# Riemannian Geometry IV

## Epiphany 2016

 Time and place: Lectures: Tue 15:00, CM107; Th 17:00, CM101 Problems classes: Tue 16:00, CM107, Weeks 13,15,17,19
Instructor: Pavel Tumarkin
e-mail: pavel dot tumarkin at durham dot ac dot uk
Office: CM110; Phone: 334-3085
Office hours: Mon 11:00 -- 12:00 and by appointment

 Textbooks:

 The content of the course can also be found in any standard textbook on Riemannian Geometry, e.g.

• F. Morgan, Riemannian Geometry.
• T. Sakai, Riemannian Geometry.
• S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry.
• P. Petersen, Riemannian Geometry.

Preliminary course content (subject to change): introduction to Lie groups, Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature; manifolds of positive curvature and Bonnet-Myers theorem; Jacobi fields and conjugate points, manifolds of nonpositive curvature and Cartan-Hadamard theorem; comparison theorems.

Schedule:

• Week 11: Lie groups: left-invariant vector fields and Lie algebras
• Week 12: Exponential map. Adjoint representation. Riemannian metrics on Lie groups
• Week 13: Riemann curvature tensor; sectional curvature, Ricci curvature
• Week 14: Bonnet - Myers theorem; second variational formula of length, applications
• Week 15: Jacobi fields; conjugate points
• Week 16: Jacobi fields and exponential map; orthogonal Jacobi fields
• Week 17: Conjugate points and minimizing geodesics; cut locus and conjugate points, examples
• Week 18: Theorem of Cartan - Hadamard; index form, Rauch comparison theorem
• Week 19: Injectivity radius, Sphere theorem; spaces of constant curvature, comparison triangles, theorem of Alexandrov - Toponogov

Handouts:

Homeworks: There will be weekly homework assignments. Selected exercises are to be handed in on weeks 13, 15, 17, and 19

• Homework 1 (due Thursday, February 4)
• Homework 2 (due Thursday, February 4)
• Homework 3 (due Thursday, February 18)
• Homework 4 (due Thursday, February 18)
• Homework 5 (due Thursday, March 3)
• Homework 6 (due Thursday, March 3)
• Homework 7 (due Thursday, March 17)
• Homework 8