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Furuhata Hitoshi
| Faculty of Science Mathematics Mathematics | Professor |
| Research Center of Mathematics for Social Creativity | Professor |
Researcher basic information
■ Degree■ URL
researchmap URLホームページURL■ Various IDs
J-Global ID■ Research Keywords and Fields
Research KeywordResearch Field■ Educational Organization
- Bachelor's degree program, School of Science
- Master's degree program, Graduate School of Science
- Doctoral (PhD) degree program, Graduate School of Science
Research activity information
■ Papers- Reduced centroaffine curves on homogeneous centroaffine hypersurfaces
Atsushi Fujioka; Hitoshi Furuhata
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 67(2026), 255, 269, Springer Science and Business Media LLC, 08 Aug. 2025, [Peer-reviewed]
English, Scientific journal - A Variation Problem for Mappings Between Statistical Manifolds
Hitoshi Furuhata; Ryu Ueno
Results in Mathematics, 80, 57, Springer Science and Business Media LLC, 25 Feb. 2025, [Peer-reviewed]
English, Scientific journal - Centroaffine surfaces of cohomogeneity one with planar curvature lines
Atsushi Fujioka; Hitoshi Furuhata
Colloquium Mathematicum, 172, 2, 173, 190, Institute of Mathematics, Polish Academy of Sciences, 2023, [Peer-reviewed]
English, Scientific journal, 40298734 - Toward differential geometry of statistical submanifolds
Hitoshi Furuhata
Information Geometry, 7, S1 (2024), 99, 108, Springer Science and Business Media LLC, 22 Nov. 2022, [Peer-reviewed], [Invited]
Scientific journal - Chen invariants and statistical submanifolds
Hitoshi Furuhata; Izumi Hasegawa; Naoto Satoh
Communications of the Korean Mathematical Society, 37, 3, 851, 864, Jul. 2022, [Peer-reviewed]
English - A characterization of the alpha-connections on the statistical manifold of normal distributions
Hitoshi Furuhata; Jun-ichi Inoguchi; Shimpei Kobayashi
Information Geometry, 4, 1, 177, 188, Springer Science and Business Media LLC, 20 Oct. 2020, [Peer-reviewed]
English, Scientific journal - Statistical submanifolds from a viewpoint of the Euler inequality
Naoto Satoh; Hitoshi Furuhata; Izumi Hasegawa; Toshiyuki Nakane; Yukihiko Okuyama; Kimitake Sato; Mohammad Hasan Shahid; Aliya Naaz Siddiqui
Information Geometry, 4, 189, 213, Springer Science and Business Media LLC, 04 Sep. 2020, [Peer-reviewed]
English, Scientific journal - Centroaffine surfaces of cohomogeneity one
FUJIOKA Atsushi; FURUHATA Hitoshi
Bull. Braz. Math. Soc., NS., 50, 1, 291, 313, 2019, [Peer-reviewed]
English - The center map of a centroaffine ruled surface
FUJIOKA Atsushi; FURUHATA Hitoshi
An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), 64, 343, 355, 2018, [Peer-reviewed]
English, Scientific journal - Kenmotsu statistical manifolds and warped product
Hitoshi Furuhata; Izumi Hasegawa; Yukihiko Okuyama; Kimitake Sato
Journal of Geometry, 108, 3, 1175, 1191, Birkhauser Verlag AG, 01 Dec. 2017, [Peer-reviewed]
English, Scientific journal - Sasakian statistical manifolds
Hitoshi Furuhata; Izumi Hasegawa; Yukihiko Okuyama; Kimitake Sato; Mohammad Hasan Shahid
JOURNAL OF GEOMETRY AND PHYSICS, 117, 179, 186, Jul. 2017, [Peer-reviewed]
English, Scientific journal - Sasakian statistical manifolds II
