Differentiable Map and Diffeomorphism
- Differential Geometry 1.4
from 「Geometrical Methods of Mathematical Physics」: Bernard F. Schutz
Differentiable map
Given an $m$-dimensional differentiable manifold $M$ and an $n$-dimensional differentiable manifold $N$, we can think of a function $f$ from $M$ to $N$. If the function from a local coordinate of $M$ to a local coordinate of $N$ is differentiable, $f$ is called a differentiable map.
- $ \exist(U,\varphi)\in\mathcal{A}_M,\; (V,\psi)\in\mathcal{A}_N $
- $ \psi\circ f\circ\varphi^{-1} : \varphi(U)\to\psi(V) \;\text{is}\; C^k \Rightarrow f\in C^k(M,N) $
For example,
- $ f\in C^0(M,N) $ : continuous map
- $ f\in C^1(M,N) $ : differentiable map
- $ f\in C^\infty(M,N) $ : smooth map
- $ f\in C^\omega(M,N) $ : analytic map
Diffeomorphism
In the same situation as above, let’s continue discussing about the function $f$. If $f$ satisfies the following properties, we call it a diffeomorphism.
- $f$ is a bijection
- $f$ is differentiable
- $f^{-1}$ is differentialbe
If such $f$ exists, $M$ and $N$ are diffeomorphic; $M \simeq N$.
This concept can be extended as $C^k$ diffeomorphicity; with $C^k$ condition instead of the differentiability condition.
In addition, we can define local diffeomorphism when the domain is restricted.
- $ x\in M,\; V\in\mathcal{N}_x,\; W=f(V) $
- $ f|_V : V \to W $ : a local diffeomorphism at $x$
