ❈❛r❧♦s ♦♥❞❡
♣✉❞❡ ❛❞q✉✐r✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥❡❝❡ssár✐♦ ♣❛r❛ ❡s❝r❡✈❡r ♠✐♥❤❛ ❞✐ss❡rt❛çã♦✳ ➚ ♣r♦❢❡ss♦r❛ ❉r❛✳ ▼❛r✐❛ ❆♣❛r❡❝✐❞❛ ❘✉❛s ❝♦♦r❞❡♥❛❞♦r❛ ❞♦ ♣r♦❥❡t♦ Pr♦❝❛❞✱ ♣♦r ❧✐❜❡r❛r ❛ ✈✐❛❣❡♠ ♣❛r❛ à ❯❙P ❛✜♠ ❞❡ q✉❡ ♣✉❞❡ss❡ ✉t✐❧✐③❛r ♦s ❧✐✈r♦s ❡ ❛rt✐❣♦s ❞❛
❜✐❜❧✐♦t❡❝❛ ❞❛ ❯❙P ❞❡ ❙ã♦ ❈❛r❧♦s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳ ❆♦ ♣r♦❢❡ss♦r ▼❛r❝❡❧♦ ❏♦sé ❙❛✐❛ ❞❛ ❯❙P ❞❡ ❙ã♦ ❈❛r❧♦s✱ ♣❡❧♦ ❛❝♦❧❤✐♠❡♥t♦ ❡ ♣❡❧❛s
❜♦❛s ❛✉❧❛s ♥❛ ❞✐s❝✐♣❧✐♥❛ ❙✐♥❣✉❧❛r✐❞❛❞❡s ❞❡ ❛♣❧✐❝❛çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ Pr♦❣r❛♠❛ ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋P❇✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ❉r✳ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ❞❡ ▼❡❞❡✐r♦s ❡ ❉r✳ ❉❛♥✐❡❧ ▼❛r✐♥❤♦ P❡❧❧❡❣r✐♥♦ q✉❡ ❢♦r❛♠ ❛♠❜♦s ❝♦♦r❞❡♥❛❞♦r❡s ❞♦ ♠❡str❛❞♦ ❞✉r❛♥t❡ ♦ ♣❡rí♦❞♦
❡♠ q✉❡ ❡r❛ ♠❡str❛♥❞♦✳
❆♦s ♠❡✉s ❛♥t✐❣♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ❉r✳ ❆♥tô♥✐♦ ❙❛❧❡s ❞❛ ❙✐❧✈❛✱❉r✳ ❊❞✉❛r❞♦ ●♦♥ç❛❧✈❡s ❞♦s ❙❛♥t♦s✱❉r✳ ❏♦ã♦ ❇❛t✐st❛ ❆❧✈❡s P❛r❡♥t❡✱ ❉r✳ ▼✐❧t♦♥ ❞❡ ▲❛❝❡r❞❛ ❖❧✐✈❡✐r❛ ❡ ❉r❛✳ ❘♦❣ér✐❛ ●❛✉❞ê♥❝✐♦ ❞♦
❘❡❣♦ ♣❡❧❛s ❜♦❛s ❛✉❧❛s ❡ ❝♦♥s❡❧❤♦s✳
❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛♦ ❘❊❯◆■ ♣❡❧❛ ❜♦❧s❛✱ ♣♦✐s s❡♠ ❡❧❛✱ ♥ã♦ t❡r✐❛ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥❝❧✉✐r ❡st❡ ❝✉rs♦✳
❊♥✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞❡ ♠❛♥❡✐r❛ ❞✐r❡t❛ ❡ ✐♥❞✐r❡t❛ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❝♦♥❝r❡t✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆❣r❛❞❡❝✐♠❡♥t♦s ✈
❘❡s✉♠♦ ✈✐✐✐
❆❜str❛❝t ✐①
■♥tr♦❞✉çã♦ ①
✶ Pr❡❧✐♠✐♥❛r❡s ✶
✶✳✶ ❙✐♥❣✉❧❛r✐❞❛❞❡s ❞❡ ❣❡r♠❡s ❞❡ ❢✉♥çõ❡s s✉❛✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ ❈❧❛ss✐✜❝❛çã♦ ❞♦s ❣❡r♠❡s ❞❡ ❝♦❞✐♠❡♥sã♦65 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✷ ❱❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❈♦♥❥✉♥t♦s s✐♥❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✹ ❈♦♥t❛t♦ ❡♥tr❡ s✉❜✈❛r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✺ ❆s ❡q✉❛çõ❡s ❞❡ ❊str✉t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷ ❙✉♣❡r❢í❝✐❡s ❡♠ R4 ✷✶
✷✳✶ ❊❧✐♣s❡ ❝✉r✈❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❖s ■♥✈❛r✐❛♥t❡s ❞❡ ❙✉♣❡r❢í❝✐❡s ❡♠ R4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ ❋♦r♠❛s ◗✉❛❞rát✐❝❛s ❉❡❣❡♥❡r❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸ ❈♦♥t❛t♦s ❞❡ ❙✉♣❡r❢í❝✐❡s ❡♠ R4 ❝♦♠ ❤✐♣❡r♣❧❛♥♦s ✸✻
✸✳✶ ❱❛r✐❡❞❛❞❡ ❝❛♥❛❧ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡♠ R4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ ❈❛r❛t❡r✐③❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛s s✐♥❣✉❧❛r✐❞❛❞❡s ❞❡ ❢✉♥çõ❡s ❛❧t✉r❛ ✳ ✳ ✳ ✳ ✹✷
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✾
◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ❛ ❣❡♦♠❡tr✐❛ ❞❛s s✉♣❡r❢í❝✐❡s ❡♠ R4 ❛tr❛✈és ❞❛ ✈❛r✐❡❞❛❞❡ ❝❛♥❛❧ ❡ ❞❛s s✐♥❣✉❧❛r✐❞❛❞❡s ❞❛s ❢❛♠í❧✐❛s ❞❡ ❢✉♥çõ❡s ❛❧t✉r❛ ❞❛s s✉♣❡r❢í❝✐❡s✳ Pr♦✈❛r❡♠♦s q✉❡ ♦s ♣♦♥t♦s ❞❡ ✐♥✢❡①ã♦ ❞❛s s✉♣❡r❢í❝✐❡ sã♦ ♦s ♣♦♥t♦s ✉♠❜í❧✐❝♦s ❞❛s ❢❛♠í❧✐❛s ❞❡ ❢✉♥çõ❡s ❛❧t✉r❛✳ ❆❧é♠ ❞✐ss♦✱ ✈❡r❡♠♦s q✉❡ ♣♦♥t♦s ❞❡ ✐♥✢❡①ã♦ ❞♦ t✐♣♦ ✐♠❛❣✐♥ár✐♦ s❡rã♦ ♣♦♥t♦s ✐s♦❧❛❞♦s ❞❛ ❝✉r✈❛ ∆−1(0)✳ ❈♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡ ❡st✉❞♦ ♣r♦✈❛r❡♠♦s q✉❡ q✉❛❧q✉❡r ♠❡r❣✉❧❤♦ ❣❡♥ér✐❝♦ ❝♦♥✈❡①♦ ❞❡ S2 ❡♠ R4 t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ❞❡ ✐♥✢❡①ã♦✳
P❛❧❛✈r❛s✲❈❤❛✈❡✿
❙✐♥❣✉❧❛r✐❞❛❞❡s✱ ❙❡❣✉♥❞❛ ❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧✱ ❊❧í♣s❡ ❞❡ ❈✉r✈❛t✉r❛✱ ❋✉♥çã♦ ❆❧t✉r❛✱ P♦♥t♦ ❞❡ ■♥✢❡①ã♦✱ P♦♥t♦ ❯♠❜í❧✐❝♦✱ ▼❡r❣✉❧❤♦ ●❡♥ér✐❝♦✳❲❡ st✉❞② t❤❡ ❣❡♦♠❡tr② ♦❢ s✉r❢❛❝❡s ✐♠♠❡rs❡❞ ✐♥ R4 t❤r♦✉❣❤ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ♦❢ t❤❡✐r ❢❛♠✐❧✐❡s ♦❢ ❤❡✐❣❤t ❢✉♥❝t✐♦♥s✳ ■♥✢❡❝t✐♦♥ ♣♦✐♥ts ♦♥ t❤❡ s✉r❢❛❝❡s ❛r❡ s❤♦✇♥ t♦ ❜❡ ✉♠❜✐❧✐❝ ♣♦✐♥ts ❢r♦♠ t❤❡✐r ❢❛♠✐❧✐❡s ♦❢ ❤❡✐❣❤t ❢✉♥❝t✐♦♥s✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ s❡❡ t❤❛t ✐♥✢❡❝t✐♦♥ ♣♦✐♥ts ♦❢ ✐♠❛❣✐♥❛r② t②♣❡ ❛r❡ ✐s♦❧❛t❡❞ ♣♦✐♥ts ♦❢ t❤❡ ❝✉r✈❡ ∆−1(0)✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ✇❡ ♣r♦✈❡ t❤❛t ❛♥② ❞✐✈❡ ❣❡♥❡r✐❝ ❝♦♥✈❡①❧② ❡♠❜❡❞❞❡❞S2 ✐♥R4 ❤❛s ✐♥✢❡①✐♦♥ ♣♦✐♥ts✳
❑❡②✇♦r❞s✿
❙✐♥❣✉❧❛r✐t✐❡s✱ ❙❡❝♦♥❞ ❋✉♥❞❛♠❡♥t❛❧ ❋♦r♠✱ ❊❧❧✐♣s❡ ❈✉r✈❛t✉r❡✱ ❍❡✐❣❤t ❋✉♥❝t✐♦♥✱ ■♥✲ ✢❡①✐♦♥ P♦✐♥t✱ ❯♠❜í❧✐❝ P♦✐♥t✱ ❊♠❜❡❞❞✐♥❣ ●❡♥❡r✐❝✳
❘❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ❞❛ ❣❡♦♠❡tr✐❛ ❞❛s s✉♣❡r❢í❝✐❡s ❡♠ R4 ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞❡ s❡✉s ❝♦♥t❛t♦s ❣❡♥ér✐❝♦s ❝♦♠ ❤✐♣❡r♣❧❛♥♦s✱ ❡ss❡s ❝♦♥t❛t♦s s❡rã♦ ❞❛❞♦s ♣❡❧❛s s✐♥❣✉❧❛r✐❞❛❞❡s ❞❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s ❛❧t✉r❛✳
