Researches on vector fields in the large by Riccati equation

Riccati方程研究大向量场

基本信息

批准号：
09640097
负责人：
INNAMI Nobuhiro
金额：
$ 1.86万
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：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640097/
关键词：
Riemannian geometry billiard theory of parallels リッカチ微分方程式ワープド積

项目摘要

Many results for Riemannian manifolds as geometry of geodesics have been obtained by using solutions of Jacobi and Riccati equations. In the light of these facts the theory of those equations can be applied to the researches of gradient vector fields, natural Lagrangian systems, differential equations and flows satisfying Huygens' principle, Finsler geometry, billiard ball problems, glued Riemannian manifolds. One of most important problems is to decide when there exists a unique solution of Riccati equation in the large.The symmetric solutions of the matrix Riccati equation have some nice properties if they are defined on the whole real numbers. The existence and uniqueness problems are important under a lot of situations. In particular, the uniqueness of solutions sometimes comes from the theory of parallels. In this research project we developed the theory of Jacobi and Riccati equations, applied it to the theory of parallels, and, as a result, made the topological and geometrical structures of manifolds clear.In 1997 we introduced the equation for Jacobi vector fields along geodesics in glued Riemannian manifolds, and we characterized the topological and geometrical structures of warped products by some properties of gradient vector fields. In 1998 and 1999 we have some results for convex billiard ball problems in a plane by applying the theory of parallels in the configuration spaces to Jacobi vector fields along billiard ball trajectories in the Euclidean plane. Mr. Hiroyuki Sakai helped us with our project, in proving some results concerning Laplace operator, eigenvalues of Laplacian on compact glued manifolds

通过使用雅可比和黎卡提方程的解，已经获得了作为测地线几何的黎曼流形的许多结果。鉴于这些事实，这些方程的理论可以应用于梯度矢量场、自然拉格朗日系统、满足惠更斯原理的微分方程和流、芬斯勒几何、台球问题、粘黎曼流形的研究。最重要的问题之一是确定Riccati方程在大范围内何时存在唯一解。矩阵Riccati方程的对称解如果定义在整个实数上，则具有一些很好的性质。存在性和唯一性问题在很多情况下都很重要。特别是，解的唯一性有时来自于相似理论。在这个研究项目中，我们发展了雅可比和黎卡提方程的理论，并将其应用于平行线理论，从而使流形的拓扑和几何结构变得清晰。1997年，我们引入了沿测地线的雅可比向量场方程在粘连黎曼流形中，我们通过梯度矢量场的一些性质来表征翘曲乘积的拓扑和几何结构。 1998 年和 1999 年，我们通过将构型空间中的平行理论应用于沿欧几里得平面中台球轨迹的雅可比矢量场，得到了平面中凸台球问题的一些结果。 Hiroyuki Sakai 先生帮助我们完成了我们的项目，证明了有关拉普拉斯算子、紧致粘连流形上拉普拉斯算子特征值的一些结果