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

伯恩的作品展示了令人印象深刻的各种技术， 这些问题源于有意义的物理应用。他 有一些优秀的出版物单独和合作， 非常能干的同事。由于他们的努力， 负曲率类黎曼流形的性质 他们都很清楚，他们正在探索 物理上有意义的现象，如熵的本质。 目前项目的两个主要目标是扩大 最近的结果与非正曲率流形， 流形没有共轭点，并进一步研究 非正曲率的流形紧流形， 非正曲率通过它们的秩来分类。测地线 流是遍历的当且仅当流形的秩为1，并且 更高秩的示例是秩1示例的乘积的乘积 对称空间。非线性流形的主要问题 正曲率是为了进一步理解例子， 等级1。对于没有共扼点的流形，它仍然是 未知秩1的类似物是否意味着遍历性 测地线流，即使在二维空间中。

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.