"Inverse Comparison Geometry"

DMS -0102776反向比较几何结构摘要： 黎曼比较定理适用于曲率以某种方式有界的流形，并通过将几何形状与众所周知的模型空间进行比较来证明。 我建议的名称逆比较几何的主题建设黎曼流形规定的曲率条件。我的建议集中在构造正曲率和非负曲率流形的问题上。在这篇文章中，我概述了我的计划：（a）在4-球面上的3-球面丛中寻找一致Pinching猜想的反例，（B）研究双凸空间和齐性空间上度量的双边Cheeger扰动，（c）寻找某些同伦5维和6维真实的射影空间上的非负曲率和正曲率。(d)在"双灵魂流形“上构造非负曲率，（e）研究”双灵魂问题“的严格版本，（f）证明我的猜想-一个完全的黎曼浸没的像的维数，正弯曲流形严格大于域的维数的一半。这些问题都解决了空间的曲率如何影响其几何形状的一般问题，拓扑学？粗略地说，曲率决定了空间的三角关系。例如，人们可以证明地球的表面是弯曲的，而不用从外太空看它。要做到这一点，有两个人开始在北极和旅行的任何两个方向是相互垂直的。如果它们以同样的速度运动，它们最终会在南极相遇。 另一方面，如果同样的实验在一个平坦的世界上进行，这两个人永远不会相遇。他们会越来越远，即使他们永远不会到达世界的“边缘”。 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，有趣的，自然的。它有着悠久的历史，可以追溯到20世纪30年代H。霍普夫、莫尔斯、勋伯格、迈耶斯和辛格。

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.