P❛r❛ ♥♦ss♦ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❛s s✉♣❡r❢í❝✐❡s ❡♠ R4 ✈❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ ✐♠❡rsã♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡♠R4✳ P❛r❛ ❝❛❞❛ ♣♦♥t♦ ❞❛ s✉♣❡r❢í❝✐❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ❡❧✐♣s❡ ♥♦ s✉❜❡s♣❛ç♦ ♥♦r♠❛❧✱ ❞❡♥♦♠✐♥❛❞❛ ❡❧✐♣s❡ ❞❡ ❝✉r✈❛t✉r❛✳ ❆ ❡❧✐♣s❡ ❞❡ ❝✉r✈❛t✉r❛ é ❞❛❞❛ ♣❡❧❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ s✉♣❡r❢í❝✐❡✳ ❯♠ ♣♦♥t♦ ❞❛ s✉♣❡r❢í❝✐❡ s❡rá ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ ❞❡ ✐♥✢❡①ã♦ q✉❛♥❞♦ ❛ ❡❧✐♣s❡ ❞❡ ❝✉r✈❛t✉r❛ ❛ss♦❝✐❛❞❛ ❛ ❡ss❡ ♣♦♥t♦ ❢♦r ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ r❛❞✐❛❧✱ ❡ss❡ ❝♦♥❝❡✐t♦ é ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✶✵❪✳
❊st❡ tr❛❜❛❧❤♦ ❜❛s❡✐❛✲s❡ ♥♦ ❛rt✐❣♦ ✑❚❤❡ ●❡♦♠❡tr② ♦❢ ❙✉r❢❛❝❡s ✐♥ ✹✲s♣❛❝❡ ❢r♦♠ ❛ ❈♦♥t❛❝t ❱✐❡✇♣♦✐♥t✑ ❡ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ❝❛♣ít✉❧♦s✳
◆♦ ❝❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♥❛ t❡♦r✐❛ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ s✉❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❡♠ ❬✽❪✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ ❣❡r♠❡s ❞❡ ❛♣❧✐❝❛çõ❡s✱ ❝♦♥❥✉♥t♦s s✐♥❣✉❧❛r❡s✱ ❝♦❞✐♠❡♥sã♦ ❞❡ ✉♠ ❣❡r♠❡✱ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ❣❡r♠❡s ❞❡ ❝♦❞✐♠❡♥sã♦≤5✱ ❝♦♥t❛t♦ ❡♥tr❡ s✉❜✈❛r✐❡❞❛❞❡s✳ ❋✐♥❛❧✐③❛♠♦s
♦ ❝❛♣ít✉❧♦ ❡st✉❞❛♥❞♦ ❡q✉❛çõ❡s ❞❡ ❡str✉t✉r❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ✐♠❡rs❛ ❡♠ R4✱ ❛tr❛✈és ❞❛s ❡q✉❛çõ❡s ❞❡ ❡str✉t✉r❛ ❞♦ Rn✳ ◆❛ s❡ssã♦ ✶✳✷ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ❣❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ r❡❧❛❝✐♦♥❛❞♦s ❛ ❝♦♥❡①ã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳
◆♦ ❝❛♣ít✉❧♦ ✷✱ ❝❛❧❝✉❧❛♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ s✉✲ ♣❡r❢í❝✐❡ ✉t✐❧✐③❛♥❞♦ ♦ r❡❢❡r❡♥❝✐❛❧ ♠ó✈❡❧✱ ♦ q✉❛❧ é ❞❡✜♥✐❞♦ ♥❛ s❡ssã♦ ✶✳✻ ♥♦ ❝❛♣ít✉❧♦ ✶✳ ❊♥❝♦♥tr❛♠♦s ❛ ❝✉r✈❛t✉r❛ ❣❛✉ss✐❛♥❛ ❞❛ s✉♣❡r❢í❝✐❡✱ ✉s❛♥❞♦ ♦ ❢❛♠♦s♦ t❡♦r❡♠❛ ❞❡ ●❛✉ss ✭ ✈❡❥❛ ❬✹❪✮✳ ❚❛♠❜é♠ ❡st✉❞❛♠♦s ❛ ❡❧✐♣s❡ ❞❡ ❝✉r✈❛t✉r❛ ❡ ♦s ✐♥✈❛r✐❛♥t❡s ❛ss♦❝✐❛❞♦s ❛
❛ss♦❝✐❛❞❛ ❛ s✉♣❡r❢í❝✐❡ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♦❜t❡r ✐♥❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ❞❛ s✉♣❡r❢í❝✐❡✳ ❚❛♠❜é♠ ✐♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❛ ✈❛r✐❡❞❛❞❡ ❝❛♥❛❧ ❛ss♦❝✐❛❞❛ ❛ s✉♣❡r❢í❝✐❡✱ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ té❝♥✐❝❛ q✉❡ ♣❡r♠✐t❡ ♦❜t❡r ✐♥❢♦r♠❛çõ❡s ❣❡♦♠étr✐❝❛s ❞❛ s✉♣❡r❢í❝✐❡ ❛ ♣❛rt✐r ❞❛ ✈❛r✐❡❞❛❞❡ ❝❛♥❛❧✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡ ❡st✉❞♦ ♣r♦✈❛r❡♠♦s q✉❡ q✉❛❧q✉❡r ♠❡r❣✉❧❤♦ ❣❡♥ér✐❝♦ ❝♦♥✈❡①♦ ❞❡ S2 ❡♠ R4 t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ❞❡ ✐♥✢❡①ã♦✳
Pr❡❧✐♠✐♥❛r❡s
◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐♠♦s ❛s ♥♦t❛çõ❡s ❡ ❞❡✜♥✐çõ❡s ❜ás✐❝❛s✱ ✉s✉❛❧♠❡♥t❡ ✉t✐❧✐✲ ③❛❞❛s ♥❛ ❚❡♦r✐❛ ❞❡ ❙✐♥❣✉❧❛r✐❞❛❞❡s ❡ ❛♣❧✐❝❛çõ❡s s✉❛✈❡s✳ ❊♠ s❡❣✉✐❞❛✱ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ❣❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✳ ❋✐♥❛❧✐③❛♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❝♦♠ ❡st✉❞♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊str✉t✉r❛ ❛ss♦❝✐❛❞❛s ❛ ✉♠❛ ✐♠❡rsã♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡♠ R4✳ ❖s r❡s✉❧t❛❞♦s ❞❡st❡ ❝❛♣ít✉❧♦ sã♦ ✐♥s♣✐r❛❞♦s ❡♠ ❬✽❪✱ ❬✶✸❪✱❬✶✷❪✱❬✹❪✳
✶✳✶ ❙✐♥❣✉❧❛r✐❞❛❞❡s ❞❡ ❣❡r♠❡s ❞❡ ❢✉♥çõ❡s s✉❛✈❡s
❯♠❛ ❛♣❧✐❝❛çã♦ f : U → Rp é ❞❡ ❝❧❛ss❡ Ck ♥♦ ❛❜❡rt♦ U ⊂ Rn q✉❛♥❞♦ ❡①✐st❡♠ ❡
sã♦ ❝♦♥tí♥✉❛s ❡♠ U t♦❞❛s ❛s ❞❡r✐✈❛❞❛s ♣❛r❝✐❛✐s ❞❡ f ❞❡ ♦r❞❡♠ ≤ k✳ ❙❡❥❛♠ U ❡ V
❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❞❡Rn❡Rp✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♠ ❣r❛♥❞❡ ♣❛rt❡ ❞♦ tr❛❜❛❧❤♦ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦✱ q✉❛♥❞♦ ♥ã♦ é ❞✐t♦ ❝♦♥trár✐♦✱ ❛♣❧✐❝❛çõ❡sf :U →V s✉❛✈❡s✱ ♦✉ C∞✱ ✐st♦
é✱ q✉❡ ♣♦ss✉✐ ❞❡r✐✈❛❞❛s ❞❡ t♦❞❛s ❛s ♦r❞❡♥s✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛ f : Rn → Rp ✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡✳ ❉✐③❡♠♦s q✉❡ x ∈Rn é ✉♠ ♣♦♥t♦ s✐♥❣✉❧❛r ❞❡ f s❡✱ ♦ ♣♦st♦ ❞❛ ♠❛tr✐③ ❥❛❝♦❜✐❛♥❛ ❞❡ f ♥♦ ♣♦♥t♦ x✱
Jf(x) =
∂fi
∂xj
(x)
, 1≤i≤p, 1≤j ≤n,
♥ã♦ é ♠á①✐♠♦✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞✐③❡♠♦s q✉❡ x é ✉♠ ♣♦♥t♦ r❡❣✉❧❛r ❞❡ f✳ ❖ ♣♦♥t♦ x
t❛♠❜é♠ ♣♦❞❡ s❡r ❝❤❛♠❛❞♦ ❞❡ ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ❞❡ f✳
➱ ❝❧❛r♦ q✉❡ ✉♠ ♣♦♥t♦ s❡r ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ é ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❧♦❝❛❧✳ ◆❡st❡ tr❛❜❛❧❤♦ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❛♣❧✐❝❛çõ❡s q✉❡ t❡♠ ✉♠ s✐♥❣✉❧❛r✐❞❛❞❡ ♥❛ ♦r✐❣❡♠✳ P♦r ❡st❡ ♠♦t✐✈♦ ✐♥tr♦❞✉③✐♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✿
❉❡✜♥✐çã♦ ✶✳✷✳ ❉❛❞❛s ❞✉❛s ❛♣❧✐❝❛çõ❡s s✉❛✈❡s f1 :U1 →Rp ❡ f2 :U2 →Rp✱ ♦♥❞❡ U1✱
