Mathematical Sciences: Geodesic Flows On Manifolds With No Conjugate Points and Related Topics

数学科学：无共轭点流形上的测地流及相关主题

• 批准号: 8702803
• 负责人: Keith Burns
• 金额: $ 0.99万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-05-01 至 1988-03-01
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702803&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geodesic; Flows; Manifolds

Burn's work displays an impressive variety of techniques and questions which stem from meaningful physical applications. He has some fine publications alone and in collaboration with some very able colleagues. Due to their efforts the structural properties of Riemannian manifolds of the negative curvature sort are remakably well understood and they are probing into the nature of physically meaningful phenomena such as entropy. The two main objectives of the current project are to extend recent results for manifolds with non-positive curvature to manifolds with no conjugate points, and to further investigate manifolds with non-positive curvature. Compact manifolds with non-positive curvature are classified by their rank. The geodesic flow is ergodic if and only if the manifold has rank 1, and higher rank examples are quotients of products of rank 1 examples and symmetric spaces. The main problem for manifolds with non- positive curvature is to further understand the examples with rank 1. For manifolds with no conjugate points, it is still unknown whether the analogue of rank 1 will imply ergodicity of the geodesic flow, even in dimension 2.

伯恩的作品展示了令人印象深刻的各种技术和问题，这些技术和问题源于有意义的物理应用。他独自以及与一些非常有能力的同事合作发表了一些优秀的出版物。由于他们的努力，负曲率类黎曼流形的结构特性得到了很好的理解，并且他们正在探索诸如熵之类的具有物理意义的现象的本质。 当前项目的两个主要目标是将非正曲率流形的最新结果扩展到没有共轭点的流形，并进一步研究非正曲率流形。具有非正曲率的紧流形按其等级进行分类。当且仅当流形具有阶 1 时，测地流才是遍历的，并且更高阶的示例是阶 1 示例与对称空间的乘积的商。具有非正曲率的流形的主要问题是进一步理解阶为 1 的示例。对于没有共轭点的流形，阶 1 的类似物是否意味着测地流的遍历性仍然未知，即使在维度 2 中也是如此。