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Hessian Geometry and Information Geometry

Research Project

Project/Area Number 15540080
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionYamaguchi University

Principal Investigator

SHIMA Hirohiko  Yamaguchi University, Faculty of Science, Prof., 理学部, 教授 (70028182)

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)
内藤 博夫  山口大学, 理学部, 教授 (10127772)
菊政 勲  山口大学, 理学部, 助教授 (70234200)
幡谷 泰史  山口大学, 理学部, 助手 (20294621)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2004: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
KeywordsHessian metrics / Hessian structures / Hessian manifolds / Codazzi structures / dual connections / Information geometry / affine differential geometry / Kaehler幾何学 / 平坦接続 / 射影的平坦接続
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.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (16 results)

All 2005 2004 Other

All Journal Article (11 results) Book (2 results) Publications (3 results)

  • [Journal Article] On computation of a coefficient of a power series root2005

    • Author(s)
      Takuya Kitamoto
    • Journal Title

      IEICE Trans.Fundamentals (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The divisibility in the cut-and-paste group of G-manifolds and fibring over the circle within a cobordism class2005

    • Author(s)
      Katsuhiro, Komiya
    • Journal Title

      Osaka J.Math 42(to appear)

    • NAID

      120004844105

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The divisibility in the cut-and-paste group of G-manifolds and fibring over the circle within a cobordism class2005

    • Author(s)
      Katsuhiro Komiya
    • Journal Title

      Osaka J.Math. 42

    • NAID

      120004844105

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On computation of a coefficient of a power series root2005

    • Author(s)
      Takuya Kitamoto
    • Journal Title

      IEICE Trans.Fundamentals

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On computation of a coefficient of a power series root2005

    • Author(s)
      Takuya Kitamoto
    • Journal Title

      IEICE Trans.Fundamentals (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The divisibility in the cut-and-paste group of G-manifolds and fibring over the circle within a cobordism class2005

    • Author(s)
      Katsuhiro Komiya
    • Journal Title

      Osaka J.Math 42(印刷中)

    • NAID

      120004844105

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On spherically symmetric solutions of the relativistic Euler equation2004

    • Author(s)
      C.H.Hsu, S.S.Lin, T.Makino
    • Journal Title

      J.Differential Equations 201

      Pages: 1-24

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On spherically symmetric solutions of the relativistic Euler equation2004

    • Author(s)
      C.H.Hsu
    • Journal Title

      J.Differential Equations 201

      Pages: 1-24

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Decaying solution of a Navier-Stokes flow without surface tension

    • Author(s)
      Yasushi Hataya
    • Journal Title

      J.Math.Kyoto Univ. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Decaying solution of a Navier-Stokes flow without surface tension

    • Author(s)
      Yasushi Hataya
    • Journal Title

      J.Math.Kyoto Univ. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Decaying solution of a Navier-Stokes flow without surface tension

    • Author(s)
      Yasushi Hataya
    • Journal Title

      J.Math.Kyoto Univ. (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Book] Geometry of Hessian structures2005

    • Author(s)
      Hirohiko Shima
    • Total Pages
      250
    • Publisher
      World Scientific
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Book] Geometry of Hessian Structures2005

    • Author(s)
      Hirohiko Shima
    • Total Pages
      250
    • Publisher
      World Scientific
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] S.Kawai, N.Nakauchi: "The first eigenvalue of the p-Laplacian on a compact Riemannian manifold"Nonlinear Analysis. 55. 33-46 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] I.Kikumasa, H.Yoshimura: "Commutative algebras with radical cube zero"Communications in Algebra. 31. 1837-1858 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] I.Kikumawsa, H.Yoshimura: "Some type of commutative artin algebras II"Proceedings of the seventh symposium on Algebra, Languages and Commutation. (to appear).

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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