Diffeomorphisms and local diffeomorphisms of manifolds

Let $M$ and $N$ be smooth manifolds. A diffeomorphism from $M$ to $N$ is a bijective smooth map $F:M\to N$ whose inverse $F^{-1}:N\to M$ is smooth. Since a bijective smooth map and its smooth inverse are both continuous (Smooth maps are continuous), every diffeomorphism is a homeomorphism, but the converse fails. The manifolds $M$ and $N$ are diffeomorphic, written $M\cong N$, when a diffeomorphism $M\to N$ exists.

Let $F:M\to N$ be a smooth map. Then $F$ is a local diffeomorphism when every $p\in M$ has an open neighbourhood $U\subseteq M$ such that $F(U)$ is open in $N$ and the corestriction $F{\mathrm{\mid }}_U^{F(U)}:U\to F(U)$ is a diffeomorphism onto the open submanifold $F(U)$ (An open subset of a smooth manifold has a canonical restricted smooth structure).

• Smoothness of the inverse is not automatic. A bijective smooth map need not be a diffeomorphism, and this is exactly why the definition demands smoothness of $F^{-1}$ outright: the map $F(x)=x^3$ on $\mathbb{R}$ is smooth and bijective, but $F^{-1}(y)=y^{1\mathrm{/}3}$ is not differentiable at $0$.

• A diffeomorphism is a local diffeomorphism. Taking $U=M$ at every point exhibits a diffeomorphism as a local diffeomorphism; no local inverse other than the global inverse is needed.