Horizontally homothetic submersions and nonnegative curvature

Abstract. We show that any horizontally homothetic submersion from a compact manifold of nonnegative sectional curvature is a Riemannian submersion. 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 manifold 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 “horizontally 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 homothetic submersion is more likely to have a positively curved image than a given Riemannian submersion. Since horizontally homothetic 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. Main Theorem. Every horizontally homothetic submersion from a compact Riemannian manifold with nonnegative sectional curvature is a Riemannian submersion (up to a change of scale on the base space). This generalizes the result in [OW] that any horizontally homothetic submersion of a round sphere with 1–dimensional fibers is a Riemannian submersion.