Definition:Smooth Local Parametrization
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Definition
Let $\tilde M$ be a smooth manifold.
Let $M \subseteq \tilde M$ be a submanifold.
Let $U \subseteq \R^n$ be an open subset of the $n$-dimensional Euclidean space.
Let $X : U \to \tilde M$ be a smooth map.
Suppose $\map X U$ is an open subset of $M$.
Suppose $X$ is a diffeomorphism onto its image.
Then $X$ is called a smooth local parametrization.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics