Tensor Algebra for Riemannian Geometry

Question

I'm trying to learn a little bit about Riemannian geometry, but the books that I'm looking through seem to assume that the reader is familiar with topics such as contracting tensors, raising and lowering indices, trace, etc... I have had a bit of a hard time finding a source for learning these topics and still do not have a clue as to why such operations are useful. I was wondering if there is a short resource that goes over all the necessary tensor algebra to study Riemannian Geometry.

Answer 1

I would recommend Lee's Riemannian Manifolds: An Introduction to Curvature. The second chapter contains a review of tensors (both on a vector space, and on a manifold) and discusses all of the topics you mentioned.