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Differentiable Manifolds and Diffeomorphisms

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Let M and N be differentiable manifolds. Let alpha: M -> N be a local diffeomorphism. Prove that if N is orientable, then M is orientable.

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Differentiable Manifolds and Diffeomorphisms are investigated. The solution is detailed and well presented.

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Inverse Function Theorem, Isomorphisms and Diffeomorphisms

Inverse Function Theorem. Let M and N be differentiable manifolds, φ : M -> N a differentiable mapping and p Є M such that dφp : TM ?> T(p)N is an isomorphism. Prove that is a local diffeomorphism at p; that is, there are neighborhoods U C M of p and V C N of p(p) such that φ : U ?> V is a diffeomorphism.
Hint: Coordinatize!

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