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 Geometry and Geometric Analysis,, by Jurgen Jost. View

Handouts:  

Homework:    HW Set 1   HW Set 2   HW Set 3   HW Set 4   HW Set 5   HW Set 6

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. A working man's introduction to elliptic theory.
6. Finding geodesics via Morse theory.
7. The Hodge Theorem and the Bochner technique.

Books on the same material:

• Riemannian Manifolds, an introduction to curvature, by John M. Lee.
• 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.