Riemannian Geometry

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:

1. The geometry of surfaces in R^3 and Riemann's thesis.
2. Riemannian metrics, length, and geodesics.
3. Connections, parallel transport, and curvature on vector bundles.
4. Jacobi fields and normal coordinates.
5. Finding geodesics via Morse theory.
6. 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.