Riemannian Geometry IV

Epiphany 2016

The Michaelmas 2015 webpage

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


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

Further (recommended) reading:

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.


  • 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


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