| Project/Area Number |
12640097
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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) |
YAMADA Kotaro Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10221657)
HAMADA Tatsuyoshi Fukuoka University, Faculty of Science, Assistant, 理学部, 助手 (90299537)
SUYAMA Yoshihiko Fukuoka University, Faculty of Science, Professor, 理学部, 教授 (70028223)
FURUHATA Hitoshi Hokkaido University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (80282036)
INOGUCHI Jun-ichi Utsunomiya University, Faculty of Education, Associate Professor, 教育学部, 助教授 (40309886)
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| Project Period (FY) |
2000 – 2002
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| Keywords | affine differential geometry / projective differential geometry / information geometry / integrable system / minimal surface / statistical manifold / conformally flat hypersurface / representation formula of Weierstrass type |
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| Research Abstract |
In this research, we studied classical differential geometries, theory of integral systems and information geometry.
1. Classical Differential Geometries (1) We characterized minimal affine hypersurfaces and minimal centroaffine immersions of codimension two. Moreover, we gave an explicit method of constructing self-dual minimal centroaffine surfaces of codimension two.
(2) We studied manifolds with projectively flat torsion-free affine connection whose Ricci curvature is symmetric and definite, and showed fundamental results on the injectivity of the projective developing maps of such manifolds and the convexity of their image.
(3) For conformally flat hypersurfaces of a 4-dimensional sphere, we defined a new conformal invariant. Using the invariant, we characterized the classical examples and constructed new examples.
(4) We developed a very concrete and comprehensive theory on curves and surfaces in 3-dimensional homogeneous spaces.
2. Integrable Systems We investigated various integrable systems appeared in classical differential geometries. We obtained representation formulae for minimal surfaces in 3-dimensional solvable Lie groups and flat surfaces in a 3-dimensional hyperbolic space. We also developed a comprehensive theory of (spacelike) surfaces with harmonic inverse mean curvature in 3-dimensional Riemannian space forms and Lorentzian space forms.
3. Information Geometry and Statistical Manifolds (1) We defined complex statistical manifolds and studied them from the view points of affine differential geometry and of information geometry, especially of quantum estimation theory.
(2) As a generalization of special Kahler manifolds, we defined statistical manifolds with compatible complex structure and investigated their fundamental properties.
(3) On (-1)-conformally flat statistical manifolds, we gave an explicit method of constructing the Volonoi diagrams.
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