Hitoshi Furuhata
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 10589, 179, 185, Springer Verlag, 2017, [Peer-reviewed]
English, International conference proceedings - Submanifold theory in holomorphic statistical manifolds
Hitoshi Furuhata; Izumi Hasegawa
Geometry of Cauchy-Riemann Submanifolds, 179, 215, Springer Singapore, 01 Jan. 2016, [Peer-reviewed], [Invited]
English, In book - Projective surfaces and pre-normalized Blaschke immersions of codimension two
FUJIOKA Atsushi; FURUHATA Hitoshi; SASAKI Takeshi
Int. Electron. J. Geom., 9, 100, 110, 2016, [Peer-reviewed]
English, Scientific journal - Projective minimality for centroaffine minimal surfaces
Atsushi Fujioka; Hitoshi Furuhata; Takeshi Sasaki
Journal of Geometry, 105, 1, 87, 102, Apr. 2014, [Peer-reviewed]
English, Scientific journal - HESSIAN MANIFOLDS OF NONPOSITIVE CONSTANT HESSIAN SECTIONAL CURVATURE
Hitoshi Furuhata; Takashi Kurose
TOHOKU MATHEMATICAL JOURNAL, 65, 1, 31, 42, Mar. 2013, [Peer-reviewed]
English, Scientific journal - Statistical hypersurfaces in the space of Hessian curvature zero II
FURUHATA Hitoshi; HU Na; Luc VRANCKEN
J. Van der Veken, I. Van de Woestyne, L. Verstraelen, L. Vrancken (eds.), Pure and Applied Differential Geometry - PADGE 2012 In Memory of Franki Dillen, Shaker Verlag GmbH, Germany, 136, 142, Shaker Verlag GmbH, 2013, [Peer-reviewed], [Invited]
English, International conference proceedings, We construct cylindrical statistical immersions between spaces of Hessian curvature zero.The words "statistical submanifold" can be found in the paper [5] in 1989, which was written by Vos in the context of statistical inference or information geometry. Although the history of this geometry is not so short, it is hard to find classical differential geometric approaches for the study of statistical submanifolds. In this paper, we would like to continue to try it after [2], and give some of basic examples of statistical submanifolds apart from applications for statistics. In other words, we will study immersions between statistical manifolds preserving statistical structures, which are called statistical immersions, in particular, called statistical hypersurfaces if the codimension equals one. We take a space Nn in Definition 1.1, which can be considered as a basic model of a statistical manifold of dimension n. The space Nn has been known as a Hessian manifold of constant Hessian curvature zero. In [2], a statistical hypersurface of a Hessian manifold of constant Hessian curvature negative into the space Nn+1 is uniquely determined. Besides, there exist no statistical hypersurfaces of a Hessian manifold of constant Hessian curvature positive into the space Nn+1. On the other hand, we have plenty of statistical hypersurfaces of Nn into Nn+1. In this paper, we determine statistical diffeomorphisms of Nn onto itself, and statistical hypersurfaces of Nn into Nn+1 with vanishing statistical second fundamental form (Propositions 2.1 and 2.2). Moreover, we explicitly construct and determine statistical immersions of a domain of N2 into N3 of cylinder type (Theorem 3.1).Pure and Applied Differential Geometry - PADGE 2012 In Memory of Franki Dillen, Shaker Verlag GmbH, Germany, ISBN: 9783844023633, (2013), 136-142. - Statistical hypersurfaces in the space of Hessian curvature zero
