EPFL - Automne 2017 M . Troyanov
نویسنده
چکیده
Solution 4.1. Soit f : U ⊂ Rn −→ Rn une application lisse et soit p ∈ U tel que dfp soit inversible. On veut montrer que f est localement un di éomorphisme. Or, comme dfp est inversible, on a rangf (p) = n et comme le rang est semi-continu inférieurement, il existe un voisinage ouvert Ũ ⊂ U de p tel que rangf (q) = n pour tout q ∈ Ũ . Alors par le théorème du rang constant, il existe un voisinage V ⊂ Ũ de p, un voisinage W ⊂ Rn de f(p) et des di éomorphismes lisses G : V → V ′ et F : W → W ′ tels que f̃ := F ◦ f ◦ G−1 : V ′ → W ′ s'écrive f̃(x, . . . , x) = (x, . . . , x).
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