Hey guys, I'm looking for a good treatment (good = concise, and clear) of affine geometry. Connections, parallel transport, etc. I'm looking for this from a mathematical P.O.V. Most of the differential geometry books I have deal only with the exterior forms, and general manifolds without this added structure. The GR books I have deal with this, but always with an assumed symmetric connection (no torsion), and they tend to go at it from a physicist's point of view. I'm not looking for anything too rigorous (by too rigorous, I mean, like at the level of a tome for professional mathematicians, I'm not looking to read 600+ pages about this material), but rigorous enough so that I'll have a good foot hold in this topic. Any suggestions? Thanks.
I would suggest Bishop and Crittenden, "Geometry of Manifolds", Chapter 5: Connections, Chapter 6: Affine Connections. Another option is Lovelock and Rund, "Tensors, Differential Forms and Variational Principles", Dover 1989, Chapter 3: Tensor Analysis on Manifolds. Personally I prefer the first one.
These are fairly advanced topics (second or third year graduate work), so I'm not sure what you can say about them without being "too rigorous". However, John Lee's Riemannian Manifolds: An Introduction to Curvature is only 224 pages long, and he's an excellent writer. This is definitely a math book, though. Another great book between physics and math is Arnol'd's Mathematical Methods of Classical Mechanics. Appendix I begins by talking about parallel transport.
A not-so-rigorous text can be good old Schouten, "Tensor Analysis for Physicists", Chapter V, some 20 pages.
By not so rigorous, I just mean I didn't want to read like several hundred pages of exposition with every caveat and potential exception explored. For example, I'm perfectly fine in assuming that partial derivatives commute without having to explicitly show that the function I'm acting them on has to fit certain criteria (continuous in a small epsilon disk around where I'm taking those derivatives I believe...so you can't have some connical section of discontinuity or something?). That's all I'm sayin. Thanks for the recommendations, I'll take a look.