Hitoshi Furuhata
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 29, S86, S90, Aug. 2011, [Peer-reviewed]
English, Scientific journal - Hypersurfaces in statistical manifolds
Hitoshi Furuhata
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 27, 3, 420, 429, Jun. 2009, [Peer-reviewed]
English, Scientific journal - The center map of an Affine immersion
Hitoshi Furuhata; Luc Vrancken
Results in Mathematics, 49, 3-4, 201, 217, Dec. 2006, [Peer-reviewed]
English, Scientific journal - Self-dual centroaffine surfaces of codimension two with constant affine mean curvature
H Furuhata; T Kurose
BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN, 9, 4, 573, 587, Oct. 2002, [Peer-reviewed]
English, Scientific journal - Codazzi structures induced by minimal affine immersions
FURUHATA Hitoshi
Banach Center Publ., 57, 17, 19, 2002, [Peer-reviewed]
English, International conference proceedings - Minimal centroaffine immersions of codimension two
H Furuhata
BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN, 7, 1, 125, 134, Jan. 2000, [Peer-reviewed]
English, Scientific journal - A conformal gauge invariant functional for Weyl structures and the first variation formula
ICHIYAMA Toshiyuki; FURUHATA Hitoshi; URAKAWA Hajime
Tsukuba Math. J., 23, 3, 551, 564, Institute of Mathematics, University of Tsukuba, 1999, [Peer-reviewed]
English, Scientific journal - An intrinsic characterization of isometric pluriharmonic immersions with codimension one
Hitoshi Furuhata
Journal of Geometry, 65, 1-2, 111, 116, Birkhauser Verlag Basel, 1999, [Peer-reviewed]
English, Scientific journal - Holomorphic centroaffine immersions and the Lelieuvre correspondence
FURUHATA Hitoshi; MATSUZOE Hiroshi
Result. Math., 33, 294, 305, 1998, [Peer-reviewed]
English, Scientific journal - Open problems in affinedifferential geometry and related topics
FURUHATA Hitoshi; MATSUZOE Hiroshi; URAKAWA Hajime
Interdiscip. Inform. Sci., 4, 2, 125, 127, Tohoku University, 1998, [Peer-reviewed]
English, Scientific journal, The following problems were raised in the workshop "Affine Differential Geometry and Related Topics" at Graduate School of Information Sciences, Tohoku University at December 16-18, 1996. - A cylinder theorem for isometric pluriharmonic immersions
H Furuhata
GEOMETRIAE DEDICATA, 66, 3, 303, 311, Jul. 1997, [Peer-reviewed]
English, Scientific journal - Moduli space of isometric pluriharmonic immersions of Kahler manifolds into indefinite Euclidean spaces
H Furuhata
PACIFIC JOURNAL OF MATHEMATICS, 176, 1, 1, 14, Nov. 1996, [Peer-reviewed]
English, Scientific journal - Isometric Pluriharmonic Immersions of Kaehler Manifolds into Semi-Euclidean Spaces
FURUHATA Hitoshi
Tohoku Math. Publ., 1, 1, 70, Tohoku University, 1995, [Peer-reviewed]
English, Doctoral thesis - CONSTRUCTION AND CLASSIFICATION OF ISOMETRIC MINIMAL IMMERSIONS OF KAHLER-MANIFOLDS INTO EUCLIDEAN SPACES
H FURUHATA
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 26, 487, 496, Sep. 1994, [Peer-reviewed]
English, Scientific journal
- 「平面上の曲線」から微分幾何学へ : 逆さサイクロイドのひみつ
古畑仁, 数学セミナー 2023年5月号 特集＝ 単元別 高校数学はこう続く, 739, 28, 32, Apr. 2023
Japanese
- 感じる数学 : ガリレイからポアンカレまで = Tangible math
数学みえる化プロジェクト; 北海道大学総合博物館; 正宗, 淳
