On complete metrics of nonnegative curvature on 2-plane bundles

Abstract

This paper is an attempt to understand the 2-plane bundle case for the converse of the soul theorem due to J. Cheeger and D Gromoll. It is shown that there is a class of 2-plane bundles over certain 52-bundles that carry complete metrics of nonnegative sectional curvature. In particular, every 2plane bundle and every S1 -bundle over the connected sum CPnφCPn of CPn with a negative CPn carries a 2-parameter family of complete metrics with nonnegative sectional curvature. A complete noncompact Riemannian manifold with nonnegative curvature (K > 0) is diffeomorphic to the normal bundle of a closed totally geodesic submanifold of the noncompact manifold according to a fundamental theorem due to J. Cheeger and D. Gromoll [4]. They also proposed the following problem: Which vector bundles over closed manifolds with K > 0 carry complete metrics with K > 0? This paper is an attempt to understand the 2-plane bundle case. For a class of 2-plane bundles and S^-bundles, we show that there exist families of complete metrics with K > 0. Our main result is the following Theorem. Every 2-plane bundle and every S1-bundle over the connected sum CP n #CP of CP n with a negative CPn carries a 2-parameter family of complete metrics with K > 0. More generally, let (£?, h) be a Hodge manifold with positive Kahlerian curvature and let M be an associated S2bundle over B. Then, there is a two dimensional subspace H2(M), generated by two integral cohomology classes, in the cohomology group H*(M) such that every principal S1-bundle and every 2-plane bundle over M supported by an Euler class in H 2{M) carries a 2-parameter family of complete metrics with K>0.