The textbook for this course is M.F. do Carmo, Riemannian geometry. It will be on reserve in the math library. Petersen's Riemannian geometry will also be on reserve, and Spivak's Comprehensive introduction to differential geometry is already on reserve.
Tentative course schedule (about three lectures each)
If there is time at the end, we will do more of the textbook, up to the Rauch comparison theorem.
In the spring semester, we will do more from the textbook (including Preissman's theorem and the sphere theorem). We will then discuss geometric topics in classical physics, including basic general relativity (using metrics of indefinite sign) and hydrodynamics (using Riemannian geometry in infinite dimensions).