共立出版, Aug. 2022, 9784320114784, x, 196p, Japanese, [Contributor] - 曲面 ---幾何学基礎講義
古畑 仁
数学書房, 2013, 9784903342382, [Single work] - 北大高校生講座 数学の並木道
北大数学科; 中村郁, 第５章 形をはかる話
日本評論社, 2004, 4535784116, Japanese, [Contributor]
- 確率単体の統計部分多様体
古畑仁
確率論・幾何学・数値解析の情報交換会, 28 Mar. 2026
28 Mar. 2026 - 30 Mar. 2026, [Invited] - Statistical Submanifolds and Statistical Biharmonic Maps
Hitoshi Furuhata
The 22nd OCAMI-RIRCM Joint Differential Geometry Workshop on Submanifolds in Symmetric Spaces and Related Topics, 22 Feb. 2026
21 Feb. 2026 - 23 Feb. 2026, [Invited] - 統計多様体にまつわる二重の調和性・非調和性
古畑仁
福岡大学微分幾何研究集会2024, 15 Feb. 2025
15 Feb. 2025 - 17 Feb. 2025, [Invited] - 統計2重調和写像
上野龍; 古畑仁
日本数学会２０２４年度年会幾何学分科会, 17 Mar. 2024, Japanese, Oral presentation
17 Mar. 2024 - 20 Mar. 2024 - A class of mappings between statistical manifolds
古畑仁
RIMS共同研究(公開型) 部分多様体と群作用の幾何学, 28 Jun. 2023
26 Jun. 2023 - 28 Jun. 2023, [Invited] - 「幾何学と言えば三角形」は本当か？
古畑仁
北海道算数数学教育会高等学校部会研究部数学教育実践研究会第125回研究会, 10 Jun. 2023, Japanese, Invited oral presentation
10 Jun. 2023 - 10 Jun. 2023, [Invited] - 数学をすべての人々へ アウトリーチの試み
古畑仁
令和4年度数学部門第1回FD研修会（北海道大学大学院理学研究院）, 10 Feb. 2023
10 Feb. 2023 - 10 Feb. 2023, [Invited] - 幾何学者たちのヘレネー
古畑仁
北大道新アカデミー, 22 Jul. 2022, Public discourse - 統計多様体は tangible か
古畑仁
北海道大学大学院理学研究院数学部門談話会, 25 May 2022, Public discourse - 双極小統計部分多様体といくつかの問題
古畑仁
Tsudoi KK, 18 Mar. 2022 - 統計部分多様体に対する不等式
古畑仁
福岡大学微分幾何セミナー, 17 Feb. 2022
[Invited] - Submanifold theory in statistical manifolds
古畑仁
日本数学会２０２１年度秋季総合分科会, 16 Sep. 2021, Japanese, Invited oral presentation
[Invited] - Centroaffine surfaces of cohomogeneity one
Atsushi Fujioka; Hitoshi Furuhata
日本数学会２０２０年度年会幾何学分科会（開催中止）, 16 Mar. 2020, Oral presentation - Statistical sectional curvature and warped product statistical manifold
Naoto Satoh; Hitoshi Furuhata; Izumi Hasegawa
日本数学会２０２０年度年会幾何学分科会（開催中止）, 16 Mar. 2020, Oral presentation - Statistical submanifolds and warped product spaces
FURUHATA Hitoshi
The 18th International Conference, Graduate School of Mathematics, Nagoya University, Information Geometry and Affine Differential Geometry III, 28 Mar. 2019, English, Oral presentation
[Invited], [International presentation] - Centroaffine surfaces of cohomogeneity one
FURUHATA Hitoshi
研究集会「幾何学のスペクトル」, 15 Dec. 2018, Japanese, Oral presentation
[Domestic Conference] - 統計多様体，具体例からの入門
古畑 仁
ミニワークショップ 統計多様体の幾何学とその周辺 (10), 16 Nov. 2018, Japanese
[Domestic Conference] - 統計多様体の定曲率空間
古畑 仁
福岡大学微分幾何研究集会（Geometry and Analysis 2018）, 04 Nov. 2018, Japanese, Oral presentation
[Invited], [Domestic Conference] - Sasakian statistical manifolds and hypersurfaces
FURUHATA Hitoshi
PNU-HU Joint Symposium, 19 Dec. 2017, English, Oral presentation
[Invited], [International presentation] - Sasakian statistical manifolds and warped product
FURUHATA Hitoshi
Seminar on Differential Geometry, 13 Nov. 2017, English, Public discourse
[Invited], [International presentation] - Sasakian statistical manifolds II
FURUHATA Hitoshi
3rd conference on Geometric Science of Information, 09 Nov. 2017, English, Oral presentation
[Invited], [International presentation] - 中心写像から見た曲面の中心アファイン幾何学
古畑 仁
第６４回幾何学シンポジウム, 28 Aug. 2017, Japanese, Oral presentation
[Invited], [Domestic Conference] - アファイン空間の曲線論入門
古畑 仁
第５回水戸幾何セミナー, 07 Jul. 2017, Japanese, Public discourse
