Differentiable Manifold

Statement

Definition. A topological manifold $M$ is said to be differentiable if there exist a family of parametrizations ${\phi }_{\alpha }:U_{\alpha }\to M$ defined on open sets $U_{\alpha }\subset {\mathbb{R}}^n$ such that

1. The coordinate neighborhoods $\{U_{\alpha }{\}}_{\alpha \in {\mathbb{N}}_{>0}}$ covers $M$;
2. For each pair of indices $\alpha ,\beta$ such that $W:={\phi }_{\alpha }(U_{\alpha })\cap {\phi }_{\beta }(U_{\beta })\ne \varnothing ,$ the overlap mapps \varphi_\beta^{-1}\circ\varphi_\alpha:&\varphi^{-1}_\alpha(W)&\to&\varphi^{-1}_\beta(W)\\ \varphi_\alpha^{-1}\circ\varphi_\beta:&\varphi^{-1}_\beta(W)&\to&\varphi^{-1}_\alpha(W)\\ \end{array}$$ are $\mathcal{C}^\infty$.
3. The family $\mathcal{A}=\{(U_{\alpha },{\phi }_{\alpha }\}$ is maximal with respect to 1 and 2, meaning that if ${\phi }_0\to M$ is a parametrization such that ${\phi }_0^{-1}\circ \phi$ and ${\phi }^{-1}\circ {\phi }_0$ are ${\mathcal{C}}^{\infty }$ for all $\phi \in \mathcal{A}$, then $(U_0,{\phi }_0)\in \mathcal{A}$. A differentiable manifold is a topological manifold with a differentiable structure.

References

• @GodNat14, Definition 2.1
• @Boo75 Chapter 3 Definition 1.2