Some References for Differential Geometry and Topology

I've included comments on some of the books I know best; this does not imply that they are better than the other books on this list. (Nor should one conclude anything from the order in which the books are listed—alphabetical by order within each group—or by comparing the lengths of different comments.)

General references that do not require too much background

Books in the next group go only briefly through manifold basics, getting to Riemannian geometry very quickly.

Books in the next group focus on differential topology, doing little or no geometry.

(Remember that differential geometry takes place on differentiable manifolds, which are differential-topological objects. Some of the deepest theorems in differential geometry relate geometry to topology, so ideally one should learn both.)

Less elementary books.

These either assume the reader is already familiar with manifolds, or start with the definition of a manifold but go through the basics too fast to be effective as an introductory text.