2005 Fiscal Year Final Research Report Summary
Classical differential geometry from the modern viewpoint and its applications
| Project/Area Number |
15540100
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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 University |
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Principal Investigator |
KUROSE Takashi Fukuoka University, Faculty of Science, Associate Professor, 理学部, 助教授 (30215107)
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| Co-Investigator(Kenkyū-buntansha) |
SUYAMA Yoshihiko Fukuoka University, Faculty of Science, Professor, 理学部, 教授 (70028223)
HAMADA Tatsuyoshi Fukuoka University, Faculty of Science, Assistant, 理学部, 助手 (90299537)
YAMADA Kotaro Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10221657)
INOGUCHI Jun-ichi Utsunomiya University, Faculty of Education, Associate Professor, 教育学部, 助教授 (40309886)
FURUHATA Hitoshi Hokkaido University, Faculty of Science, Lecturer, 大学院・理学研究科, 講師 (80282036)
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| Project Period (FY) |
2003 – 2005
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| Keywords | classical differential geometry / affine hypersphere / discretization / conformally flat hypersurface / real hypersurface / singularity / flat front / maxface |
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| Research Abstract |
In this research, we planned to give a now development of the theories of classical differential geometry by restructuring them from the modern viewpoint, particularly, of the theories of integrable systems and of singularities. Our main results are the following :
1.(1)In affine differential geometry, one of the core theories of classical differential geometry, we mainly studied the geometry of affine hyperspheres and their representation formulae, and showed a relationship with the geometry of holomorphic statistical manifolds and the several properties of the center maps. We also studied the discretization of affine or centroaffine plane curves and gave a description of their time-evolution following discrete soliton equations ; (2)we characterized the classical examples of conformally flat hypersurfaces in 4-dimensional Euclidean space and constructed new examples ; (3)for real hypersurfaces in complex space forms, we introduced a new geometric invariant and classified Hopf real hypersurfaces using the invariant.
2.We studied the geometric properties of surfaces with singularities and obtained the following results : (1)We constructed the theory of flat fronts, the flat surfaces with singularities of a certain kind in 3-dimensional hyperbolic space. In particular, we defined (weak) completeness of flat fronts and showed their global properties ; (2)investigating the properties of the singularities of maximal surfaces in 3-dimensional Minkowski space, we constructed the theory of maxfaces, the spacelike maximal surfaces allowing singularities of a certain kind.
3.We studied transformations of surfaces and showed that the transformations given by the sphere congruences in Moebius geometry are obtained by the complexified line congruences in Euclidean space. We also investigated biharmonic curves in 3-dimensional homogeneous spaces and determined such curves when the homogeneous spaces are irreducible and reductive.
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