Instructor: A.Freire, Ayres 304, 974-4313, freire@math.utk.edu
Office hours: Tu-Th 11:00-12:00, or by appointment
Section: 69158, Tu-Th 9:40-10:55, Ayres 102

TEXT: Riemannian Geometry: A Modern Introduction, by Isaac Chavel
 

COURSE OUTLINE
 

1.}GEODESICS and RIEMANNIAN CURVATURE
geodesic flow/exponential map/local properties/Hopf-Rinow
curvature tensor/sectional curvature/isometric immersions/submanifolds
hyperbolic space: models, geodesics, isometries

2.}SYMMETRIC SPACES
Killing fields/ curvature of symmetric spaces
examples and rough classification
holonomy and curvature/de Rham decomposition

3.}JACOBI FIELDS
first and second variations of arc length/ index form and Jacobi fields
basic comparison theorems; thms. of Bonnet-Myers, Synge-Weinstein
Riemannian coverings, thms of Cartan-Hadamard and Cartan-Ambrose-Hicks
Jacobi fields and exponential map/ conjugate and cut loci
Bishop's volume comparison theorem

4.}CURVATURE AND TOPOLOGY
volume growth and growth of the fundamental group: K<0
Preissman's theorem
Bochner's method
Gauss-Bonnet-Chern in even dimensions

5.}COMPARISON AND FINITENESS THEOREMS
Rauch's comparison theorem/ Toponogov's triangle comparison
Cheeger's finiteness theorem
Cheeger-Gromoll's splitting theorem

GRADING:Based on problem sets, a written midterm and a written final