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2011 Fiscal Year Final Research Report

Riemannian Geometry and Information Geometry of Poisson kernels and heat kernels

Research Project

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Project/Area Number 21540065
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionUniversity of Tsukuba

Principal Investigator

ITO Mitsuhiro  筑波大学, 名誉教授 (40015912)

Co-Investigator(Renkei-kenkyūsha) SATO Hiroyasu  東京電機大学, 情報環境学部, 助教 (00375396)
Project Period (FY) 2009 – 2011
Keywordsフィシャー情報計量 / ポアソン核 / 熱核 / 確率測度 / ビューズマン関数 / 漸近調和空間 / 体積エントロピー
Research Abstract

We investigated Riemannian geometry and Fisher information geometry of the Poisson kernel map φ from an Hadamard manifold X into the space P(∂X) of probability measures of positive density function, absolutely continous with respect to the Riemannian volume, defined on the ideal boundary of X. We obtained the following results : if the mapφ is homothetic and minimal, then the Poisson kernel is represented in terms of the Busemann function so that X turns out to be asymptotically harmonic and satisfies the axiom of visibility and furthermore we can show that this homothetic constant coincides with the volume entropy of X.

  • Research Products

    (20 results)

All 2012 2011 2010 2009

All Journal Article (7 results) (of which Peer Reviewed: 7 results) Presentation (13 results)

  • [Journal Article] Poisson kernel and Asymptotical Harmonicity2011

    • Author(s)
      M. Itoh, H. Satoh, Fisher Information Geometry
    • Journal Title

      Diff. Geom. Appl

      Volume: 29 Pages: S107-S115

    • Peer Reviewed
  • [Journal Article] Information Geometryof Poisson kernels on Damek-Riccispaces2010

    • Author(s)
      M. Itoh, H. Satoh
    • Journal Title

      Tokyo Math. Journal

      Volume: 33 Pages: 129-144

    • Peer Reviewed
  • [Journal Article] 確率測度の空間の調和性とその周辺2010

    • Author(s)
      M. Itoh, H. Satoh
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 1720 Pages: 99-111

    • Peer Reviewed
  • [Journal Article] Fisher Information Geometry, Poisson Kernel and Harmonicity2010

    • Author(s)
      M. Itoh
    • Journal Title

      Journal of Geometry

      Volume: 97 Pages: 131

    • Peer Reviewed
  • [Journal Article] Seiberg-Wittentheory and the geometric structureR x H^22009

    • Author(s)
      M. Itoh, T. Yamase
    • Journal Title

      Hokkaido Math. Journ

      Volume: 38 Pages: 67-81

    • Peer Reviewed
  • [Journal Article] Harmonic Spaces and Fisher Information Geometry2009

    • Author(s)
      M. Itoh, H. Satoh
    • Journal Title

      Proc. 13^<th> Intern. Workshop for Diff. Geometry

      Pages: 27-36

    • Peer Reviewed
  • [Journal Article] Geometry of Bochner Curvature of Kaehler Manifold2009

    • Author(s)
      M. Itoh, R. Kawakami
    • Journal Title

      Proc. 13^<th> Intern. Workshopfor Diff. Geometry

      Pages: 15-26

    • Peer Reviewed
  • [Presentation] ホロ球と双曲空間の幾何2012

    • Author(s)
      伊藤光弘, 佐藤弘康
    • Organizer
      日本数学会年会
    • Place of Presentation
      東京理科大学
    • Year and Date
      2012-03-26
  • [Presentation] Mean curvature of Horospheresand Volume Entropy of an Hadamard Manifold2012

    • Author(s)
      M. Itoh
    • Organizer
      Miniworkshop on Horospheres and Four Dimensional Manifolds
    • Place of Presentation
      Kyungpook National Univ., Korea
    • Year and Date
      2012-03-03
  • [Presentation] Geometry of Horospheres and Hyperbolic Spaces2012

    • Author(s)
      M.Itoh
    • Organizer
      Miniworkshop on Horospheres and Four Dimensional Manifolds
    • Place of Presentation
      Kyungpook National Univ., Korea
    • Year and Date
      2012-03-02
  • [Presentation] Information Geometry of Hadamard Manifolds2011

    • Author(s)
      M. Itoh
    • Organizer
      Poisson Kernelsand Horospheres
    • Place of Presentation
      Academia Sinica, Taipei, Taiwan
    • Year and Date
      2011-12-12
  • [Presentation] ホロ球の平均曲率と漸近調和Hadamard多様体の体積エントロピー2011

    • Author(s)
      伊藤光弘, 佐藤弘康
    • Organizer
      日本数学会秋季分科会
    • Place of Presentation
      信州大学
    • Year and Date
      2011-09-21
  • [Presentation] Geometry of Horospheres, Helgason-Fourier Transform over an Hadamard Manifold2011

    • Author(s)
      M. Itoh
    • Organizer
      Miniworkshop in Diff. Geometry
    • Place of Presentation
      Taegu, KyungpookNational Univ., Korea
    • Year and Date
      2011-06-03
  • [Presentation] Damek-Ricci空間内のホロ球面の主曲率について2010

    • Author(s)
      伊藤光弘, 佐藤弘康
    • Organizer
      日本数学会秋季分科会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-24
  • [Presentation] Fisher Information Geometry2010

    • Author(s)
      M. Itoh, H. Satoh
    • Organizer
      Poisson kernel and Asymptotical Harmonicity, Conferenceof DGA2010
    • Place of Presentation
      Brno, Czech
    • Year and Date
      2010-08-28
  • [Presentation] Fisher Information Metrics, Poisson Kernels and harmonic Spaces2010

    • Author(s)
      伊藤光弘
    • Organizer
      九大微分幾何学セミナー
    • Place of Presentation
      九州大学, 福岡
    • Year and Date
      2010-03-16
  • [Presentation] Einstein Self-Dual 4-manifolds and Related Topics2009

    • Author(s)
      伊藤光弘
    • Organizer
      筑波大学微分幾何学研究会
    • Place of Presentation
      つくば
    • Year and Date
      2009-12-05
  • [Presentation] Geometry of Bochner Curvature of Kaehler Manifolds2009

    • Author(s)
      M. Itoh, R. Kawakami
    • Organizer
      13^<th> Intern. Workshop of Diff. Geometry, Kyungpook Univ
    • Place of Presentation
      Taegu, Korea
    • Year and Date
      2009-11-07
  • [Presentation] harmonic Spaces and Fisher Information Geometry of Poisson Kernel2009

    • Author(s)
      M. Itoh, H. Satoh
    • Organizer
      13^<th> Intern. Workshop of Diff. Geometry, Kyungpook Univ
    • Place of Presentation
      Taegu, Korea
    • Year and Date
      2009-11-05
  • [Presentation] Fisher Information Geometry, Poisson Kernel and Harmonicity2009

    • Author(s)
      M. Itoh
    • Organizer
      9^<th> Intern. Conference on Geometry andAppl
    • Place of Presentation
      Varna, Bulgaria
    • Year and Date
      2009-09-07

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Published: 2013-07-31  

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