ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53 Sormani, Christina, Johns Hopkins University, U. S. A. The Topology of Noncompact Manifolds with Positive Ricci Curvature Let M^{n} be a complete noncompact manifold. Cheeger and Gromoll proved that if M^{n} has nonnegative sectional curvature then it is diffeomorphic to the normal bundle over a compact totally geodesic submanifold (1972). If M^{n} has positive sectional curvature then it is diffeomorphic to R^{n}. However, if M^{n} only has positive Ricci curvature then it can have a nontrivial fundamental group (Nabonnand, 1980). We will say that M^{n} has the loops to infinity property if given any noncontractible closed curve C, any ray and any compact set K, there exists a closed geodesic contained in M\K which is homotopic to C along the given ray. It is proven that if M^{n} has positive Ricci curvature then it has this loops to infinity property. Counter examples and a discussion of manifolds with nonnegative Ricci curvature will be presented as well.