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.

伯恩的作品显示出源于有意义的物理应用的各种技术和问题。他独自与一些非常有能力的同事合作，只有一些出色的出版物。由于他们的努力，对负曲率排序的riemannian流形的结构特性得到了很好的理解，并且正在探究物理有意义的现象（例如熵）的性质。 当前项目的两个主要目标是将具有非阳性曲率的歧管扩展到没有共轭点的歧管，并进一步研究具有非阳性曲率的歧管。具有非阳性曲率的紧凑型歧管由其等级分类。当且仅当歧管具有等级1时，而较高的等级示例是等级1示例和对称空间的产品的商。具有非正曲率的流形的主要问题是进一步了解等级1的示例。对于没有共轭点的歧管，即使在维度2中，等级1的类似物的类似物是否会暗示地球流量的奇质性。