Project/Area Number  09640097 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  Niigata University 
Principal Investigator 
INNAMI Nobuhiro Graduate School of Science, Niigata University Prof., 大学院・自然科学研究所, 教授 (20160145)

CoInvestigator(Kenkyūbuntansha) 
WATABE Tsuyoshi Faculty of Science, Niigata University Prof., 理学部, 教授 (60018257)
SEKIGAWA Kouei Faculty of Science, Niigata University Prof., 理学部, 教授 (60018661)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,900,000 (Direct Cost : ¥2,900,000)
Fiscal Year 1999 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1997 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  Riemannian geometry / billiard / theory of parallels / リーマン幾何 / ビリヤード / 平行線の幾何 / リッカチ微分方程式 / ワープド積 
Research Abstract 
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
