Math 548/549: Geometric Structures

Winter and Spring Quarters:

Comparison Geometry and Compactness Theorems

Daniel Pollack

Winter/Spring 1999, MWF 1:30-2:20
The Winter and Spring quarters will focus on the geometry and topology of manifolds whose sectional or Ricci curvatures satisfy certain bounds (e.g. non-negative curvature). We will cover a number of important results, some of which are listed below. We have listed a number of texts below as references. These will be on reserve, however there here will be no required text as we will draw on a number of different sources.

     Critical points of the distance function
     Hessian and Laplacian comparison theorems
     Toponogov's theorem
     Bishop Volume comparison and Gromov's relative volume comparison Theorems
     Cheeger-Gromoll Splitting Theorem
     Gromov-Hausdorff distance
     Cheeger's Finiteness Theorem

 REFERENCES:
          Riemannian Geometry, A Modern Introduction by Isaac Chavel
          (Cambridge University Press 1995)
          Comparison theorems in Riemannian geometry by Jeff Cheeger and David Ebin
          (North-Holland 1975)
          Lectures on differential geometry by Richard Schoen and Shing-Tung Yau
          (International Press 1994)