Definition:Local Diffeomorphism

Definition

Let $n$ and $k$ be natural numbers.

Let $U \subset \R^n$ be an open set.

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Let $f: U \to \R^n$ be a mapping.

Then $f$ is a local $C^k$-diffeomorphism if and only if every $a \in U$ has a open neighborhhood such that the restriction of $f$ to it is a $C^k$-diffeomorphism on its image.

Smooth Manifold

Let $S$ and $T$ be smooth manifolds with or without boundary.

Let $f: S \to T$ be a mapping.

Then $f$ is a local diffeomorphism if and only if every point $p \in S$ has a open neighborhood $U$ such that:

• $f \sqbrk U$ is open in $T$
• $f {\restriction_U}: U \to f \sqbrk U$ is a diffeomorphism.