Positive and nonnegative curvature on bundles

束上的正曲率和非负曲率

• 批准号: 0355120
• 负责人: Kristopher Tapp
• 金额: $ 6.05万
• 依托单位: Williams College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-09-08 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355120&HistoricalAwards=false
• 关键词: Positive; nonnegative; curvature; bundles

Proposal DMS-0303326P.I.: Kristopher Tapp (Bryn Mawr College)Title: Positive and nonnegative curvature on bundlesABSTRACT:In recent research, the PI developed tools for studying the following two general problems in Riemannian geometry: (1) For which vector bundles does the total space admit a complete Riemannian metric with nonnegative sectional curvature? (2) For which sphere bundles does the total space admit a Riemannian metric with positive curvature? The PI plans to use these tools to find new examples, obstructions, and rigidity theorems. In particular, the PI plans to construct metrics of nonnegative curvature on holomorphic vector bundles over complex projective space, and determine whether Einstein-Hermitian connections can appear as the connections in the normal bundles of their souls. Also, the PI intends to study nonnegatively curved metrics on trivial vector bundles over spheres. This problem is related to the Hopf conjecture, or more specifically to the question: how large is the family of nonnegatively curved metrics on the product of two spheres? The PI will work with undergraduate students studying the metrics on such products obtained by re-scaling the product metric by a compact Lie group which acts by isometries. Finally, the PI proposes to find obstructions to metrics of nonnegative curvature on vector bundles over spheres with prescribed soul metrics.The question of which manifolds can have nonnegative curvature is central to Riemannian geometry. Nonnegative curvature is a visually natural restriction on the way in which an object curves about in space. All known examples come from compact Lie groups with bi-invariant metrics, which are indispensable tools in diverse fields of mathematics, physics, cosmology, and other disciplines in which simplification is achieved through symmetry. The search for new examples of manifolds with nonnegative (or positive) curvature has a long history, yet very few constructions are known. Since his tools represent a construction which is substantially different form known methods, the PI believes they deserve further study.

提案DMS-0303326 P.I.：Kristopher Tapp（Bryn Mawr College）题目：黎曼几何中的正曲率和非负曲率摘要：在最近的研究中，PI开发了研究黎曼几何中以下两个一般问题的工具：（1）对于哪些向量丛，全空间允许具有非负截面曲率的完备黎曼度量？ (2)对于哪些球丛，全空间允许具有正曲率的黎曼度量？ PI计划使用这些工具来寻找新的例子、障碍和刚性定理。 特别地，PI计划构造复射影空间上全纯向量丛的非负曲率度量，并确定Einstein-Hermitian联络是否可以作为其灵魂的法丛中的联络出现。 此外，PI打算研究球上平凡向量丛上的非负弯曲度量。 这个问题是有关的霍普夫猜想，或更具体地说，问题：有多大的家庭非负弯曲度量的产品的两个领域？ PI将与本科生一起研究这些产品的度量，这些产品是通过一个紧凑的李群来重新缩放产品度量而获得的。 最后，PI提出在具有指定灵魂度量的球面上的向量丛上寻找非负曲率度量的障碍。哪些流形可以具有非负曲率的问题是黎曼几何的核心。 非负曲率是对物体在空间中弯曲的方式的视觉自然限制。 所有已知的例子都来自具有双不变度量的紧致李群，它们是数学、物理、宇宙学和其他通过对称性实现简化的学科中不可或缺的工具。 寻找具有非负（或正）曲率的流形的新例子有很长的历史，但已知的构造很少。 由于他的工具代表了一种与已知方法截然不同的结构，PI认为它们值得进一步研究。