2004 Fiscal Year Final Research Report Summary
Hessian Geometry and Information Geometry
| Project/Area Number |
15540080
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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 | Yamaguchi University |
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Principal Investigator |
SHIMA Hirohiko Yamaguchi University, Faculty of Science, Prof., 理学部, 教授 (70028182)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAUCHI Nobumitsu Yamaguchi University, Faculty of Sciences, Assoc.Prof., 理学部, 助教授 (50180237)
YOSHIMURA Hiroshi Yamaguchi University, Faculty of Sciences, Assoc.Prof., 理学部, 助教授 (00182824)
MAKINO Tetsu Yamaguchi University, Faculty of Engineering, Prof., 工学部, 教授 (00131376)
KITAMOTO Takuya Yamaguchi University, Faculty of Education, Assoc.Prof., 教育学部, 助教授 (30241780)
KOMIYA Katsihiro Yamaguchi University, Faculty of Science, Prof., 理学部, 教授 (00034744)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Hessian metrics / Hessian structures / Hessian manifolds / Codazzi structures / dual connections / Information geometry / affine differential geometry |
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| Research Abstract |
Let M be a flat manifold with flat connection D. A Riemannian metric g on M is said to be a Hessian metric if it is locally expressed by the Hessian with respect to the flat connection D. Hessian geometry (the geometry of Hessian manifolds) is a very close relative of Kahlerian geometry, and may be placed among, and finds connection with important pure mathematical fields such as affine differential geometry, homogeneous spaces, cohomology and others. Moreover, Hessian geometry, as well as being connected with these pure mathematical areas, also, perhaps surprisingly, finds deep connections with information geometry. The notion of flat dual connections, which plays an important role in information geometry, appears in precisely the same way for our Hessian structures. Thus Hessian geometry offers both an interesting and fruitful area of research. In this project we study Hessian geometry putting together Kahlerian geometry, affine differential geometry and information geometry, and obtained the following results.
1.We constructed new Hessian metrics applying a method of information geometry. Conversely, we obtained families of probability distributions using a differential geometric method.
2.We developed affine differential geometry of level surfaces of potential functions of Hessian metrics, and investigating Laplacians of gradient mappings we proved a certain problem similar to the affine Bernstein's problem proposed by S.S. Chern.
3.We obtained a duality theorem and vanishing theorems for Hessian manifolds similar to that of Kahlerian geometry.
4.Since a Hessian structure satisfies the Codazzi equation, the notion of Hessian structures is naturally extended to the Codazzi structures. We proved that a manifold with a constant Codazzi structure has an immersion into a certain homogeneous Hessian manifold of codimension 1.
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