"Inverse Comparison Geometry"

DMS-0102776逆比较几何：黎曼比较定理涉及曲率以某种方式有界的流形，并通过与一个著名的模型空间的几何比较来证明。对于构造具有规定曲率条件的黎曼流形这一课题，我提出了逆比较几何的概念。我的建议集中在构造正曲率和非负曲率流形的问题上。在它中，我概述了我的计划：(A)在4-球面上的3-球丛中寻找一致Pinching猜想的反例；(B)研究双商和齐次空间上度量的双边Cheeger扰动；(C)在某些同伦的5维和6维实射影空间上寻找非负和正曲率。(D)在“双灵魂流形”上构造非负曲率，(E)研究“双灵魂问题”的刚性版本，以及(F)证明我的猜想-完全正曲线流形的黎曼浸没的像的维度严格大于域的维度的一半。这些问题都解决了空间的曲率如何影响其几何和拓扑的一般问题？粗略地说，曲率决定了空间的三角关系。例如，一个人可以证明地球表面是弯曲的，而不是从外太空看它。要做到这一点，需要两个人从北极出发，沿着彼此垂直的任意两个方向行进。如果它们以相同的速度行进，它们最终会在南极再次相遇。另一方面，如果同样的实验是在一个平坦的世界上进行的，两个人永远不会相遇。即使他们永远不会到达世界的“边缘”，他们之间的距离也会越来越远。研究这个一般性问题的主要理由是，它看起来本质上是美丽的、耐人寻味的和自然的。它有着悠久的历史，可以追溯到1930年霍普夫、莫尔斯、勋伯格、迈耶斯和辛格的S的作品。

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.