Horizontally homothetic submersions and nonnegative curvature

Abstract

We show that any horizontally submersion from a compact manifold of nonnegative sectional curvature is a Riemannian sub- mersion. The lack of examples of manifolds with positive sectional curvature has been a major obstacle to their classification. Apart from S n , every known compact mani- fold with positive sectional curvature is constructed as the image of a Riemannian submersion of a compact manifold with nonnegative sectional curvature. Here we study a generalization of Riemannian submersions called horizon- tally homothetic submersions. For this larger class of submersions, the analog of O'Neill's horizontal curvature equation has exactly one extra term ((Gu1) and (KW)). This extra term is always nonnegative and can potentially be positive. So the horizontal curvature equation suggests that a single horizontally submersion is more likely to have a positively curved image than a given Rie- mannian submersion. Since horizontally submersions are (a priori) more abundant, one is lead to believe that they have much more potential for creating positive curvature than Riemannian submersions. Unfortunately, our main result suggests that this is an illusion.