Differentiable Manifolds

Definition: Smooth Compatibility

Let $({\mathcal{D}}_{\phi },\phi )$ and $({\mathcal{D}}_{\psi },\psi )$ be charts on a manifold.

We say that $({\mathcal{D}}_{\phi },\phi )$ and $({\mathcal{D}}_{\psi },\psi )$ are smoothly compatible if the transition map $\psi \circ {\phi }^{-1}$ is a diffeomorphism or ${\mathcal{D}}_{\phi }\cap {\mathcal{D}}_{\psi }=\varnothing$.

Definition: Smooth Atlas

An atlas for a manifold is smooth if all of its charts are pairwise smoothly compatible.

Definition: Maximal Smooth Atlas

A smooth atlas $\mathcal{A}$ is maximal if there is no chart $(U,\phi )$ which is pairwise smoothly compatible with the charts in $\mathcal{A}$ such that $\mathcal{A}\cup \{(U,\phi )\}$ is still a smooth atlas.

Definition: Differentiable Manifold

A differentiable manifold or smooth manifold $(M,\mathcal{A})$ is a manifold $M$ equipped with a maximal smooth atlas $\mathcal{A}$ on it.

We call $\mathcal{A}$ the differentiable structure or smooth structure of $(M,\mathcal{A})$.