U2 ⊂ Rn✱ ❝♦♠ x ∈ U1 ❡ x ∈ U2✳ ❉✐③❡♠♦s q✉❡ f1 ∼ f2 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ⊂U1∩U2 ❞❡ x t❛❧ q✉❡ f1(x) = f2(x)✱ ∀ x∈U
❆s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s♦❜r❡ ❡ss❛ r❡❧❛çã♦ sã♦ ❝❤❛♠❛❞❛s ❞❡germes❞❡ ❛♣❧✐❝❛çõ❡s
❡♠ x✳ ❉❡♥♦t❡♠♦s ♦ ❣❡r♠❡ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ f : Rn → Rp ❡♠ x ♣♦r f : (Rn, x) →
(Rp, y)✱ ♦♥❞❡y=f(x)✳ ❉✐③❡♠♦s q✉❡x❡ysã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❢♦♥t❡ ❡ ♠❡t❛ ❞♦ ❣❡r♠❡✳
P❛r❛ ❝❛❞❛ ❣❡r♠❡ f : (Rn, x)→(Rp, y)✱ ❛ss♦❝✐❛♠♦s ❛ s✉❛ ❞❡r✐✈❛❞❛df
x :Rn →Rp q✉❡
é ❞❡✜♥✐❞♦ ❝♦♠♦ s❡♥❞♦ ❛ ❞❡r✐✈❛❞❛ ❡♠ x ❞❡ q✉❛❧q✉❡r ✉♠ r❡♣r❡s❡♥t❛♥t❡✳ ❯♠ ❣❡r♠❡ é
✐♥✈❡rtí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ s✉❛ ❞❡r✐✈❛❞❛ é ✐♥✈❡rtí✈❡❧✳ ❖ ♣♦st♦ ❞❡ ✉♠ ❣❡r♠❡ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ♣♦st♦ ❞❡ s✉❛ ❞❡r✐✈❛❞❛ ❡♠ x✳ ◗✉❛♥❞♦ ♦ ♣♦st♦ ❞❡ f : Rn → Rp é ✐❣✉❛❧ ❛ n
❞✐③❡♠♦s q✉❡ ♦ ❣❡r♠❡ é ✉♠❛ ✐♠❡rsã♦✳ ◆♦ ❝❛s♦ ❡♠ ♦ ♣♦st♦ é ✐❣✉❛❧ ❛ p✱ ❞✐③❡♠♦s q✉❡ ♦
❣❡r♠❡ é ✉♠❛ s✉❜♠❡rsã♦✳
❉❡✜♥✐çã♦ ✶✳✸✳ ❉♦✐s ❣❡r♠❡sf : (Rn, x
1)→(Rp, y1)❡g : (Rn, x2)→(Rp, y2)sã♦ ❡q✉✐✲ ✈❛❧❡♥t❡s q✉❛♥❞♦ ❡①✐st❡♠ ❣❡r♠❡s ✐♥✈❡rtí✈❡✐s h : (Rn, x1) → (Rn, x2) ❡ k : (Rp, y1) →
(Rp, y2) ♣❛r❛ ♦s q✉❛✐s ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛✱
(Rn, x
1)
f
/
/
h
(Rp, y
1)
k
(Rn, x2)
g
/
/(Rp, y 2),
✐st♦ é✱ k◦f =g◦h✳
❉❡♥♦t❛♠♦s ♣♦r En,p ♦ ❝♦♥❥✉♥t♦ ❞♦s ❣❡r♠❡s ❞❡ ❛♣❧✐❝❛çõ❡s f : (Rn,0) → Rp ❞❡
❝❧❛ss❡ C∞✳ ◗✉❛♥❞♦ p = 1✱ ❡st❡ ❝♦♥❥✉♥t♦ é ❞❡♥♦t❛❞♦ ♣♦r E
n✳ ❖❜s❡r✈❡♠♦s q✉❡ εn é
✉♠ ❛♥❡❧ ❧♦❝❛❧ ❝✉❥♦ ✐❞❡❛❧ ♠❛①✐♠❛❧ é mn:={f ∈ En;f(0) = 0}✳ ❆❧é♠ ❞✐ss♦ é ♣♦ssí✈❡❧
✈❡r✐✜❝❛r q✉❡ mn é ♦ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦rx1, ..., xn✳
❉❡✜♥✐çã♦ ✶✳✹✳ ❙❡❥❛♠ f✱ g ∈ En✳ ❉✐③❡♠♦s q✉❡ ❢ ❡ ❣ sã♦ R✲❡q✉✐✈❛❧❡♥t❡s ❡ ❞❡♥♦t❛♠♦s
♣♦r f ∼ g s❡ ❡①✐st❡ ✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ h : (Rn,0) → (Rn,0) t❛❧ q✉❡ f =
g◦h−1✳
P❛r❛ ♥♦ss♦ ❡st✉❞♦ é ✐♠♣♦rt❛♥t❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ ❛♥á❧✐s❡ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✳
❉❡✜♥✐çã♦ ✶✳✺✳ ❙❡❥❛ f : Rn → R ✉♠❛ ❢✉♥çã♦ s✉❛✈❡✳ ❯♠ ♣♦♥t♦ x0 ❡♠ Rn é ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ ❞❡❣❡♥❡r❛❞♦ s❡ x0 é ✉♠ ♣♦♥t♦ s✐♥❣✉❧❛r ❞❡ f ❡ ❛ ❍❡ss✐❛♥❛✱ q✉❡ é ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③
∂2f
∂xi∂xj
(x0)
, 1≤i, j ≤n,
é ♥ã♦ ♥✉❧♦✳
❉❡✜♥✐çã♦ ✶✳✻✳ ❯♠❛ ❢✉♥çã♦ s✉❛✈❡ f :Rn →R é ❞✐t❛ s❡r ✉♠❛ ✉♠❛ ❢✉♥çã♦ ❞❡ ▼♦rs❡ s❡ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s s✐♥❣✉❧❛r❡s sã♦ ♣♦♥t♦s ❝rít✐❝♦s ♥ã♦ ❞❡❣❡♥❡r❛❞♦s✳
❖❜s❡r✈❛çã♦ ✶✳✼✳ ◆♦t❡ q✉❡ ✉♠❛ ❢✉♥çã♦ r❡❣✉❧❛r f : Rn → R é t❛♠❜é♠ ✉♠❛ ❢✉♥çã♦
❞❡ ▼♦rs❡✳
➱ ❜❡♠ ❝♦♥❤❡❝✐❞♦ ❞♦ ❝á❧❝✉❧♦ q✉❡ ❛s ❢✉♥çõ❡s ❞❡ ▼♦rs❡ ❞❡s❡♠♣❡♥❤❛♠ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ❡♠ s✉❛s ❛♣❧✐❝❛çõ❡s ❡ ♣♦ss✉❡♠ ✉♠❛ ❢♦r♠❛ ♥♦r♠❛❧ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ ❞❡❣❡♥❡r❛❞♦ ❝♦♠♦ ✈❡r❡♠♦s ❛ s❡❣✉✐r✳
▲❡♠❛ ✶✳✽✳ ❙❡❥❛ f : (Rn, x0)→R ✉♠ ❣❡r♠❡ s✉❛✈❡✳ ❊♥tã♦✿
✶✮ ❙❡ x0 é ✉♠ ♣♦♥t♦ r❡❣✉❧❛r ❞❡ f✱ ❡♥tã♦ ♦ ❣❡r♠❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ π : (Rn,0) → R✱ ❞❛❞❛ ♣♦r π(x1, ..., xn) = x1✳
✷✮ ❙❡ x0 ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ♥ã♦ ❞❡❣❡♥❡r❛❞♦ ❞❡ f✱ ❡♥tã♦ ♦ ❣❡r♠❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ g :
(Rn,0)→R✱ ❞❛❞♦ ♣♦r
g(x) =x21+x22· · ·+x2λ−x2λ+1− · · · −x2n.
❉❡♥♦t❛r❡♠♦s ♣♦r Pk(Rn,Rp) ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ ❞❛s ❛♣❧✐❝❛çõ❡s f : Rn → Rp
t❛❧ q✉❡ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ fi ❞❡ f = (f1, f2..., fp) é ✉♠ ♣♦❧✐♥ô♠✐♦ ❞❡ grau 6 k ♥❛s
❝♦♦r❞❡♥❛❞❛s x1, x2, ..., xn ❞❡ Rn ❝♦♠ t❡r♠♦ ❝♦♥st❛♥t❡ ♥✉❧♦✳ ❆ ♥♦çã♦ ❞❡ ❡s♣❛ç♦ ❞❡ ❦✲
❥❛t♦ ❞❡ ❛♣❧✐❝❛çõ❡s s✉❛✈❡s é ✐♥tr♦❞✉③✐❞❛ ❡♠ ❬✶✶❪✳ ◆❡st❡ tr❛❜❛❧❤♦ ✉t✐❧✐③❛♠♦s ❛ s❡❣✉✐♥t❡ ✐❞❡♥t✐✜❝❛çã♦✿
Pr♦♣♦s✐çã♦ ✶✳✾✳ ❙❡❥❛ Jk(Rn,Rp) ♦ ❡s♣❛ç♦ ❞♦s ❦✲❥❛t♦s✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦
❝❛♥ô♥✐❝❛ ❡♥tr❡ ♦ ❡s♣❛ç♦ ❞❡ ❦✲❥❛t♦s ❡ ♦ ❝♦♥❥✉♥t♦ Rn×Rm×Pk(Rn,Rp).
❉❡✜♥✐çã♦ ✶✳✶✵✳ P❛r❛ ❝❛❞❛ ❛♣❧✐❝❛çã♦f = (f1, f2, ..., fp)∈C∞(Rn,Rp)❡ ❝❛❞❛a∈Rn✱
❞❡✜♥✐♠♦s ❛ ❛♣❧✐❝❛çã♦
Jkf :Rn −→Jk(Rn,Rp)
a 7−→Jkf(a) = (a, f(a), P
1(a), ..., Pn(a)),
✭✶✳✶✮
♦♥❞❡ Pi(a) é ♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ❚❛②❧♦r ❞❛ ❢✉♥ç❛♦ fi ❞❡ ♦r❞❡♠ k ❡♠ a✱ s❡♠ ♦ t❡r♠♦
❝♦♥st❛♥t❡✳
❉❡♥♦t❛r❡♠♦s ♣♦r jkf(a) = (P
1(a), ..., Pn(a))✳ ❆ ❛♣❧✐❝❛çã♦ Jkf é ❞❡ ❝❧❛ss❡ C∞ ❡
jkf(a)é ❝❤❛♠❛❞♦ ♦ ❦✲❥❛t♦ ❞❡ f ❡♠ ❛✳
❊①❡♠♣❧♦ ✶✳✶✶✳ ❙❡❥❛ f :R→R ✉♠❛ ❢✉♥çã♦ s✉❛✈❡✳ ◆❡st❡ ❝❛s♦ t❡♠♦s q✉❡✿
jkf(a) = f′
(a)x+f
′′
(a) 2! x
2+· · ·+ fk(a)
k! x
k,
❡ Jkf(a) ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❡s♣❛ç♦Rk+2 ❝♦♠ ❛ ❝♦rr❡s♣♦♥❞ê♥✲ ❝✐❛
(a, f(a), f′(a) + f
′′
(a) 2! x
2+· · ·+fk(a)
k! x
k)↔(a, f(a), f′
(a),f
′′
(a) 2! ,· · · ,
fk(a)
k! ).
