Fisher information geometry of Riemannian manifolds and Poisson kernel, heat kernel

• Project/Area Number: 24540065
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: University of Tsukuba
• Principal Investigator: ITOH Mitsuhiro 筑波大学, 名誉教授 (40015912)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• Keywords: Fisher 計量 / Hadamard 多様体 / 理想境界 / Busemann 関数 / Poisson 核 / 漸近調和 / 双曲空間 / 体積エントロピー / Fisher 情報計量 / 重心 / ホロ球 / ヤコビテンソル / ビューズマン関数 / 確率測度 / 情報幾何学 / ブーゼマン関数 / ダメック-リッチ空間

Outline of Final Research Achievements

We developed geometry of horospheres, level hypersurfaces of Busemann function and obtained rigidity theorems of real, complex, quaternionic hyperbolic spaces in terms of volume entropy These results are consequence of the identity theorem of volume entropy and horosphere mean curvature for an asymptotically harmonic Hadamard manifold. We obtained information geometry of barycenter map on ideal boundary of an Hadamard manifold by using theory developed by T.Friedrich.

Research Products

• [Journal Article] Geometry of Fisher Information Metric and the Barycenter Map (2015)
  • Author(s): M. Itoh and H. Satoh
  • Journal Title: Entropy Volume: 17(4) Issue: 4 Pages: 1814-1849
  • DOI: 10.3390/e17041814
  • Peer Reviewed / Open Access
• [Journal Article] Information Geometry of Busemann-Barycenter for Probability Measures (2015)
  • Author(s): M. Itoh and H. Satoh
  • Journal Title: International Journal of Mathematics Volume: 27 Pages: 1-20
  • Peer Reviewed / Open Access
• [Journal Article] Fisher Information Geometry of The Barycenter Map (2014)
  • Author(s): M. Itoh and H. Satoh
  • Journal Title: AIP Conference Proceedings Volume: 1641 Pages: 90-96
  • DOI: 10.1063/1.4905967
  • Peer Reviewed / Open Access
• [Journal Article] Information Geoemtry of Barycenter Map (2014)
  • Author(s): M. Itoh and H. Satoh
  • Journal Title: Springer Proceedings in Math. and Statistics Volume: 106 Pages: 79-88
  • DOI: 10.1007/978-4-431-55215-4_7
  • ISBN: 9784431552147, 9784431552154
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Horospheres and hyperbolicity of Hadamard Manifold (2014)
  • Author(s): Mitsuhiro Itoh, Hiroyasu Satoh and Young Jin Suh
  • Journal Title: Differential Geometry and its Applications Volume: 87 Pages: 144-155
  • DOI: 10.1016/j.difgeo.2014.02.001
  • Peer Reviewed / Open Access
• [Journal Article] Lecture note on Differential Geometry of Horospheres (2013)
  • Author(s): Mitsuhiro Itoh
  • Journal Title: Proceedings of the Seventieth International Workshop on Differential Geometry Volume: 17 Pages: 215-247
• [Journal Article] Horospheres and hyperbolic spaces (2013)
  • Author(s): Mitsuhiro Itoh, Hiroyasu Satoh
  • Journal Title: Kyushu Mathematical Journal Volume: 67 Pages: 309-326
  • Peer Reviewed
• [Journal Article] Horospheres and Hyperbolic Spaces (2013)
  • Author(s): Mitsuhiro Itoh and Hiroyasu Satoh
  • Journal Title: Kyushu Math. Journal Volume: 66 Pages: 1-10
  • Peer Reviewed
• [Journal Article] ホロ球面の幾何による双曲空間の特徴付けについて (2012)
  • Author(s): 伊藤光弘 佐藤弘康
  • Journal Title: 数理解析研究所講究録 Volume: 1817 Pages: 38-51
• [Presentation] Osserman Conjecture and Related Topics with Four Dimensional Manifolds (2015)
  • Author(s): M. Itoh
  • Organizer: Colloquium of Mathematics Department, Sungkyunkwan University
  • Place of Presentation: Sungkyunkwan Univ. Korea
  • Year and Date: 2015-03-26
  • Invited
• [Presentation] 測地線の Fisher 情報幾何と重心写像 (2015)
  • Author(s): 伊藤 光弘, 佐藤 弘康
  • Organizer: 日本数学会年会
  • Place of Presentation: 明治大学リバティータワー, 東京都
