Normal holonomy group, characteristic classes and rigidity in nonnegative curvature
by
Luis Guijarro
Universidad Autónoma de Madrid, Spain
Coauthors: Gerard Walschap (University of Oklahoma)

Oral Communication

The soul theorem of Cheeger and Gromoll states that any noncompact, complete manifold M with nonnegative sectional curvature is diffeomorphic to a vector bundle over a compact totally geodesic submanifold S named the soul of M. A well-known question of Riemannian geometry is to what extent the converse to this theorem holds: i.e, if E --> S is a vector bundle, can we put a nonnegatively curved metric on E? The question is most interesting when S is simply connected, where we do not know of any counterexample.

The natural approach is then to ask how the twisting of the bundle influences possible metrics of nonnegative curvature. In this work, we attack this problem by studying topological conditions on S and E that force the normal holonomy group of any Riemannian connection on E to act transitively on the unit bundle E₁.

As a consequence, we manage to give simple characterizations on vector bundles over some simple S (spheres included) that force any metric with nonnegative curvature on E to satisfy

1. The normal exponential map exp:\nu(S)=E --> M is a diffeomorphism.
2. The metric projection \pi:M --> S (also known as the Sharafutdinov map) is C^\infty
3. The ideal boundary M(\infty) is a point.

http://www.adi.uam.es/~guijarro

Date received: February 20, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-12.