2006 Fiscal Year Final Research Report Summary
Research on global structures of Lorentzian semi-symmetric spaces
| Project/Area Number |
16540089
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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 | Fukuoka Institute of Technology |
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Principal Investigator |
GOTO Midori Fukuoka Institute of Technology, Information Engineering, Professor, 情報工学部, 教授 (60162161)
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| Co-Investigator(Kenkyū-buntansha) |
SUGAHARA Kunio Osaka Kyoiku University, Education, Professor, 教育学部, 教授 (20093255)
NISHIHARA Masaharu Fukuoka Institute of Technology, Information Engineering, Professor, 情報工学部, 教授 (20112287)
NISHIYAMA Takahiro Yamaguchi University, Graduate School of Science and Engineering, Associate Professor, 大学院理工学研究科, 助教授 (60333241)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Lorentzian metric / Lorentz manifold / semi-symmetric space / Liouville metric / Liouville manifold / Liouville surface / generalized Liouville manifold |
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| Research Abstract |
In the study of Riemannian geometry, it is important to investigate geodesic behaviors, which effect on topological structures of manifolds. Curvatures (local notion) of Riemannian manifolds often have influence on topological structures (global notion). What about in Lorentz geometry? From the viewpoints, we investigate Lorentz geometry. Global behaviors of geodesics are not well-known for most manifolds. Hence, investigations of Liouville structure, which have been initiated by C.G.F.Jacobi in 1866, turn out to be important.
Our main results which we obtained during the period are in the following two forthcoming papers, both of which are joint works by Midori Goto and Kunio Sugahara :
1. Generalized Liouville manifolds.
In this paper, we defined a new notion, the generalized Liouville structure, and succeeded to prove that all quadric surfaces in the n+1 dimensional Cartesian space possess the generalized Liouville structures with respect to the indefinite/definite metrics, besides the Eucledian metric. In the classification of Riemann-Liouville surfaces the number of singularities plays a key role in Kiyohara's and Igarashi-Kiyohara-Sugahara's papers about Liouville surfaces. We proved that there is no such kind of singularity in the Lorentz-Liouville surface. For generalized Liouville maniflods of dimension >2, we determined the set of singularities. The classification problems for generalized Liouville manifolds are supposed to be made clear.
2. A remark on Lorentzian metrics of 3-dimensional manifolds.
In the paper, we see that there are 3-dimensional Riemannian manifolds whose Riemannian connections coincide with the Lorentzian connections associated with the naturally defined Lorentzian metrics. Reviewing the result, we conjecture that, except for the case of constant curvature, there may not be expected any relations between curvatures and topological structures.
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