## Smooth/Differentiable Manifold

Smooth Manifold
A smooth manifold is a pair $(M,\mathcal{A})$, where $M$ is a topological manifold and $\mathcal{A}$ is a smooth structure on $M$.

Topological Manifold
A topological $n$-manifold $M$ is a topological space such that:
1) $M$ is Hausdorff: For every distinct pair of points $p,q\in M$, there are disjoint open subsets $U,V\subset M$ such that $p\in U$ and $q\in V$.
2) $M$ is second countable: There exists a countable basis for the topology of $M$.
3) $M$ is locally Euclidean of dimension $n$: Every point of $M$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^n$. For each $p\in M$, there exists:
– an open set $U\subset M$ containing $p$;
– an open set $\widetilde{U}\subset\mathbb{R}^n$; and
– a homeomorphism $\varphi: U\to\widetilde{U}$.

Smooth structure
A smooth structure $\mathcal{A}$ on a topological $n$-manifold $M$ is a maximal smooth atlas.

Smooth Atlas $\mathcal{A}=\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in J}$ is called a smooth atlas if $M=\bigcup_{\alpha\in J}U_\alpha$ and for any two charts $(U,\varphi)$, $(V,\psi)$ in $\mathcal{A}$ (such that $U\cap V\neq\emptyset$), the transition map $\displaystyle \psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V)$ is a diffeomorphism.

Source:
Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218) by John Lee  Differentiable Manifolds (Modern Birkhäuser Classics) by Lawrence Conlon  These two books are highly recommended books for Differentiable Manifolds. John Lee’s book has almost become the standard book. Its style is similar to Hatcher’s Algebraic Topology, it can be wordy but it has detailed description and explanation of the ideas, so it is good for those learning the material for the first time.

Lawrence Conlon’s book is more concise, and has specialized chapters that link to Algebraic Topology. 