❆♦ ❝♦♥❥✉♥t♦ C∞(Rn,Rp) ✈❛♠♦s ❛ss♦❝✐❛r ✉♠❛ t♦♣♦❧♦❣✐❛✱ ❝❤❛♠❛❞❛ ❚♦♣♦❧♦❣✐❛ ❞❡
❲❤✐t♥❡②✳
❉❡✜♥✐çã♦ ✶✳✶✷ ✭❚♦♣♦❧♦❣✐❛ ❞❡ ❲❤✐t♥❡②✮✳ ❙❡❥❛ f ∈ C∞(Rn,Rp)✳ ❯♠❛ ❜❛s❡ ♣❛r❛ ❛
t♦♣♦❧♦❣✐❛ ❞❡ Ck ❞❡ ❲❤✐t♥❡② ❞❡ C∞(Rn,Rp) é ❞❛❞❛ ♣❡❧♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s
V(f, δ) ={g ∈C∞(Rn,Rp);Jkg(x)−Jkf(x)< δ(x)}, ♦♥❞❡ δ :Rn →R é ❝♦♥tí♥✉❛ ❡ ♣♦s✐t✐✈❛✳
❆ ❚♦♣♦❧♦❣✐❛ C∞ ❞❡ ❲❤✐t♥❡② C∞(Rn,Rp) t❡♠ ❝♦♠♦ ❜❛s❡ ❛ ✉♥✐ã♦ ❞❡ t♦❞♦s ♦s
❛❜❡rt♦s ❞❛s t♦♣♦❧♦❣✐❛s Ck ❞❡ ❲❤✐t♥❡②✱ ❝♦♠ k ≥0✳
❉❡✜♥✐çã♦ ✶✳✶✸✳ ❯♠ ❣❡r♠❡ f ∈ En é ❦✲❞❡t❡r♠✐♥❛❞♦ s❡ ♣❛r❛ q✉❛❧q✉❡r ❣❡r♠❡ g ∈ En
❝♦♠ jk(g)(0) =jk(f)(0) t❡♠♦s q✉❡ ❢ é R✲❡q✉✐✈❛❧❡♥t❡ ❛ ❣✳
❯♠ ❣❡r♠❡ f ∈ En é ✜♥✐t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦ s❡ ❡①✐st✐r ✉♠ k ∈ N t❛❧ q✉❡ ❢ s❡❥❛
❦✲❞❡t❡r♠✐♥❛❞♦✳
❉❡✜♥✐çã♦ ✶✳✶✹✳ ❙❡❥❛ f : (Rn,0)→(Rn,0) ✉♠ ❣❡r♠❡ t❛❧ q✉❡ 0 é ✐s♦❧❛❞♦ ❞❡ f−1(0)✳ ❆ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ µ0[f] ❞❡ f ❡♠ 0 é ❞❡✜♥✐❞❛ ♣♦r
µ0[f] = dimR[En/hf1, . . . , fni],
♦♥❞❡ hf1, . . . , fni é ♦ ✐❞❡❛❧ ❣❡r❛❞♦ ♣❡❧❛s ❝♦♠♣♦♥❡♥t❡s fi ❞❡ f ❡♠ 0✳ ❉✐③❡♠♦s q✉❡ f é
✜♥✐t♦ s❡ µ0[f]<∞✳
❉❛❞❛ ✉♠❛ ❛♣❧✐❝❛çã♦ g : Rn → Rn✱ ♦♥❞❡ g = (g
1, . . . , gn) ❝♦♠ ❝❛❞❛ gi s❡♥❞♦ ✉♠
♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ t❛❧ q✉❡0é ✐s♦❧❛❞♦ ❡♠ g−1(0)✱ t❡♠♦s q✉❡ µ
0[g] =Qni=1di✱ ♦♥❞❡
di é ♦ ❣r❛✉ ❞❡ ❝❛❞❛ gi✳
Pr♦♣♦s✐çã♦ ✶✳✶✺ ✭❬✶✻❪✮✳ ❙❡❥❛ f : (Rn,0) → (Rn,0)✱ ✉♠ ❣❡r♠❡ ✜♥✐t♦✳ ❈♦♥s✐❞❡r❡
f = (f1, . . . , fn) ❡ fi = fiki +q ✱ ♦♥❞❡ f ki
i é ❛ ♣❛rt❡ ❤♦♠♦❣ê♥❡❛ ❞❡ fi ❝♦♠ ❣r❛✉ ki ❡
jkiq(0) = 0✳ ❊♥tã♦✿ ✐✮ µ0[f]≥Qni=1ki✳
✐✐✮ µ0[f] =Qni=1ki s❡✱ ❡ s♦♠❡♥t❡✱ s❡ ♦ s✐st❡♠❛ fiki = 0♣❛r❛ i = 1, . . . , n t❡♠ ❛♣❡♥❛s
s♦❧✉çã♦ tr✐✈✐❛❧ ❡♠ Cn✳
❉❡✜♥✐çã♦ ✶✳✶✻✳ ❙❡❥❛ f : (Rn,0)→R ✉♠❛ ❣❡r♠❡✳ ❆Re✲❝♦❞✐♠❡♥sã♦ ❞❡ f✱ ❞❡♥♦t❛❞❛ ♣♦r cod(f,Re) é ❞❡✜♥✐❞❛ ❝♦♠♦✿
cod(f,Re) = µ0[∇f].
❆ Re✲❝♦❞✐♠❡♥sã♦✱ q✉❡ ❢♦✐ ❞❡✜♥✐❞❛ ❛❝✐♠❛✱ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✽❪✳
Pr♦♣♦s✐çã♦ ✶✳✶✼ ✭❬✽❪✮✳ ❙❡❥❛♠ ❞♦✐s ❣❡r♠❡s ❢ ❡ ❣ ❡♠ En✳ ❚❡♠♦s q✉❡✱
✐✮ ❙❡ ❢ ❡ ❣ sã♦ R✲❡q✉✐✈❛❧❡♥t❡s ❡♥tã♦ cod(f,Re) = cod(g,Re)✳
✐✐✮ cod(f,Re) = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ 0 é ✉♠ ✈❛❧♦r r❡❣✉❧❛r ❞❡ f✳
✶✳✶✳✶ ❈❧❛ss✐✜❝❛çã♦ ❞♦s ❣❡r♠❡s ❞❡ ❝♦❞✐♠❡♥sã♦
6
5
❉❡✜♥✐çã♦ ✶✳✶✽✳ ❯♠ ❣❡r♠❡ f ∈ m2
n ✭✐st♦ é✱ ❛ ♦r✐❣❡♠ é ✉♠ ♣♦♥t♦ s✐♥❣✉❧❛r✮ é ♥ã♦
❞❡❣❡♥❡r❛❞♦ q✉❛♥❞♦ ❛ ♠❛tr✐③ ❍❡ss✐❛♥❛ Hf =
∂2
f ∂xi∂xj(0)
é ♥ã♦ s✐♥❣✉❧❛r✳ ▲❡♠❛ ✶✳✶✾ ✭▲❡♠❛ ❞❡ ▼♦rs❡✮✳ ❙❡❥❛ f ∈m2
n✳ ❊♥tã♦✱ cod(f,Re) = 1 s❡✱ ❡ s♦♠❡♥t❡ s❡✱
f é ♥ã♦ ❞❡❣❡♥❡r❛❞♦✳ ◆❡st❡ ❝❛s♦ ❢ s❡rá R✲❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❣❡r♠❡ ❞❛ ❢♦r♠❛
x21+...+x2s−x2s+1−...−x2n.
❉❡✜♥✐çã♦ ✶✳✷✵✳ ❙❡❥❛♠ f ∈m2
n ❡ cod(f,Re)≥2✳ ❉✐③❡♠♦s q✉❡ ❢ t❡♠ ❝♦♣♦st♦ ❝ s❡ ♦
♣♦st♦ ❞❛ ♠❛tr✐③ ❍❡ss✐❛♥❛ é n−c✳
❖❜s❡r✈❛çã♦ ✶✳✷✶✳ ❖ ❝♦♣♦st♦ ❞❛s ❢✉♥çõ❡s ❞❡ ▼♦rs❡ é ♥✉❧♦✳ ▲❡♠❛ ✶✳✷✷ ✭▲❡♠❛ ❞❛ ❙❡♣❛r❛çã♦✮✳ ❙❡❥❛ f ∈m2
n ✉♠ ❣❡r♠❡ ✜♥✐t❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦
❞❡ ❝♦♣♦st♦ ❝✳ ❊♥tã♦✱ ❢ é R✲❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❣❡r♠❡
(x1, ..., xn)→g(x1, ..., xc)±x2c+1±...±x2n,
❝♦♠ g ∈m3
c.
Pr♦♣♦s✐çã♦ ✶✳✷✸✳ ❙❡❥❛♠ f ∈ m2
n ❞❡ ❝♦♣♦st♦ 1 ❡ cod(f,Re) = k✳ ❊♥tã♦✱ ❢ é R✲
❡q✉✐✈❛❧❡♥t❡ ❛♦ ❣❡r♠❡
(x1, ..., xk)→ ±xk1+1±x22±...±x2n.
❊st❡ ❣❡r♠❡ é ❝❤❛♠❛❞♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡ Ak✳
❉❡♠♦♥str❛çã♦✿ ❱❡r r❡❢❡rê♥❝✐❛ ❬✽❪ ▲❡♠❛ ✶✳✷✹✳ ❙❡❥❛ f ∈ m2
n ✉♠ ❣❡r♠❡ ❞❡ Re✲❝♦❞✐♠❡♥sã♦ ✜♥✐t❛ ❡ ❞❡ ❝♦♣♦st♦ ❝✱ ❡♥tã♦
cod(f,Re)≥ c(c2+1) + 1✳
P❡❧♦ ❧❡♠❛ ❛❝✐♠❛✱ t❡♠♦s q✉❡ ♣❛r❛ ❝❧❛ss✐✜❝❛r ♦s ❣❡r♠❡s ❞❡ ❝♦❞✐♠❡♥sã♦≤5✱ ❝♦♥s✐❞❡r❛✲
s❡ ❛♣❡♥❛s ♦s ❣❡r♠❡s ❞❡ ❝♦♣♦st♦ ≤2✳
Pr♦♣♦s✐çã♦ ✶✳✷✺ ✭❬✽❪✮✳ ❙❡❥❛ f ∈ m2
n ❞❡ ❝♦♣♦st♦ ✷ ❡ cod(f,Re) ≤ 5✳ ❊♥tã♦✱ ❢ é
❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❣❡r♠❡s
±(x31−x1x22)±x23±...±x2n
±(x31+x32)±x23±...±x2n
±(x21x2+x42)±x23±...±x2n.
