Updates of Serge Lang — Differential manifolds

Question

I have Serge Lang — differential manifolds. An interesting read. But the book is 50 years old. Are there newer books that give a better and more comprehensive treatment of the material, or this the best of its kind?

Answer 1

It sounds like you have the version of this book from 1971. It was updated and expanded in the 1990's and the most recent version is titled "Fundamentals of Differential Geometry."

Lang first put out "Introduction to Differentiable Manifolds" in 1962, which was a very useful reference on the basics of Banach and Hilbert manifolds. He updated and and expanded the text multiple times, giving it a different name each time instead of calling later versions the $n$th edition for $n \geq 2$. Your 1971 version went by "Differential Manifolds," a later version was called "Differential and Riemannian Manifolds," and the latest edition is called "Fundamentals of Differential Geometry." There is actually an "Introduction to Differentiable Manifolds, 2nd Ed." by Lang, which is not any of the above, but rather a version of the original 1962 text that was adapted to be more introductory by only addressing finite-dimensional topics.

Answer 2

I don't have much experience with Lang's book, but some other books that are in vogue amongst graduate students right now are:

1. John Lee, Introduction to Smooth Manifolds
2. Loring Tu, An Introduction to Manifolds
3. Guillemin and Pollack, Differential Topology
4. Milnor, Topology from the Differential Viewpoint
5. Do Carmo, Riemannian Geometry
6. Bott and Tu, Differential Forms in Algebraic Topology
7. Milnor, Morse Theory

The first four are of a more introductory nature, while the last three draw on material from the first four.