Riemannian Submersions and Positive Curvature

摘要建议：DMS-9803258主要调查者：Frederick Wilhelm Frederick Wilhelm建议解决三个问题。第一个是创建具有许多正曲率的黎曼度量的新例子的程序。这些指标是在特定的光纤束上进行的。该计划似乎特别有可能在四个球体上成功实现三个光纤束。其次，他将致力于证明他的新猜想-完全正曲线流形的黎曼浸没的像的维度严格大于区域维度的一半。第三，他将尝试完成圆球的黎曼沉没的分类。这三个问题都解决了一般的问题，即空间的曲率如何影响其几何和拓扑？粗略地说，曲率决定了空间的三角。例如，一个人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，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。即使它们从未到达世界的“边缘”，它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 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.