❚❡♦r❡♠❛ ✶✳✷✻ ✭❚❡♦r❡♠❛ ❞❡ ❚❤♦♠✮✳ ❙❡❥❛ f ∈m2
n ❞❡ ♠♦❞♦ q✉❡ 1≤ cod(f, Re)≤ 5✳
❊♥tã♦✱ ❛ ♠❡♥♦s ❞❛ s♦♠❛ ❞❡ ✉♠❛ ❢♦r♠❛ q✉❛❞rát✐❝❛ ♥❛s ♦✉tr❛s ✈❛r✐á✈❡✐s✱ ❡ ♠✉❧t✐♣❧✐✲ ❝❛çã♦ ♣♦r ±1✱ ❢ é R✲❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❞♦s s❡❣✉✐♥t❡s ❣❡r♠❡s ❧✐st❛❞♦s ♥❛ t❛❜❡❧❛ ❛❜❛✐①♦✳
❙í♠❜♦❧♦ ◆♦♠❡ ●❡r♠❡ ❈♦♣♦st♦ cod(f,Re)
A1 ▼♦rs❡ x2 ✵ ✶
A2 ❉♦❜r❛ x3 ✶ ✷
A3 ❈ús♣✐❞❡ x4 ✶ ✸
A4 ❘❛❜♦ ❞❡ ❛♥❞♦r✐♥❤❛ x5 ✶ ✹
A5 ❇♦r❜♦❧❡t❛ x6 ✶ ✺
D4− ❯♠❜í❧✐❝♦ ❡❧í♣t✐❝♦ (x3−xy2) ✷ ✹
D4+ ❯♠❜í❧✐❝♦ ❤✐♣❡r❜ó❧✐❝♦ (x3+y3) ✷ ✹
D±5 ❯♠❜í❧✐❝♦ ♣❛r❛❜ó❧✐❝♦ (x2y+y4) ✷ ✺
❚❛❜❡❧❛ ✶✳✶✿ s✐♥❣✉❧❛r✐❞❛❞❡s
✶✳✷ ❱❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛
❚r❛♥s✈❡rs❛❧✐❞❛❞❡ é ✉♠❛ ✐❞é✐❛ ✐♠♣♦rt❛♥t❡ ❡ ♣r♦❢✉♥❞❛ ♥♦ ❡st✉❞♦ ❞❛ t❡♦r✐❛ ❞❛s s✐♥❣✉❧❛r✐❞❛❞❡s✳ ●r❛♥❞❡s r❡s✉❧t❛❞♦s s♦❜r❡ ❣❡♥❡r✐❝✐❞❛❞❡ ❡♠ ❝♦♥❥✉♥t♦s ❢♦r❛♠ ♦❜t✐❞♦s ❝♦♠❜✐♥❛❞♦✲s❡ ♦s t❡♦r❡♠❛s ❞❡♠♦♥str❛❞♦s ♣♦r ❘❡♥é ❚❤♦♠ ❝♦♠ ❛ ✐❞é✐❛ ❞❡ tr❛♥s✈❡r✲ s❛❧✐❞❛❞❡ ❡♥tr❡ s✉❜✈❛r✐❞❛❞❡s✳ ◆❡st❡ tr❛❜❛❧❤♦ ❛ tr❛♥s✈❡rs❛❧✐❞❛❞❡ ❛♣❛r❡❝❡rá ❞✐✈❡rs❛s ✈❡③❡s✳
❉❡✜♥✐çã♦ ✶✳✷✼✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ n é ✉♠ ❝♦♥❥✉♥t♦ M ❡
✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❛♣❧✐❝❛çõ❡s ❜✐✉♥í✈✉❝❛s xα :Uα ⊂Rn→M ❞❡ ❛❜❡rt♦s Uα ❞❡ Rn ❡♠ M
t❛✐s q✉❡✿
✶✳ S
α
xα(Uα) = M.
✷✳ P❛r❛ t♦❞♦ ♣❛r α✱ β ❝♦♠ xα(Uα)∩xβ(Uβ) = W 6= ∅✱ ♦s ❝♦♥❥✉♥t♦s xα−1(W) ❡
xβ−1(W) sã♦ ❛❜❡rt♦s ❡♠ Rn ❡ ❛s ❛♣❧✐❝❛çõ❡s xβ−1◦xα sã♦ s✉❛✈❡s✳
✸✳ ❆ ❢❛♠í❧✐❛ {(Uα, xα)} é ♠á①✐♠❛ r❡❧❛t✐✈❛♠❡♥t❡ às ❝♦♥❞✐çõ❡s 1 ❡ 2✳
❖ ♣❛r (Uα, xα) ✭♦✉ ❛♣❧✐❝❛çã♦ xα✮ ❝♦♠ p ∈ xα(Uα) é ❝❤❛♠❛❞♦ ❞❡ ♣❛r❛♠❡tr✐③❛çã♦
✭♦✉ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✮ ❞❡ M ❡♠ p❀ xα(Uα) é ❡♥tã♦ ❝❤❛♠❛❞❛ ✉♠❛ ✈✐③✐♥❤❛♥ç❛
❝♦♦r❞❡♥❛❞❛ ❞❡p✳ ❯♠❛ ❢❛♠í❧✐❛{(Uα, xα)}s❛t✐s❢❛③❡♥❞♦1❡2é ❝❤❛♠❛❞❛ ✉♠❛ ❡str✉t✉r❛
❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ M✳
❉❡✜♥✐çã♦ ✶✳✷✽✳ ❙❡❥❛♠M1 ❡ M2 ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ϕ:M1 →
M2 é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ p ∈ M1 s❡ ❞❛❞❛ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ y : V ⊂ Rp → M2 ❡♠
ϕ(p) ❡①✐st❡ ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ x:U ⊂Rn →M
1 ❡♠ p t❛❧ q✉❡ ϕ(x(U))⊂y(V) ❡ ❛ ❛♣❧✐❝❛çã♦
y−1◦ϕ◦x:U ⊂Rn →Rp é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ x−1(p)✳
❉❡✜♥✐çã♦ ✶✳✷✾✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ α: (−ε, ε)→M é ❝❤❛♠❛❞❛ ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ M✳ ❙✉♣♦♥❤❛ q✉❡α(0) =p∈
M✱ ❡ s❡❥❛ D ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❞❡ M ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ p✳ ❖ ✈❡t♦r t❛♥❣❡♥t❡ à
❝✉r✈❛ α ❡♠ t = 0 é ❛ ❢✉♥çã♦ α′(0) : D →R ❞❛❞❛ ♣♦r
α′(0)f = d(f◦α)
dt |t=0 f ∈ D.
❯♠ ✈❡t♦r t❛♥❣❡♥t❡ ❡♠pé ♦ ✈❡t♦r t❛♥❣❡♥t❡ ❡♠t= 0 ❞❡ ❛❧❣✉♠❛ ❝✉r✈❛α: (−ε, ε)→M
❝♦♠ α(0) =p✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s ❛ M ❡♠ p s❡rá ✐♥❞✐❝❛❞♦ ♣♦r TpM✳
❖ ❝♦♥❥✉♥t♦ TpM✱ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ✉s✉❛✐s ❞❡ ❢✉♥çõ❡s✱ ❢♦r♠❛ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧
❞❡ ❞✐♠❡♥sã♦ n ❡ é ❝❤❛♠❛❞♦ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❞❡ M ❡♠ p✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s
✈❡❥❛ ❬✹❪✳
❖❜s❡r✈❛çã♦ ✶✳✸✵✳ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ s❡❥❛ T M = {(p, v);p ∈
M, v ∈TpM}✳ ❖ ❝♦♥❥✉♥t♦ T M ♠✉♥✐❞♦ ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ s❡rá ❝❤❛♠❛❞♦
✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡ M✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡❥❛ ❬✹❪✳
❉❡✜♥✐çã♦ ✶✳✸✶✳ ❙❡❥❛♠M ❡ N ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✲
✈❡❧ ϕ : M → N é ✉♠❛ ✐♠❡rsã♦ s❡ dϕp :TpM → Tϕ(p)N é ✐♥❥❡t✐✈❛ ♣❛r❛ t♦❞♦ p∈ M✳ ❙❡✱ ❛❧é♠ ❞✐ss♦✱ ϕ é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ ϕ(M)⊂N✱ ♦♥❞❡ ϕ(M)t❡♠ ❛ t♦♣♦❧♦❣✐❛
✐♥❞✉③✐❞❛ ♣♦r N✱ ❞✐③✲s❡ q✉❡ ϕ é ✉♠ ♠❡r❣✉❧❤♦✳ ❙❡ M ⊂N ❡ ❛ ✐♥❝❧✉sã♦ i: M →N é
✉♠ ♠❡r❣✉❧❤♦✱ ❞✐③✲s❡ q✉❡ M é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ N✳
❉❡✜♥✐çã♦ ✶✳✸✷✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ✉♠❛
❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ ❛ ❝❛❞❛ ♣♦♥t♦ p ∈ M ❛ss♦❝✐❛ ✉♠ ✈❡t♦r X(p) ∈ TpM✳ ❊♠ t❡r✲
♠♦s ❞❡ ❛♣❧✐❝❛çõ❡s✱ X é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ M ♥♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ T M✳ ❖ ❝❛♠♣♦ é
❞✐❢❡r❡♥❝✐á✈❡❧ s❡ ❛ ❛♣❧✐❝❛çã♦ X :M →T M é ❞✐❢❡r❡♥❝✐á✈❡❧✳
Pr♦♣♦s✐çã♦ ✶✳✸✸✳ ❙❡❥❛♠ U ⊂ Rm+n ❛❜❡rt♦ ❡ f : U → Rn ✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦
M ={p∈U;f(p) =c e dfp :Rn+m →Rn sobrejetora}
❊♥tã♦✱
✭✐✮ M é ❛❜❡rt♦ ❡♠ f−1(c)✳
✭✐✐✮ ❙✉♣♦♥❞♦ q✉❡M é ♥ã♦ ✈❛③✐♦✱ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ s✉❛✈❡ ❞❡ ❞✐♠❡♥sã♦ m ❞♦Rm+n✱ ❡
✭✐✐✐✮ TpM = ker (df)p ♣❛r❛ t♦❞♦ p∈M✳