  • Year and Date: 2015-03-24
• [Presentation] 調和 Hadamard 多様体と Gauss 超幾何微分方程式 (2015)
  • Author(s): 伊藤 光弘, 佐藤 弘康
  • Organizer: 日本数学会年会
  • Place of Presentation: 明治大学リバティータワー, 東京都
  • Year and Date: 2015-03-24
• [Presentation] Fisher 情報計量の幾何学とその重心写像への応用 (2015)
  • Author(s): 伊藤 光弘
  • Organizer: ミニワークショップ 統計多様体の幾何学とその周辺
  • Place of Presentation: 北海道大学理学研究科, 北海道
  • Year and Date: 2015-03-10
• [Presentation] Fisher Information Geometry of The Barycenter Map (2014)
  • Author(s): M. Itoh, H. Satoh
  • Organizer: Conference of MaxEnt 2014
  • Place of Presentation: アンボワーズ, フランス
  • Year and Date: 2014-09-21
• [Presentation] Information Geometry of Barycenter Map (2014)
  • Author(s): M. Itoh, H. Satoh
  • Organizer: Satellite Conference of ICM2014, Real and Complex Submanifolds
  • Place of Presentation: NIMS, Daejeon, Korea
  • Year and Date: 2014-08-20
  • Invited
• [Presentation] Geometry of Horospheres and Related Topics (2013)
  • Author(s): Mitsuhiro Itoh
  • Organizer: Hot Topics Workshop on Differential Geometry and Mathematical Physics
  • Place of Presentation: National Institute for Mathematical Sciences,, Korea
  • Invited
• [Presentation] Intensive Lectures on Differential Geometry of Horospheres (2012)
  • Author(s): Mitsuhiro Itoh
  • Organizer: Research Group in Real and Complex Grassmann Manifolds
  • Place of Presentation: Kyungpook National University, Korea
  • Invited
• [Presentation] Self-Dual Einstein metrics and Pointwise Osseman 4-manifolds
  • Author(s): Mitsuhiro Itoh
  • Organizer: DGAConferenceBrno2013
  • Place of Presentation: Brno, Czech
• [Presentation] Fisher information geometry of the barycenter of probability measures
  • Author(s): Mitsuhiro Itoh
  • Organizer: Geometric Sciences of Information
  • Place of Presentation: Paris, France
• [Presentation] ホロ球の四元数平均曲率と四元数Kaehler Hadamard 多様体
  • Author(s): 伊藤光弘 佐藤弘康 Young Jin Suh
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 愛媛大学 松山市、愛媛
• [Presentation] Rigidity of Quaternionic Hyperbolic Spaces, part I
  • Author(s): Mitsuhiro Itoh
  • Organizer: International Miniworkshop of Differential Geomety
  • Place of Presentation: Kyungpook University, Korea
  • Invited
• [Presentation] Rigidity of Quaternionic Hyperbolic Spaces, part II
  • Author(s): Mitsuhiro Itoh
  • Organizer: International Miniworkshop of Differential Geometry
  • Place of Presentation: Kyungpook University, Korea
  • Invited
• [Presentation] Information Geometry of Busemann barycenter
  • Author(s): Mitsuhiro Itoh
  • Organizer: International Miniworkshop of Differential Geometry
  • Place of Presentation: Kyungpook University , Korea
  • Invited
• [Presentation] Pointwise Osserman 4-manifolds and Self-Dual Einstein metrics
  • Author(s): Mitsuhiro Itoh
  • Organizer: International Miniworkshop of Differential Geometry
  • Place of Presentation: Kyungpook University, Korea
  • Invited
• [Presentation] Barycenter and Information geometry
  • Author(s): 伊藤光弘 佐藤弘康
  • Organizer: 日本数学会年会
  • Place of Presentation: 学習院大学，東京
• [Presentation] Rigidity, volume entropy and Kaehler, quaternionic Kaehler Hadamard manifolds
  • Author(s): 伊藤光弘 佐藤弘康 Young Jin Suh
  • Organizer: 日本数学会
  • Place of Presentation: 九州大学（福岡県）
• [Presentation] Complex Hyperbolic Space and Horospheres
  • Author(s): 伊藤光弘
  • Organizer: 筑波大学微分幾何学火曜セミナー
  • Place of Presentation: 筑波大学（茨城県）
• [Book] 倉西数学への誘い (2014)
  • Author(s): 伊藤 光弘
  • Total Pages: 189
  • Publisher: 岩波書店
• [Book] 曲面の幾何学 (2013)
  • Author(s): 伊藤 光弘
  • Total Pages: 190
  • Publisher: 遊星社