Smooth Manifold

A subset of Manifold, part of Topology and What is Differential Geometry

It is a class of manifolds that you can do calculus on.

Formal Definition: Smooth Manifold

A smooth manifold (or $C^{\infty }$ manifold) is a topological manifold $M$ together with a maximal smooth atlas.

An atlas $\mathcal{A}={(U_{\alpha },{\varphi }_{\alpha })}_{\alpha \in I}$ is a collection of charts such that:

• ${\bigcup }_{\alpha \in I}{U}_{\alpha }=M$ (the charts cover $M$)
• Each ${\varphi }_{\alpha }:U_{\alpha }\to V_{\alpha }\subseteq {\mathbb{R}}^n$ is a homeomorphism

An atlas is smooth if for all $\alpha ,\beta \in I$ with $U_{\alpha }\cap U_{\beta }\ne \varnothing$, the transition map:

${\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}:{\varphi }_{\alpha }(U_{\alpha }\cap U_{\beta })\to {\varphi }_{\beta }(U_{\alpha }\cap U_{\beta })$

is a smooth ($C^{\infty }$) map between open subsets of ${\mathbb{R}}^n$.

A maximal smooth atlas is a smooth atlas that contains every chart compatible with it.