Project/Area Number  10440022 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
Geometry

Research Institution  Yamaguchi University 
Principal Investigator 
SHIMA Hirohiko Yamaguchi University, Mathematics, Professor, 理学部, 教授 (70028182)

CoInvestigator(Kenkyūbuntansha) 
河津 清 山口大学, 教育学部, 教授 (70037258)
柳 研二郎 山口大学, 工学部, 教授 (90108267)
NAKAUCHI Nobumitsu Yamaguchi University, Mathematics, Associate Professor, 理学部, 助教授 (50180237)
INOUE Toru Yamaguchi University, Mathematics, Professor, 理学部, 教授 (00034728)
NAITOH Hiroo Yamaguchi University, Mathematics, Professor, 理学部, 教授 (10127772)
KOMIYA Katsuhiro Yamaguchi University, Mathematics, Professor, 理学部, 教授 (00034744)
HATAYA Yasushi Yamaguchi University, Mathematics, Assistant, 理学部, 助手 (20294621)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥6,600,000 (Direct Cost : ¥6,600,000)
Fiscal Year 1999 : ¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1998 : ¥3,300,000 (Direct Cost : ¥3,300,000)

Keywords  Hessian manifold / Kahler manifold / dual connection / Codazzi structure / information geometry / affine differential geometry / Hessian多様体 / Kahler多様体 / 双対接続 / 双対構造 / 情報幾何学 / 射影的平坦接続 / Hessian構造 
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 centroaffine hypersurface immersion. (2) There is a bijective correspondence between semisimple symmetric spaces with invariant projectively flat affine connections and centralsimple 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.
