1999 Fiscal Year Final Research Report Summary
Researches on vector fields in the large by Riccati equation
| Project/Area Number |
09640097
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Niigata University |
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Principal Investigator |
INNAMI Nobuhiro Graduate School of Science, Niigata University Prof., 大学院・自然科学研究所, 教授 (20160145)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1997 – 1999
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| Keywords | Riemannian geometry / billiard / theory of parallels |
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| 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
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