❉❡✜♥✐çã♦ ✶✳✸✹✳ ❙❡❥❛♠ M ❡ N s❡♥❞♦ ✈❛r✐❡❞❛❞❡s s✉❛✈❡s ❡ f : M → N s❡♥❞♦ ✉♠❛
❛♣❧✐❝❛çã♦ s✉❛✈❡✳ ❈♦♥s✐❞❡r❡ S s❡♥❞♦ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ N ❡ s❡❥❛ x∈ M✳ ❊♥tã♦ f
✐♥t❡rs❡❝t❛ S tr❛♥s✈❡rs❛❧♠❡♥t❡ ❡♠ x s❡❀
✐✮ f(x)∈/ S ♦✉
✐✐✮ f(x)∈S ❡ (df)x(TxM) +Tf(x)S =TxN✳
♦♥❞❡ TxM é ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ à M ❡♠ x✳
❉✐r❡♠♦s q✉❡ f é tr❛♥s✈❡rs❛❧ ❛ S✱ ❞❡♥♦t❛❞♦ ♣♦r f ⋔S✱ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦x∈M✱ f ❢♦r tr❛♥s✈❡rs❛❧ ❛ S ♥❛ ♣♦♥t♦ x✳
❚❡♦r❡♠❛ ✶✳✸✺✳ ✭❚r❛♥s✈❡rs❛❧✐❞❛❞❡ ❞❡ ❚❤♦♠✮ P❛r❛ t♦❞❛ s✉❜✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ S ❞❡ Jk(Rn,Rp)✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❛♣❧✐❝❛çõ❡s F ❡♠ C∞(Rn,Rp) t❛❧ q✉❡ jkF ⋔S é ❛❜❡rt♦ ❡✱
♣♦rt❛♥t♦ ❞❡♥s♦ ♥❛ Cr✲t♦♣♦❧♦❣✐❛✱q✉❛❧q✉❡r q✉❡ s❡❥❛ r >k+ 1✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ t❡♦r❡♠❛ ❞❡ tr❛♥s✈❡rs❛❧✐❞❛❞❡ ❞❡ ❚❤♦♠✱ t❡♠♦s ♦s s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
▲❡♠❛ ✶✳✸✻✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ❞❡ ▼♦rs❡ é ❞❡♥s♦ ❡♠ C∞(Rn,R)✳
❉❡✜♥✐çã♦ ✶✳✸✼✳ ❯♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ✭♦✉ ❡str✉t✉r❛ ❘✐❡♠❛♥♥✐❛♥❛✮ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ p ❞❡ M
✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h,ip ✭✐st♦ é✱ ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r s✐♠étr✐❝❛✱ ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✮ ♥♦
❡s♣❛ç♦ t❛♥❣❡♥t❡ TpM✱ q✉❡ ✈❛r✐❛ ❞✐❢❡r❡♥❝✐❛✈❡❧♠❡♥t❡ ♥♦ s❡❣✉✐♥t❡ s❡♥t✐❞♦✿ P❛r❛ t♦❞♦
♣❛r X ❡ Y ❞❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V ❞❡ ▼✱ ❛ ❢✉♥çã♦
hX, Yi é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ V✳
❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ ✉♠❛ ❞❛❞❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❝❤❛♠❛✲s❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛✳
❆s ❞❡✜♥✐çõ❡s ❡ ♦s r❡s✉❧t❛❞♦s s♦❜r❡ ❝♦♥❡①ã♦ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✹❪✳ ■♥❞✐❝❛r❡♠♦s ♣♦r X(M) ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❞❡ ❝❧❛ss❡C∞ ❡♠ M✳
❉❡✜♥✐çã♦ ✶✳✸✽✳ ❯♠❛ ❝♦♥❡①ã♦ ❛✜♠ ∇ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ✉♠❛
❛♣❧✐❝❛çã♦
∇:X(M)× X(M)→ X(M)
(X, Y) 7→ ∇XY
q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✐✮ ∇f X+gYZ =f∇XZ +g∇YZ,
✐✐✮ ∇X(Y +Z) =∇XY +∇XZ,
✐✐✐✮ ∇X(f Y) =f∇XY +X(f)Y,
♦♥❞❡ X, Y, Z ∈ X(M) ❡ f, g∈ D(M)✳
❉❡✜♥✐çã♦ ✶✳✸✾✳ ❙❡❥❛♠ X✱ Y ∈ X(Rn) ❡ p∈Rn✱ ❛ ❝♦♥❡①ã♦ ❡♠ Rn s❡rá ❞❛❞❛ ♣♦r
(∇XY)(p) = (dY)p(X(p)).
❈♦r♦❧ár✐♦ ✶✳✹✵✳ ❯♠❛ ❝♦♥❡①ã♦ ∇ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ M é ❝♦♠♣❛tí✈❡❧
❝♦♠ ❛ ♠étr✐❝❛ s❡ ❡ só s❡
XhY, Zi=h∇XY, Zi+hY,∇XZi, X, Y, Z ∈ X(M).
❉❡✜♥✐çã♦ ✶✳✹✶✳ ❯♠❛ ❝♦♥❡①ã♦ ❛✜♠ ∇ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ❞✐t❛
s✐♠étr✐❝❛ q✉❛♥❞♦
∇XY − ∇YX = [X, Y]
♣❛r❛ t♦❞♦ X, Y ∈ X(M)✳
❚❡♦r❡♠❛ ✶✳✹✷ ✭▲❡✈✐✲❈✐✈✐t❛✮✳ ❉❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ M✱ ❡①✐st❡ ✉♠❛
ú♥✐❝❛ ❝♦♥❡①ã♦ ❛✜♠ ∇ ❡♠ M s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
❛✮ ∇ é s✐♠étr✐❝❛✳
❜✮ ∇ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛✳
❆ ❝♦♥❡①ã♦ ❞❛❞❛ ♣❡❧♦ t❡♦r❡♠❛ ❛❝✐♠❛ é ❞❡♥♦♠✐♥❛❞❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ✭♦✉ ❞❡ ▲❡✈✐✲❈✐t❛✮ ❞❡ M✳
❙❡❥❛ f : M → M¯ ✉♠❛ ✐♠❡rsã♦ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ s✉❛✈❡ M ❞❡ ❞✐♠❡♥sã♦ n ❡♠
✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❞✐♠❡♥ã♦n+m✳ ❆ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡M¯ ✐♥❞✉③
❞❡ ♠❛♥❡✐r❛ ♥❛t✉r❛❧ ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛ ❡♠ M✿ s❡ v1, v2 ∈ TpM✱ ❞❡✜♥❡✲s❡
hv1, v2i = hdfp(v1), dfp(v2)i✳ ◆❡st❛ s✐t✉❛çã♦ ❛ ❛♣❧✐❝❛çã♦ ❢ é ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ M ❡♠ M¯✳ ◆♦t❡ q✉❡ f é ❧♦❝❛❧♠❡♥t❡ ✉♠ ♠❡r❣✉❧❤♦✱ ✐st♦ é✱ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛
U ⊂ M ❞❡ p t❛❧ q✉❡ f : U → R4 é ✉♠ ♠❡r❣✉❧❤♦✱ ♦ q✉❛❧ ✐♠♣❧✐❝❛ q✉❡ f(U) ⊂ M¯ é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ M¯✳ ❉❡♥♦t❛♠♦s f(U) = M✳ ❆❣♦r❛✱ ✐r❡♠♦s ✐♥tr♦❞✉③✐r ❛
s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥s✐❞❡r❛♥❞♦✲❛ r❡❧❛t✐✈❛♠❡♥t❡ ❛ ✉♠ ❝❛♠♣♦ ξ ♥♦r♠❛❧ ❛ M✳ ◆♦ss❛ ✈❛r✐❡❞❛❞❡M s❡rá ♠✉♥✐❞❛ ❞❛ ❝♦♥❡①ã♦ r✐❡♠❛♥♥✐❛♥❛∇✐♥❞✉③✐❞❛ ❞❛ ❝♦♥❡①ã♦
❘✐❡♠❛♥♥✐❛♥❛ ∇❞❡ M¯✳
❙❡❥❛♠ X✱ Y ❝❛♠♣♦s ❧♦❝❛✐s ❞❡ ✈❡t♦r❡s ❡♠ M✳ ❉❡♥♦t❛♠♦s ♣♦r X✱ Y ❛s ❡①t❡♥sõ❡s
❧♦❝❛✐s ❞♦s ❝❛♠♣♦s X ❡Y ❛M¯✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ✳ ❆ ❝♦♥❡①ã♦ r✐❡♠❛♥♥✐❛♥❛ ∇ ❡♠ M é
❞❡✜♥✐❞❛ ❝♦♠♦
∇XY = (∇XY) T,
♦♥❞❡ ( ¯∇X¯Y¯)T é ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ∇¯X¯Y¯ ♥♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡
❞❡M✳
❉❡✜♥✐çã♦ ✶✳✹✸✳ ❙❡❥❛♠ X ❡ Y ❝❛♠♣♦s ❧♦❝❛✐s ❞❡ ✈❡t♦r❡s ❡♠ M✳ ❉❡✜♥✐♠♦s ♦ ❝❛♠♣♦
❧♦❝❛❧ ❞❡ ✈❡t♦r❡s ❡♠ R4 ♥♦r♠❛❧ ❛ M✳ ❈♦♠♦
B(X, Y) = ¯∇X¯Y¯ − ∇XY = ( ¯∇X¯Y¯)N.
