Introductory Text about Riemannian Manifolds

Question

Some of my friends and me want to study the subject of Riemannian manifolds, and we are looking for an introductory text to study that subject. We studied differential geometry, and are about to finish out B.A. this year.

We consider studying afterwards about covering spaces of Riemannian manifolds.

Do you have any recommandations for such a text, that would be comfortable to study from and that will deal also with more advanced material on that subject?

Thanks in advance.

Answer 1

Definitely check Petersen's Riemannian Geometry. It covers the basics very well, but also connects to more advanced material. Since you've already studied some differential geometry, you should find it reasonably paced and delightfully easy to follow.

Do Carmo's Riemannian Geometry is a classic. I initially learned out of it. However, in my opinion, it lacks the "vibe" of most modern approaches. I'd suggest using this as a supplement.

I do not recall how much detail each gives to covering spaces. I know they're mentioned somewhere in Petersen's book, but I don't remember seeing it in do Carmo's book.

Answer 2

I like Lee's Riemannian Manifolds (An Introduction to Curvature). It doesn't cover as much as Petersen's book, but I think it has some great examples and lots of good exercises. As with Lee's Introduction to Smooth Manifolds, the exposition is excellent.