Studies of glued Riemannian manifolds

粘合黎曼流形的研究

基本信息

批准号：
13640068
负责人：
INNAMI Nobuhiro
金额：
$ 2.18万
依托单位：
Niigata University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

We find what rendition on gradient vector fields characterizes warped products, Riemannian products and round spheres. To do this we apply the theory of Jacobi equations without conjugate points to the differential maps of the local one-parameter groups generated by gradient vector fields.We say that a manifold M is a glued manifold if M is a union of complete connected manifolds which are glued at their boundary Geodesies in a glued Riemannian manifold M are by definition locally minimizing curves in M. The variation vector fields through geodesies satisfy the Jacobi equation in each component manifold In this project we find the equation which show how Jacobi vector fields change in passing across the boundary of a component manifold into the neighboring component As an application we characterize glued Riemannian manifolds whose glued boundary separates conjugate points.Circles and Ellipses has been characterized by some properties of billiard ball trajectories. Those properties have been discussed in connection with the characterization of flat metrics on tori by some families of geodesics and tori of revolution. The main method is the geometry of geodesies due to H.Busemann which was reconstructed in the configuration space by V.Bangert. In particular, the theory of parallels plays an important role in this work. Roughly speakin, there exists a foliation of parallels in the configuration space for billiards if and only if there exists a foliation of non-null homotopic curves in the phase space which is invariant under the billiard ball map.

我们发现梯度矢量场上的表现表征了扭曲乘积、黎曼乘积和圆形球体。为此，我们将没有共轭点的雅可比方程理论应用于由梯度向量场生成的局部单参数组的微分映射。如果 M 是完全连通流形的并集，则我们说流形 M 是胶合流形，其中粘在它们的边界上粘连黎曼流形 M 中的大地测量根据定义是 M 中的局部最小化曲线。通过大地测量的变化向量场满足每个分量流形中的雅可比方程在这个项目中，我们找到了显示雅可比向量场如何变化的方程通过穿过组件流形的边界进入相邻组件作为一个应用，我们描述了粘连黎曼流形，其粘连边界将共轭点分开。圆和椭圆已经用台球轨迹的一些属性来表征。一些测地线和旋转环面族已经结合环面平面度量的特征讨论了这些属性。主要方法是 H.Busemann 提出的大地测量几何学，由 V.Bangert 在位形空间中重建。特别是，平行理论在这项工作中发挥着重要作用。粗略地说，当且仅当在台球映射下不变的相空间中存在非零同伦曲线的叶状结构时，台球的配置空间中才存在平行线的叶状结构。