Coordinate chart as an isomorphism of smooth manifolds

Statement

Let $M$ be an $n$ -dimensional smooth manifold and let $(U,\varphi)$ be a smooth coordinate chart. Give $U\subseteq M$ its open-submanifold structure and $\varphi(U)\subseteq\mathbb R^n$ its standard smooth structure. Then

\varphi:U\longrightarrow\varphi(U)

is a diffeomorphism, hence an isomorphism in the category of smooth manifolds.

What the chart itself is

Strictly, a chart is the pair $(U,\varphi)$ , not only the arrow $\varphi$ . The coordinate map is generally not a morphism with domain all of $M$ ; its domain is the open submanifold $U$ . It sits beside the smooth inclusion

U\xrightarrow{\ \varphi\ }\varphi(U), \qquad U\hookrightarrow M.

Thus charts provide local isomorphisms, not usually global isomorphisms $M\cong\mathbb R^n$ .

Transition maps

For overlapping charts $(U,\varphi)$ and $(V,\psi)$ , the transition map

\psi\circ\varphi^{-1}: \varphi(U\cap V)\longrightarrow\psi(U\cap V)

is likewise an isomorphism between open submanifolds of Euclidean space. The pairwise smoothness of these local isomorphisms is the compatibility condition in a smooth atlas.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth structures, charts, and smooth maps.