I'm taking a course on Riemannian geometry using Lee and Petersen, and I'm falling a bit behind mostly because it has been a long time since I thought about differential geometry.
I took a course on differential geometry using do Carmo about two years ago, and remember a little bit, but it did not cover much manifold theory (and did not talk about tangent spaces, derivations, connections, differentials, etc).
Is there a good text that has a reasonably concise introduction to the essentials of manifold theory that I'd need to survive and get something out of the Riemannian geometry course? I've got a fairly strong analysis/ point-set topology and algebra background; it's mostly the geometry aspect that I'm lacking. I've been relying on Bishop/Goldberg's Tensor Analysis on Manifolds to get by, but am wondering if there are other (consise) texts/sets of notes I should be looking at.
