Goals: This course is an introduction to Riemannian geometry. The emphasis is on quickly acquiring a working knowledge of the tools and ideas of the subject. **Course Syllabus**

Prerequisites: Math 868 or a familiarity with manifolds (vector fields, differential forms, tangent and tensor bundles).

Textbook: *Riemannian Manifolds, an introduction to curvature, * by John M. Lee.

Handouts: Lie Groups

Homework: HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9

Preliminary list of topics:

- The geometry of surfaces in R^3 and Riemann's thesis.
- Riemannian metrics, length, and geodesics.
- Connections, parallel transport, and curvature on vector bundles.
- Jacobi fields and normal coordinates.
- Finding geodesics via Morse theory.
- The Hodge Theorem and the Bochner technique.

Books on the same material:

- Riemannian Geometry, by M. do Carmo.
- A Comprehensive Introduction to Differeintial Geometry, Vols. 1,2,3, by M. Spivak.
- Riemannian Geometry, by S. Gallot, D. Hulin, and J. Lafontaine.
- Comparison Theorems in Riemannian Geometry, by J. Cheeger and D. Ebin.