❖ ❝❛♠♣♦ ❧♦❝❛❧ ❞❡ ✈❡t♦r❡s B(X, Y) ♥ã♦ ❞❡♣❡♥❞❡ ❞❛s ❡①t❡♥sõ❡s X✱ Y✳
❱❛♠♦s ✐♥❞✐❝❛r ♣♦r X(M)⊥ ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s s✉❛✈❡s ♥♦r♠❛✐s ❛ f(U)✳
Pr♦♣♦s✐çã♦ ✶✳✹✹✳ ❙❡ X✱ Y ∈ X(M)✱ ❡♥tã♦ ❛ ❛♣❧✐❝❛çã♦ B : X(M)× X(M) → X(M)⊥ ❞❛❞❛ ♣♦r
B(X, Y) =∇XY − ∇XY
é ❜✐❧✐♥❡❛r ❡ s✐♠étr✐❝❛✳
❖❜s❡r✈❛çã♦ ✶✳✹✺✳ ❖ ✈❛❧♦r ❞❡ B(X, Y)(p) ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❡ X(p) ❡ Y(p)✳
❙❡❥❛ p∈M ❡ ξ∈(TpM)⊥✳ ❆ ❛♣❧✐❝❛çã♦ Kξ:TpM ×TpM →R ❞❛❞❛ ♣♦r
Kξ(x, y) = hB(x, y), ξi, x, y ∈TpM,
é ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r ❡ s✐♠étr✐❝❛✳
❉❡✜♥✐çã♦ ✶✳✹✻✳ ❙❡❥❛ x∈TpM✳ ❆ ❢♦r♠❛ q✉❛❞rát✐❝❛ IIξ ❞❡✜♥✐❞❛ ❡♠ TpM ♣♦r
IIξ(x) = Kξ(x, x)
é ❝❤❛♠❛❞❛ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ❢ ❡♠ ♣ s❡❣✉♥❞♦ ♦ ✈❡t♦r ξ✳
❙❡ x✱ y ∈ Tf(q)f(M) ⊂ Tf(q)M✱ sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ✐♥❞✐❝❛r❡♠♦s ♣♦r
K(x, y)❡K(x, y) ❛s ❝✉r✈❛t✉r❛s s❡❝❝✐♦♥❛✐s ❞❡M ❡M✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P❛r❛ ♠❛✐♦r❡s
❞❡t❛❧❤❡s s♦❜r❡ ❛ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ✈❡❥❛ ❬✹❪✳ ❖ t❡♦r❡♠❛ ❛❜❛✐①♦ ❡①♣r✐♠❡ ❛s ❞✐❢❡r❡♥ç❛s ❞❛s ❝✉r✈❛t✉r❛s s❡❝❝✐♦♥❛✐s ❞❡ M ❡ M ♣♦r ♠❡✐♦ ❞❡ ❡①♣r❡ssõ❡s ❝♦♥str✉í❞❛s ❛ ♣❛rt✐r ❞❛
s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧✳
❚❡♦r❡♠❛ ✶✳✹✼ ✭●❛✉ss✮✳ ❙❡❥❛♠ q ∈ M ❡ x✱ y ✈❡t♦r❡s ♦rt♦♥♦r♠❛✐s ❞❡ Tf(q)f(M)✳ ❊♥tã♦
K(x, y)−K(x, y) =hB(x, x), B(y, y)i − kB(x, y)k2.
❉❡♠♦♥str❛çã♦✿ ❱❡❥❛ ❬✹❪
✶✳✸ ❈♦♥❥✉♥t♦s s✐♥❣✉❧❛r❡s
❙❡❥❛ f : Rn → Rp ✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡✳ ❖ ❝♦♥❥✉♥t♦ s✐♥❣✉❧❛r Σ(f) é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s s✐♥❣✉❧❛r❡s ❞❡ f✳ ❆ ✐♠❛❣❡♠ ❞❡ Σ(f)✱ f(Σ(f))✱ é ❝❤❛♠❛❞♦ ❞❡
❞✐s❝r✐♠✐♥❛♥t❡ ♦✉ ❝♦♥❥✉♥t♦ ❞❡ ❜✐❢✉r❝❛çã♦✳
❊①❡♠♣❧♦ ✶✳✹✽✳ ❆ ❛♣❧✐❝❛çã♦ ❝ús♣✐❞❡ ❞❡ ❲❤✐t♥❡② ♥♦ ♣❧❛♥♦ é ✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡
f :R2 →R2 ❞❛❞❛ ♣♦r (x, y)7→(u, v) ♦♥❞❡ u=x✱ v =y3−xy✳ ❖ ❝♦♥❥✉♥t♦ s✐♥❣✉❧❛r é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ♦♥❞❡ ❛ ♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ t❡♠ rank < 2✱ ✐st♦ é ❛
♣❛r❛❜ó❧❛ x= 3y2✳ ❊ ♦ ❝♦♥❥✉♥t♦ ❜✐❢✉r❝❛çã♦ é ❛ ✐♠❛❣❡♠ ❞❡st❛ ♣❛r❛❜ó❧❛ s♦❜ ❢✱ ♦✉ s❡❥❛✱ ❛ ❝ú❜✐❝❛ ❝✉s♣✐❞❛❧ q✉❡ t❡♠ ❛ ❡q✉❛çã♦ 4u3 −27v2 = 0✳
❋✐❣✉r❛ ✶✳✶✿ P❛r❛❜ó❧❛ ❡ ❈ús♣✐❞❡✳
❉❡✜♥✐çã♦ ✶✳✹✾✳ ❙❡❥❛f :Rn→Rp✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡✳ P❛r❛ ❝❛❞❛i= 1, ...,min{n, p}✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ Σi(f) é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡
♠❛♥❡✐r❛✿
Σi(f) = {x∈Rn: dim(ker(df
x)) =i}.
❊①❡♠♣❧♦ ✶✳✺✵✳ ❙❡❥❛ f : (R2,0) → (R2,0) ❞❡✜♥✐❞❛ ♣♦r f(x, y) = (x2, y2)✱ ✈❛♠♦s ❝❛❧❝✉❧❛r Σi(f)✱ i= 0,1,2✳
Pr✐♠❡✐r❛♠❡♥t❡✱ t❡♠♦s
df(x,y)= "
2x 0
0 2y
#
❡ ❞❛í✱ ♥♦t❡♠♦s q✉❡ dim(ker(df(x,y))) = 2 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (x, y) = (0,0)✳ ❉❡st❛ ❢♦r♠❛✱ Σ2(f) = {(0,0)}✳ ❖ ❝♦♥❥✉♥t♦ Σ1(f) é ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛s ❡q✉❛çõ❡s x 6= 0 ❡
y= 0 ♦✉ x= 0 ❡ y6= 0✳ P♦rt❛♥t♦✱ Σ1(f) = {{(x,0)} ∪ {(0, y)} − {(0,0)}✳
❊ ✜♥❛❧♠❡♥t❡ t❡♠♦s q✉❡ Σ0(f) = {(x, y) ∈ R;x 6= 0, y 6= 0}✱ ♣♦✐s✱ ♣❛r❛ ❡ss❡s ♣♦♥t♦s dim(ker(df(x,y))) = 0✳
❖❜s❡r✈❡ q✉❡ t♦❞♦s ♦s Σi(f) ❞❡st❡ ❡①❡♠♣❧♦ sã♦ s✉❜✈❛r✐❡❞❛❞❡s ❞♦ R2✳
❉❡✜♥✐çã♦ ✶✳✺✶✳ ❉❛❞❛ ✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡ f :Rn→Rp t❡♠♦s ♦s ❝♦♥❥✉♥t♦s ❞❡ s✐♥✲
❣✉❧❛r✐❞❛❞❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ Σi(f)✳ ❙❡ ❡ss❡s sã♦ s✉❜✈❛r✐❡❞❛❞❡s ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r
♦s ❝♦♥❥✉♥t♦s ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ Σi,j(f) = Σj(f|Σi(f))✳ ❊ ❡st❡ ♣r♦✲
❝❡ss♦ ♣♦❞❡ s❡r ❝♦♥t✐♥✉❛❞♦✳ ❙❡ ❡ss❡s ❝♦♥❥✉♥t♦s sã♦ s✉❜✈❛r✐❡❞❛❞❡s ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r ♦s ❝♦♥❥✉♥t♦s ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠ Σi,j,k(f) = Σk(f|Σi,j(f))✳ ❊ ❛ss✐♠
♣♦r ❞✐❛♥t❡✳ ❖s ❝♦♥❥✉♥t♦s ♦❜t✐❞♦s ❞❡ss❛ ♠❛♥❡✐r❛ sã♦ ♦s ❝♦♥❥✉♥t♦s ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❞❡ ❢✳
❊①❡♠♣❧♦ ✶✳✺✷✳ ❉❛❞♦ ✉♠ ε >0❝♦♥s✐❞❡r❡ ✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡ f :R2 →R2 ❞❡✜♥✐❞❛ ♣♦r f(x, y) = (u, v) ♦♥❞❡ u=x2−y2 + 2εx ❡ v = 2xy−2εy✳
❆ ♠❛tr✐③ ❥❛❝♦❜✐❛♥❛ ❞❡ f é
"
2x+ 2ε −2y
2y 2x−2ε
#
,
q✉❡ t❡♠ rank <2 q✉❛♥❞♦ s❡✉ ❞❡t❡r♠✐♥❛t❡ s❡ ❛♥✉❧❛✱ ♦✉ s❡❥❛✱ ♥♦ ❝ír❝✉❧♦x2+y2 =ε2✳ ❊♥tã♦✱ t❛❧ ❝ír❝✉❧♦ é ♦ ❝♦♥❥✉♥t♦ s✐♥❣✉❧❛r ❞❡f✳ ❙❡ ♣❛r❛♠❡tr✐③❛r♠♦s ♦ ❝♦♥❥✉♥t♦ s✐♥❣✉❧❛r✱
❝♦❧♦❝❛♥❞♦
x=εcosθ y =εsinθ
❡♥tã♦ ♦❜t❡♠♦s ✉♠❛ ♣❛r❛♠❡tr✐③❛çã♦ ❞♦ ❞✐s❝r✐♠✐♥❛♥t❡ ♥❛ ❢♦r♠❛
u=ε2(cos 2θ+ 2 cosθ) v =ε2(sin 2θ−2 sinθ)
q✉❡ é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✉s✉❛❧ ❞❡ ✉♠ ❤✐♣♦❝✐❝❧ó✐❞❡ tr✐❝✉s♣✐❞❛❧✳
◆❛ ✈❡r❞❛❞❡ ♥♦ss♦ ❝ír❝✉❧♦ x2+y2 =ε2 é ♣r❡❝✐s❛♠❡♥t❡ ♦ ❝♦♥❥✉♥t♦ Σ1(f) ❞❡ s✐♥❣✉✲ ❧❛r✐❞❛❞❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ♣♦✐s ♥♦t❡ q✉❡ ❛ ♠❛tr✐③ ❥❛❝♦❜✐❛♥❛ ♥ã♦ ♣♦❞❡ t❡rrank = 0✳
❚❡♠♦s q✉❡ ❡①✐st❡♠ três ♣♦♥t♦s ♥♦ ❝ír❝✉❧♦ q✉❡ ♣r❡❝✐s❛♠ s❡r ❞✐st✐♥❣✉✐❞♦s ❞♦s ♦✉tr♦s ♥❛ ♠❡❞✐❞❛ ❡♠ q✉❡ sã♦ ❧❡✈❛❞♦s ♣♦r f ❛ ❝ús♣✐❞❡s ♥♦ ❤✐♣♦❝✐❝❧ó✐❞❡✳
❆♥❛❧✐s❛r❡♠♦s ❛❣♦r❛ ❛ r❡str✐çã♦ f|Σ1(f)✳ ❱❛♠♦s ❝❛❧❝✉❧❛r ♦ rank ❞❛ r❡str✐çã♦ ♥✉♠ ♣♦♥t♦ (x, y) ♥♦ ❝ír❝✉❧♦✳ ❘❡❧❡♠❜r❡ q✉❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❛ r❡str✐çã♦ é ❛ r❡str✐çã♦ ❞❛
❞✐❢❡r❡♥❝✐❛❧ ❞❡ f ♣❛r❛ ❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦✳ ❆❣♦r❛ ❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦
♥✉♠ ♣♦♥t♦ (x, y)é ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧❛ ♦r✐❣❡♠ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ❡st❡ ✈❡t♦r✳ ❯♠ ✈❡t♦r
t❛♥❣❡♥t❡ ✉♥✐tár✐♦ s❡rá (−y/ε, x/ε) ❡ ❛ ✐♠❛❣❡♠ ❞❡st❡ s♦❜ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ f ❡♠ (x, y)
❋✐❣✉r❛ ✶✳✷✿ ❍✐♣♦❝✐❝❧ó✐❞❡✳
s❡rá ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦ ❞❛ ♠❛tr✐③ ❥❛❝♦❜✐❛♥❛ ❛ ❡❧❡✱ ♦❜t❡♥❞♦✲s❡ ♦ ✈❡t♦r "
2x+ 2ε −2y
2y 2x−2ε
# "
−y/ε
x/ε
#
= 2/ε
"
−2xy−εy
−y2+x2−εx #
.
