Math 226A: Differential Geometry
Riemannian geometry is the study of smooth manifolds equipped with Riemannian
metrics. It is the language of Einstein's theory of general relativity, and of many
of the modern developments in physics, such as gauge theory and string theory. It
has also had an impact on numerous areas of mathematics, from analysis to algebraic
geometry. The Ricci flow, a concept coming from differential geometry, was recently
used by Perelman to prove the Poincare Conjecture.
Time and Place: MWF 1-1:50 pm in Geology 4645
Instructor: Ciprian Manolescu
Office: MS 6921
Office Hours: Mon 2-3 and Thu 11-12
In this course we will cover the following topics:
- Riemannian metrics, connections, the curvature tensor;
- Sectional, Ricci, and scalar curvature;
- Einstein manifolds;
- Geodesics, distance, variational methods, Jacobi fields;
- Completeness, the Hopf-Rinow theorem;
- The Cartan-Hadamard theorem on non-positive sectional curvature;
- Myers' theorem on positive Ricci curvature;
- Hodge theory;
- The Bochner technique.
Prerequisite: A working knowledge of smooth manifolds, differential
forms and de Rham cohomology, at the level of Math 225A and Math 225B.
Grading: Based on a few problem sets.
Textbook: P. Petersen, Riemannian Geometry, 2nd edition,
Springer Verlag, 2006.
Other recommended books:
- M. P. do Carmo, Riemannian Geometry, Birkhauser, 1992.
- J. Cheeger, D. G. Ebin, Comparison Theorems in Riemannian
Geometry, AMS Chelsea, 2008.
- J. Milnor, Morse Theory, Annals of Mathematics Studies No.
51, Princeton Univ. Press, 1969.
- F. Warner, Foundations of Differentiable Manifolds and Lie Groups,
Springer Verlag, 1983.