📂Geometry

Differentiable Homomorphism

Definition¹

Let’s call $M_{1}, M_{2}$ a differential manifold. A function $\varphi : M_{1} \to M_{2}$ is called a diffeomorphism if it satisfies the following conditions:

1. $\varphi$ is differentiable.
2. $\varphi$ is a bijective function.
3. $\varphi ^{-1}$ is differentiable.

If for the neighborhoods $U$ and $V$ of points $p \in M_{1}$ and $\varphi(p)$, the contraction mapping $\varphi|_{U} : U \to V$ is a diffeomorphism, then $\varphi$ is called a local diffeomorphism.

Theorem

Let’s call $M_{1}^{n}, M_{2}^{n}$ an $n$-dimensional differential manifold. Let $\phi : M_{1} \to M_{2}$ be a differentiable function, and suppose that for $p \in M_{1}$, $d\phi_{p} : T_{p}M_{1} \to T_{\phi (p)}M_{2}$ is an isomorphism. Then $\phi$ is a local diffeomorphism at $p$.

Proof

This is established by the Inverse Function Theorem.

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