Definition:Smooth Local Parametrization

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.