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Differentiable Map and Diffeomorphism
- Differential Geometry 1.4

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 $x$


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