"Inverse Comparison Geometry"

《逆比较几何》

• 批准号: 0102776
• 负责人: Frederick Wilhelm
• 金额: $ 8.7万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102776&HistoricalAwards=false
• 关键词: Inverse; Comparison; Geometry

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.

DMS - 0102776的逆比较几何：黎曼比较定理是关于曲率以某种方式有界的流形，并通过与已知模型空间的几何比较来证明。我建议将构造具有规定曲率条件的黎曼流形这门学科命名为“逆比较几何”。我的建议集中于构造正曲率和非负曲率流形的问题。在这篇文章中，我概述了我的计划：(a)在4球上的3球束中找到一致夹紧猜想的反例，(b)研究双商和齐次空间上度量的两面Cheeger摄动(c)在某些同伦5维和6维实射影空间上寻找非负曲率和正曲率。(d)在“双魂流形”上构造非负曲率，(e)研究“双魂问题”的刚性版本，以及(f)证明我的猜想——一个完整的、正弯曲流形的黎曼淹没像的维数严格大于该域的一半维数。这些问题都解决了一个普遍的问题，即空间的曲率如何影响其几何和拓扑结构？粗略地说，曲率决定了空间的三角性质。例如，人们可以证明地球表面是弯曲的，而不需要从外太空看它。要做到这一点，让两个人从北极出发，在相互垂直的任意两个方向上旅行。如果它们以同样的速度行进，它们最终会在南极再次相遇。另一方面，如果同样的实验在一个平坦的世界上进行，这两个人永远不会相遇。即使他们从未到达世界的“边缘”，他们的距离也会越来越远。研究这个一般性问题的主要理由是，它似乎本质上是美丽的、有趣的和自然的。它有着悠久的历史，可以追溯到20世纪30年代H. Hopf、Morse、Schoenberg、Meyers和Synge的工作。