❆ ❞✐❢❡r❡♥❝✐❛❧ ❞❛ r❡str✐çã♦ ❝❡rt❛♠❡♥t❡ t❡♠ rank ≤ 1❀ ❡ ❡❧❛ t❡♠ rank ✵ s♦♠❡♥t❡
q✉❛♥❞♦ ❡st❡ ú❧t✐♠♦ ✈❡t♦r ❢♦r ♥✉❧♦✱ ♦✉ s❡❥❛✱ ❡①❛t❛♠❡♥t❡ ♥❛s r❛í③❡s ❝ú❜✐❝❛s ❞❡ ε3✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈❛r❛s ♥♦ss♦s três ♣♦♥t♦s sã♦ ❞✐st✐♥❣✉✐❞♦s ♣r❡❝✐s❛♠❡♥t❡ ♣❡❧♦ ❢❛t♦ q✉❡ ❡❧❡s sã♦ ♣♦♥t♦s Σ1(f) ♣❛r❛ ❛ r❡str✐çã♦ f|Σ1(f)✱ ♦✉ s❡❥❛✱ ♣♦♥t♦s Σ1,1(f)✳
✶✳✹ ❈♦♥t❛t♦ ❡♥tr❡ s✉❜✈❛r✐❡❞❛❞❡s
❙❡❥❛♠ U ❡ V ❞✉❛s s✉❜✈❛r✐❡❞❛❞❡s ❡♠Rn✱ ❞❡✜♥✐❞❛s ❧♦❝❛❧♠❡♥t❡ ❛tr❛✈és ❞❛ ✐♠❡rsã♦
f : Rm → Rn ❡ ❞❛ s✉❜♠❡rsã♦ g : Rn → Rk✱ ♦♥❞❡ U = f(Rm) ❡ V = g−1(0)✱ ❝♦♠
p∈U∩V✱ ♦✉ s❡❥❛✱p=f(x0)✱x0 ∈Rm ❡g◦f(x0) = 0✳ ❙✉♣♦♥❞♦m≥k✱ ❝♦♥s✐❞❡r❛♠♦s q✉❡ ❡①✐st❡ ❝♦♥t❛t♦ ❡♥tr❡ U ❡ V ❡♠ p s❡ ❛s ❞✉❛s s✉❜✈❛r✐❡❞❛❞❡s ♥ã♦ sã♦ tr❛♥s✈❡rs❛✐s
♥❡ss❡ ♣♦♥t♦✳ ■st♦ ❡q✉✐✈❛❧❡ ❛ ❞✐③❡r q✉❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ dx0(g◦f) ♥ã♦ é s♦❜r❡❥❡t✐✈❛ ❬✶✶❪❀
♣♦rt❛♥t♦ ❛ ❛♣❧✐❝❛çã♦ g◦f t❡♠ ✉♠❛ s✐♥❣✉❧❛r✐❞❛❞❡ ♦✉ ✉♠ ♣♦♥t♦ ❝r✐t✐❝♦ ❡♠ x0✳
❖ t✐♣♦ ❞❡ ❝♦♥t❛t♦ ❡♥tr❡ ❛s s✉❜✈❛r✐❡❞❛❞❡s U ❡V s❡rá ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ t✐♣♦ ❞❡ s✐♥✲
❣✉❧❛r✐❞❛❞❡ q✉❡ ❛ ❛♣❧✐❝❛çã♦g◦f t❡♠ ♥♦ ♣♦♥t♦x0✳ ❊st❡ é ♦ ♠♦t✐✈♦ q✉❡ ❛ ❞❡♥♦♠✐♥❛♠♦s
❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ❝♦♥t❛t♦✳
❙❡❣✉❡ ❛❜❛✐①♦ ❛ ❞❡✜çã♦ ❞❡ ❑✲❡q✉✐✈❛❧ê♥❝✐❛ ✭♦✉ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❝♦♥t❛t♦✮✳
❉❡✜♥✐çã♦ ✶✳✺✸✳ ✭ ▼♦♥t❛❧❞✐✮ ❉❛❞♦s ❞♦✐s ❣❡r♠❡sf, g : (Rm,0)→(Rn,0)❞✐③❡♠♦s q✉❡
❢ ❡ ❣ sã♦ ❦✲❡q✉✐✈❛❧❡♥t❡s ❡ ❞❡♥♦t❛♠♦s ♣♦r f∼Kg✱ s❡ ❡①✐st❡♠ ❞✐❢❡♦♠♦r✜s♠♦s ❞❡ ❣❡r♠❡s h: (Rm,0)→(Rm,0)❡ H : (Rm×Rn,(0,0))→(Rm×Rn,(0,0)) t❛✐s q✉❡ ♦ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛
(Rm,0) (I
m
R,f) //
h
(Rm×Rn,(0,0))
H
(Rm,0) (IRm,g) //(Rm×Rn,(0,0)),
♦✉ s❡❥❛✱ H(x,0) = (h(x),0) ❡ H(x, f(x)) = (h(x), g◦h(x)) ♣❛r❛ t♦❞♦ x∈Rm✳
❙❡❥❛ M ✉♠❛ s✉❢❡r❢í❝✐❡ ✐♠❡rs❛ ❡♠ Rn✱ n≥4✱ ❧♦❝❛❧♠❡♥t❡ ❞❡✜♥✐❞❛ ♣♦r M =φ(R2)✱ ♦♥❞❡φ :R2 →Rné ✉♠❛ ✐♠❡rsã♦✳ ❖s ❝♦♥t❛t♦s ❞❡M ❞❡ ❝♦♠ ❤✐♣❡r♣❧❛♥♦s ❡ ❤✐♣❡r❡s❢❡r❛s
sã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧♦ s✉❜❝♦♥❥✉♥t♦ ψ−1(0) ⊂ Rn✱ n ≥ 4✱ ♦♥❞❡ ψ : Rn → R é ✉♠❛
s✉❜♠❡rsã♦✳
❙❡ ❛ s✉❜✈❛r✐❡❞❛❞❡ é ✉♠ ❤✐♣❡r♣❧❛♥♦ ❞❡ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ v ∈Sn−1 ❡ ❞✐stâ♥❝✐❛ à ♦r✐❣❡♠ ρ∈R+✳ ❆ s✉❜♠❡rsã♦ s❡rá ❞❛❞❛ ♣♦r
ψ(x1, ..., xn) = x1v1+· · ·+xnvn+ρ.
P♦rt❛♥t♦✱ ♦s ❝♦♥t❛t♦s ❞❡ ▼ ❝♦♠ ❛ ❢❛♠í❧✐❛ ❞❡ ❤✐♣❡r♣❧❛♥♦s sã♦ ❞❛❞♦s ♣❡❧❛s s✐♥❣✉❧❛r✐❞❛❞❡s ❞❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s ❛❧t✉r❛✿
λ(φ) :R2×Sn−1 →R
((x, y), v)7→λ(φ)((x, y), v) = hφ(x, y), vi.
❉❡✜♥✐çã♦ ✶✳✺✹✳ ❙❡❥❛♠ φ:Rm →Rn ✐♠❡rsã♦ ❡ ψ :Rn→R s✉❜♠❡rsã♦ q✉❡ ❞❡✜♥❡♠ ❧♦❝❛❧♠❡♥t❡ ❛s s✉❜✈❛r✐❡❞❛❞❡s U = φ(Rm) ❡ V = ψ−1(0)✳ ❉✐③❡♠♦s q✉❡ U ❡ V t❡♠ ❝♦♥t❛t♦ ❞❡ ♦r❞❡♠ ≥ 2 ❡♠ p ∈ U ∩V s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞❛s ❞❡r✐✈❛❞❛s ❞❡ ψ◦φ ❞❡
♦r❞❡♠ ≤2 s❡ ❛♥✉❧❛♠ ❡♠ ♣✱ ♦✉ s❡❥❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱
∂ψ◦φ
∂x1 (p) =· · ·=
∂ψ◦φ
∂xm(p) = 0
∂2
ψ◦φ ∂x2
1 (p) = · · ·=
∂2
ψ◦φ ∂x2
m (p) =
∂2
ψ◦φ
∂x1∂x2(p) =· · ·=
∂2
ψ◦φ
∂xm−1∂xm(p) = 0
.
