Horizontally homothetic submersions and nonnegative curvature.pdf

a r X i v : m a t h / 0 6 0 2 0 2 4 v 2 [ m a t h .D G ] 7 F e b 2 0 0 6 HORIZONTALLY HOMOTHETIC SUBMERSIONS AND NONNEGATIVE CURVATURE YE-LIN OU AND FREDERICK WILHELM? Abstract. We show that any horizontally homothetic 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 Sn, 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 homothetic submersion is more likely to have a positively curved image than a given Rie- mannian 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 sub- mersion (up to a change of scale on the base space). This generalizes the result in [OW] that any horizontally homothetic submer- sion of a round sphere with 1–dimensional fibers is a Riemannian submersion. 1991 Mathematics Subject Classification. 53C20, 53C21. Key words and phrases. Horizontally homothetic submersion, Riemannian submersion, non- negative curvature. ?Support from NSF grant DMS-0102776 is gratefully acknowledged by the second author. 1 2 Y