2005 Fiscal Year Final Research Report Summary
Hyperbolic structures on manifolds and their deformations
| Project/Area Number |
15540069
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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 | Kyoto University |
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Principal Investigator |
FUJII Michihiko Kyoto University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (60254231)
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| Co-Investigator(Kenkyū-buntansha) |
IMANISHI Hideki Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90025411)
UE Masaaki Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80134443)
SAITO Hiroshi Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20025464)
MORIMOTO Yoshinori Kyoto University, Graduate School of Human and Environmental Studies, Professor, 大学院・人間・環境学研究科, 教授 (30115646)
OCHIAI Hiroyuki Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (90214163)
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| Project Period (FY) |
2003 – 2005
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| Keywords | manifolds / hyperbolic structure / deformation / cone manifold / hypergeometric equation / regular singular point / elliptic curve / rational point |
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| Research Abstract |
The main purpose of this research was to study deformations of a 3-dimensional hyperbolic cone manifold M with non-empty singular set. The head investigator Fujii and the investigator Ochiai constructed an algorithm for solving ordinary differential equations of Fuchsian type which describe deformations of M. This algorithm was reported in the journal, Interdisciplinary Information Sciences 9(2003). Fujii found some relation between the degeneration of hyperbolic structures on a hyperbolic knot and some rational points of an elliptic curve. This result was reported in the journal, J.Math.Kyoto Univ.45(2005). Fujii succeeded in drawing such a rational point on the modular curve of the elliptic curve. This result was reported in the conference, "Riemann Surfaces and Discontinuous Groups" at Tokyo Institute of Technology in December 2004. Furthermore, Fujii found the possibility that these rational points are located on some circle which constitute the boundary of some fundamental region of the modular curve. This result was reported at the conference, "Topology and Computer 2005", at Osaka Sangyo University in November 2005.
The investigator Ue proved that the Fukumoto-Furuta invariant for Seifert 3-manifolds coincides with the Neumann-Siebenmann invariant, and also proved its spin rational homology cobordism invariance. Ue obtained the conditions for Seifert 3-manifolds to be obtained by Dehn surgery on knots in the 3-sphere. The investigator Saito succeeded to give a concrete description of admissible representations on non-Archimedes local fields in terms of intertwining operators.
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Research Products
(12 results)
