Differential Geometry I

Department: 
MATH
Course Number: 
6455
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every Spring Semester

Core topics in differential and Riemannian geometry including Lie groups, curvature, relations with topology.

Prerequisites: 

MATH 4441 or MATH 6452 or permission of the instructor.

Course Text: 

Text at the level of Riemannian Geometry of do Carmo's or Gallot-Hulin-Lafontaine.

Topic Outline: 
  1. Riemannian metrics and geodesics
  2. Examples of Riemannian manifolds (submanifolds, submersions, warped products, homogeneous spaces, Lie groups)
  3. Covariant derivative, parallel transport
  4. Geodesics, exponential map, Hopf-Rinow
  5. Curvature tensor, Ricci, sectional, mean, scalar curvatures, spaces of constant curvature, curvature computaions for examples listed in 2).
  6. 1st and 2nd variation formulas, Jacobi fields, Rauch and Ricatti comparison, and applications such as Myers and Cartan-Hadamard theorems
  7. Selections from more advanced topics such as: volume comparision and Ricci curvature, minimal surfaces, spectral geometry, Hodge theory, symmetric spaces and holonomy, comparison geometry and Lorentz geometry