Ma 157a: Riemannian Geometry I

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Instructor: Alexander Gorodnik
Office: 172 Sloan
Phone: 395-4325
E-mail: gorodnik "at" caltech "dot" edu
Office Hours: TBA and by appointment

Course Description:
Ma 157a is an introductory course in Riemannian geometry. The course will begin with the basic notions of Riemannian geometry (such as metrics, geodesic flow, connections, curvature, Jacobi fields, etc.), which describe local structure of Riemannian manifolds. It is one of the fundamental problems of modern Riemannian geometry is to understand how local properties of a manifold are related to its global geometry and topology. We illustrate this relations by the classical results of Cartan-Hadamard, Myers and Synge. In the end we discuss some of more advanced topics (comparison theorems, Klingenberg's sphere theorem, dynamics of geodesic flow, etc.)

Prerequisites: A background in the basic topology of smooth manifolds, for example, Ma109c.

Grading Policy: Homework problems will be collected.

Homeworks:

• Homework 1 (due Wed., Jan. 25)
• Homework 2 (due Wed., Feb. 1)
• Homework 3 (due Wed., Feb. 15)
• Homework 4 (due Wed., Feb. 22)
• Homework 5 (due Wed., March 1)
• Homework 6 (due Sat., March 11)

Textbooks:
We do not follow any single book closely. There are two recommended introductory textbooks

• Do Carmo, Riemannian Geometry, Springer-Verlag, 1992
• Gallot, Hulin and Lafontaine, Riemannian Geometry, Springer-Verlag, Universitext, 2004

• Berger, A panoramic view of Riemannian geometry. Springer-Verlag, Berlin, 2003
This is a nice survey of modern Riemannian Geometry with lots of examples and intuitive explanations.
• Sakai, Riemannian Geometry, AMS, 1996
This is a good reference, but it contains much more material that we will cover.