2004 Fiscal Year Final Research Report Summary
Studies of glued Riemannian manifolds
| Project/Area Number |
13640068
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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 Niigata University, Faculty of Sciences, Professor, 理学部, 教授 (20160145)
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| Co-Investigator(Kenkyū-buntansha) |
SEKIGAWA Kouei Niigata University, Faculty of Sciences, Professor, 理学部, 教授 (60018661)
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| Project Period (FY) |
2001 – 2004
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| Keywords | geometru of geodesics / Riemannian geometry |
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| Research Abstract |
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.
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