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