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*

- Title page, Preface and Chapter 1:
*What is a connection and why should we care?*(20 pages, 251 KB)

one page per side two pages per side - Chapter 2:
*Bundles*(52 pages, 418 KB)

one page per side two pages per side - Chapter 3:
*Connections*(26 pages, 270 KB)

one page per side two pages per side - Chapter 4:
*Natural Constructions on Vector Bundles*(20 pages, 209 KB)

one page per side two pages per side - Chapter 5:
*Curvature on Bundles*(14 pages, 174 KB)

one page per side two pages per side - Chapter 6:
*Curvature in Riemannian Geometry*(33 pages, 289 KB)

one page per side two pages per side - Appendix A:
*Multilinear algebra and index notation*(31 pages, 269 KB)

one page per side two pages per side - Appendix B:
*Lie groups and Lie algebras*(21 pages, 211 KB)

one page per side two pages per side