"Inverse Comparison Geometry"

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Abstract for DMS - 0102776Inverse Comparison Geometry: Riemannian Comparison Theorems pertain to manifolds whose curvatures are bounded in some way, and are proven by comparing the geometry to that of a well known model space. I propose the name Inverse Comparison Geometry for the subject of constructing Riemannian manifolds with prescribed curvature conditions. My proposal focuses on the problem of constructing manifolds of positive and nonnegative curvature. In it, I outline my plans to (a) find counterexammples to the Uniform Pinching Conjecture among the 3-sphere bundles over the 4-sphere,(b) study two sided Cheeger perturbations of the metrics on biquotients and homogeneous spaces(c) search for nonnegative and positive curvature on certain homotopy 5 and 6 dimensional real projective spaces. (d) construct nonnegative curvature on ``double soul manifolds'',(e) study a rigid version of the ``double soul problem'', and(f) prove my conjecture---that the dimension of the image of a Riemannian submersion of a complete, positively curved manifold is strictly greater than half the dimension of the domain.These problems all address the general question of how does the curvature of a space effect its geometry and topology? Roughly speaking, curvature is what determines the trigonometry of a space. For example one can prove that the surface of the earth is curved with out looking at it from outer space. To do this have two people start at the north pole and travel in any two directions that are perpendicular to each other. If they travel at the same speed, they will eventually meet again at the south pole. On the other hand, if the same experiment were conducted on a flat world, the two people would never meet. They would keep getting further apart, even if they never reached the "edge" of the world. The main justification for studying this general question is that it seems intrinsically beautiful, intriguing, and natural. It has a long history, that dates back to the 1930's work of H. Hopf, Morse, Schoenberg, Meyers, and Synge.