7 - Nonnegative curvature on disk bundles

Summary

Introduction

The search for manifolds of nonnegative curvature is one of the classical problems in Riemannian geometry. While general obstructions are scarce, there are relatively few general classes of examples and construction methods. Hence, it is unclear how large one should expect the class of closed manifolds admitting a nonnegatively curved metric to be. For a survey of known examples, see e.g. [12].

Apart from taking products, there are only two general methods to construct new nonnegatively curved metrics out of given spaces. One is the use of Riemannian submersions which do not decrease curvature by O'Neill's formula. The other is the glueing of two manifolds (which we call halves) along their common boundary. Typically, the boundary of each half is assumed to be totally geodesic or, slightly more restrictively, a collar metric. This in turn implies by the Soul theorem ([2]) that each half is the total space of a disk bundle over a totally geodesic closed submanifold. In addition, the glueing map of the two boundaries must be an isometry.

While many examples can be constructed by such a glueing, its application is still limited. On the one hand, there is not too much known on the question of which disk bundles over a nonnegatively curved compact manifold admit collar metrics of nonnegative curvature, and on the other hand, even if such metrics exist, the metric on the boundary is not arbitrary. Thus, glueing together two such disk bundles to a nonnegatively curved closed manifold is possible in special situations only.