I'm looking for a textbook to supplement a course in general relativity. I have been looking for textbooks on introductory differential geometry covering the subject material in the 2/3-space, in line with Cartan et al.'s original work. I came across the following books:
Elementary Differential Geometry bu Barrett O' Neill
Differential Geometry: Curves - Surfaces - Manifolds by Wolfgang Kunhnel
Elementary Differential Geometry by Edward Pressley
These books cover differential geoemtry in the 2-3 space. On the other hand, the course in GR will give me a crash course in differential geometry, discussing manifolds, curvature etc. and the discussion shall only be intituively motivated.
Also, I'm not sure of the prerequisites. For instance, Wolfgang's, though seemingly relevant, looks to be very terse and expects one to know the theory of multivariable calculus, which I'm currently covering.
To that end, what is a good companion resource to keep with me on the side, under the assumption that after working a text on multivariable analysis, I'd want to start working out a text on manifolds and I can then go all the stuff seriously. Would keeping one of the aforementioned books be a good idea? If not, which one should I use?