Riemannian Submersions and Positive Curvature

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Abstract Proposal: DMS-9803258Principal Investigator: Frederick WilhelmFrederick Wilhelm proposes to work on three problems. The first is aprogram for creating new examples of Riemannian metrics that have lotsof positive curvatures. The metrics are on certain fiber bundles.The program seems particularly likely to succeed for three spherebundles over the four sphere. Second he will work on proving his newconjecture---that the dimension of the image of a Riemanniansubmersion of a complete, positively curved manifold is strictlygreater than half of the dimension of the domain. Third he will try tocomplete the classification of Riemannian submersions of roundspheres.The three problems all address the general question of how does thecurvature of a space effect its geometry and topology? Roughlyspeaking, curvature is what determines the trigonometry of a space.For example one can prove that the surface of the earth is curved without looking at it from outer space. To do this have two people startat the north pole and travel in any two directions that areperpendicular to each other. If they travel at the same speed, theywill eventually meet again at the south pole. On the other hand, ifthe same experiment were conducted on a flat world, the two peoplewould never meet. They would keep getting further apart, even if theynever reached the "edge" of the world. The main justification forstudying this general question is that it seems intrinsicallybeautiful, intriguing, and natural. It has a long history, that datesback to the 1930's work of H. Hopf, Morse, Schoenberg, Meyers, andSynge.