Riemannian Submersions and Positive Curvature

黎曼淹没和正曲率

• 批准号: 9803258
• 负责人: Frederick Wilhelm
• 金额: $ 6.8万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803258&HistoricalAwards=false
• 关键词: Riemannian; Submersions; Positive; Curvature

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.

摘要提案：DMS-9803258首席研究员：弗雷德里克·威廉弗雷德里克·威廉提出了三个问题的工作。 第一个是一个程序，用于创建具有大量正曲率的黎曼度量的新例子。 这些指标是在某些纤维束上的。该计划似乎特别有可能在四个球体上的三个球体束上成功。 其次，他将致力于证明他的新猜想-一个完整的正曲流形的黎曼浸没的像的维数严格大于域的维数的一半。第三，他将试图完成分类黎曼淹没roundspheres。这三个问题都解决了一般问题如何弯曲的空间影响其几何和拓扑结构？ 粗略地说，曲率决定了空间的三角学。例如，人们可以证明地球的表面是弯曲的，而不需要从外太空去看。 要做到这一点，有两个人从北极开始，并在任何两个相互垂直的方向旅行。如果它们以同样的速度旅行，它们最终会在南极相遇。 另一方面，如果同样的实验是在一个平面世界中进行的，这两个人将永远不会相遇。 他们会越来越远，即使他们永远不会到达世界的“边缘”。 研究这个普遍问题的主要理由是，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。