A smooth manifold is a pair , where is a topological manifold and is a smooth structure on .
A topological -manifold is a topological space such that:
1) is Hausdorff: For every distinct pair of points , there are disjoint open subsets such that and .
2) is second countable: There exists a countable basis for the topology of .
3) is locally Euclidean of dimension : Every point of has a neighborhood that is homeomorphic to an open subset of . For each , there exists:
– an open set containing ;
– an open set ; and
– a homeomorphism .
A smooth structure on a topological -manifold is a maximal smooth atlas.
is called a smooth atlas if and for any two charts , in (such that ), the transition map is a diffeomorphism.
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.