The following is a book-in-progress that I wrote as lecture notes for the second half of MIT's advanced undergraduate course in Differential Geometry (18.950) in Spring 2007. The notes assume some basic knowledge of differentiable manifolds, vector fields, tensors and differentiable forms (as in e.g. Volume 1 of Spivak's Comprehensive Introduction...) and proceed toward the fundamental notions of Riemannian geometry, using connections on bundles as a central theme. The point is to provide a preferably intuitive and well-motivated introduction to some concepts that are of crucial importance in modern differential geometry, but often seem overly abstract to the beginner.
These notes can be used for any noncommercial purpose without my explicit permission, though if you use them to teach a class, I'd be curious for you to let me know how it goes. Various updates and expansions are in the works and will occasionally be posted on this page: in particular, I hope in the future to add some discussion of Killing vectors and isometries, as well as new chapters on topics related to physics and a new appendix reviewing the prerequisites on differentiable manifolds. In the mean time, feel free to e-mail me with comments or corrections; you'll find my contact info on my home page.
Each of the following is in PDF format with hyperlinks, with a choice of either one or two pages per side.
Last update: September 26, 2008
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