Definition:Local Diffeomorphism/Smooth Manifold

Definition

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.

Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): Chapter $4$: Submersions, Immersions, and Embeddings: $\S$ Maps of Constant Rank