Researches on vector fields in the large by Riccati equation

Riccati方程研究大向量场

• 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

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

利用Jacobi和Riccati方程的解，得到了作为测地线几何的黎曼流形的许多结果。根据这些事实，这些方程的理论可以应用于梯度向量场、自然拉格朗日系统、满足惠更斯原理的微分方程和流动、Finsler几何、台球问题、胶合黎曼流形的研究。Riccati方程的一个重要问题是确定其解在大范围内是否存在唯一性，矩阵Riccati方程的对称解如果定义在全真实的实数上，则具有很好的性质。解的存在唯一性问题在很多情况下都是重要的。特别是，解的唯一性有时来自平行理论。在本研究项目中，我们发展了Jacobi和Riccati方程的理论，并将其应用到平行线理论中，从而使流形的拓扑和几何结构更加清晰，1997年我们引入了胶合黎曼流形中沿着测地线的Jacobi向量场方程，并利用梯度向量场的一些性质刻画了翘曲积的拓扑和几何结构。在1998年和1999年，我们有一些结果凸台球问题在一个平面上的应用理论的平行线的配置空间的雅可比向量场沿着台球轨迹在欧几里德平面。坂井博之先生帮助我们完成了我们的项目，证明了关于紧胶合流形上的拉普拉斯算子和拉普拉斯算子的特征值的一些结果

项目成果

Nobuhiro INNAMI: "Gradient vector fields which characterize warped products"Mathematica Scandinavica. (発表予定).

Nobuhiro INNAMI：“表征扭曲产品的梯度矢量场”Mathematica Scandinavica（待公布）。

Takashi Oguro: "Four-dimensional almost Kahler Einstein and *-Einstein manifolds"Geom. Dedicata. 69, 1. 91-112 (1998)

Takashi Oguro：“四维几乎卡勒爱因斯坦和*-爱因斯坦流形”Geom。

S.Ogose 他: "Anote on Polysurface groups" Nihonkai Math.J.8・1. 7-18 (1997)

S.Ogose 等：“多曲面群的注释”Nihonkai Math.J.8・1（1997）。

T. Ogura: "Four-dimensional almost kahler Einstein and *-Einstein manifolds"Geom. Dedicata. 69. 91-112 (1998)

T. Ogura：“四维几乎卡勒爱因斯坦和*-爱因斯坦流形”Geom。

Nobuhiro INNAMI: "Integral formulas for polyhedral and spherical billiards" J.Math.Soc.Japan. 50・2. 339-357 (1998)

Nobuhiro INAMI：“多面体和球形台球的积分公式”J.Math.Soc.Japan 50・2（1998）。