1999 Fiscal Year Final Research Report Summary
Differential Geometry and Information Geometry II
| Project/Area Number |
10440022
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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, Mathematics, Professor, 理学部, 教授 (70028182)
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| Co-Investigator(Kenkyū-buntansha) |
KOMIYA Katsuhiro Yamaguchi University, Mathematics, Professor, 理学部, 教授 (00034744)
INOUE Toru Yamaguchi University, Mathematics, Professor, 理学部, 教授 (00034728)
NAITOH Hiroo Yamaguchi University, Mathematics, Professor, 理学部, 教授 (10127772)
HATAYA Yasushi Yamaguchi University, Mathematics, Assistant, 理学部, 助手 (20294621)
NAKAUCHI Nobumitsu Yamaguchi University, Mathematics, Associate Professor, 理学部, 助教授 (50180237)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Hessian manifold / Kahler manifold / dual connection / Codazzi structure / information geometry / affine differential geometry |
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| Research Abstract |
A pair (D, g) of an affine connection D and a Riemann metric g is said to be a Codazzi structure if g satisfies Codazzi equattion with respect to D. For a Codazzi structure (D, g) , D is flat if and only if (D, g) is a Hessian structure, that is, g is locally expressed by a Hessian with respect to affine coordinate systems for D. Hessian (Codazzi) structures are deeply connected with Kahler geometry and affine differential geometry, and play important, essential and central roles in information geometry. In this project we engaged in fundamental researches for Hessian (Codazzi) structures from both differential geometric and information geometric viewpoints, and obtained the following results.
1 We relate the existence of invariant projectively flat affine connections to that of certain affine representation of Lie algebras. Using such affine representation we proved :
(1) A homogeneous space G/K admits an invariant projectively flat affine connection if and only if G/K has an equivariant centro-affine hypersurface immersion.
(2) There is a bijective correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.
(3) A homogeneous space admits an invariant Codazzi structure of constant curvature c=0 if and only if it has an equvariant immersion of codimension 1 into a certain homogenous Hessian manifolds.
2 For a linear mapping ρ of a domain Ω into the space of positive definite symmetric matrices we conatructed an exponential family of probability distributions parametrized by the elements in RィイD1nィエD1×Ω, and studied a Hessian structure on RィイD1nィエD1×Ω given by the exponential family. Using ρ we introduced a Hessian structure on a vector bundle over a compact hyperbolic affina manifold and proved a certain vanishing theorem.
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