[Invited], [Domestic Conference] - 剱持統計多様体
古畑 仁
関大微分幾何研究会, 25 Jun. 2017, Japanese, Oral presentation
[Invited], [Domestic Conference] - 佐々木構造をもつ統計多様体
古畑 仁
名城研究集会「多様体上の計量と幾何構造」, 04 Mar. 2017, Japanese, Oral presentation
[Invited], [Domestic Conference] - もう一度はじめから，本当のアファイン曲線
古畑 仁
北海道大学幾何学コロキウム, 22 Apr. 2016, Japanese, Public discourse
[Invited], [Domestic Conference] - 正則統計多様体のＣＲ部分多様体
古畑 仁
福岡大学微分幾何学研究集会, 30 Oct. 2015, Japanese, Oral presentation
[Invited], [Domestic Conference] - Affine differential geometry and surfaces in the projective space
FURUHATA Hitoshi
研究集会「Differential Geoemetry with Dajczer」, 06 Oct. 2015, English, Oral presentation
[Invited], [International presentation] - 統計多様体の曲率
古畑 仁
ミニワークショップ 統計多様体の幾何学とその周辺 (7), 01 Sep. 2015, Japanese, Oral presentation
[Domestic Conference] - 正則統計多様体とその曲率
古畑 仁
RIMS共同研究「統計多様体の諸分野への応用」, 19 Nov. 2014, Japanese, Oral presentation
[Invited], [Domestic Conference] - ヘッセ多様体とアファイン球面
古畑 仁
ミニワークショップ 統計多様体の幾何学とその周辺 (5), 10 Jan. 2014, Japanese, Oral presentation
[Domestic Conference] - 中心アファイン極小曲面の射影極小性
古畑 仁
福岡大学微分幾何学研究集会, 02 Nov. 2013, Japanese, Oral presentation
[Invited], [Domestic Conference] - 接続の幾何学における部分多様体論にむけて
古畑 仁
ミニワークショップ 統計多様体の幾何学とその周辺 (4), 02 Mar. 2013, Japanese, Oral presentation
[Domestic Conference] - Statistical immersions of spaces of constant Hessian curvature
FURUHATA Hitoshi
The 8th HU and SNU Symposium on Mathematics -Recent developments of Geometry and Topology-, 07 Dec. 2012, English, Oral presentation
[Invited], [International presentation] - Geometry of statistical submanifolds
FURUHATA Hitoshi
PADGE2012, 29 Aug. 2012, English, Oral presentation
[Invited], [International presentation] - Submanifold theory for statistical manifolds
FURUHATA Hitoshi
Workshop on Differential Geometry, 08 Jun. 2012, English, Oral presentation
[Invited], [International presentation] - A realization problem for statistical manifolds
FURUHATA Hitoshi
福岡大学微分幾何学研究集会, 03 Nov. 2011, English, Oral presentation
[International presentation] - Statistical Immersions
FURUHATA Hitoshi
RIKEN Workshop on Information Geometry, 31 Aug. 2011, English, Oral presentation
[Invited], [International presentation] - 部分多様体の微分幾何学と統計多様体
古畑 仁
第１３回ＤＣＳセミナー, 25 May 2011, Japanese, Public discourse
[Invited], [Domestic Conference] - 定曲率 Hesse 多様体のモデルについて
古畑 仁
研究集会「擬リーマン幾何学の展開III」, 19 Dec. 2010, Japanese, Oral presentation
[Invited], [Domestic Conference] - Statistical manifolds --- Hypersurfaces and complex structures
古畑 仁
ミニワークショップ 統計多様体の幾何学とその周辺 (2), 29 Oct. 2010, Japanese, Oral presentation
[Domestic Conference] - Statistical submanifolds and spaces of constant Hessian sectional curvature
FURUHATA Hitoshi
Differential Geometry and its Applications, 28 Aug. 2010, English, Oral presentation
[Invited], [International presentation] - Centroaffine Surfaces with Special Center
古畑 仁
松江微分幾何学研究会２００９, 11 Dec. 2009, Japanese, Oral presentation
[Invited], [Domestic Conference] - 統計多様体の微分幾何学入門
古畑 仁
北海道大学特異点論セミナー, 27 Nov. 2009, Japanese, Public discourse
[Invited], [Domestic Conference] - Spaces of non-positive constant Hessian curvature
古畑 仁
福岡大学微分幾何学研究集会, 20 Nov. 2009, Japanese, Oral presentation
[Invited], [Domestic Conference] - Examples of Statistical Manifolds
古畑 仁
ミニワークショップ 統計多様体の幾何学とその周辺 (1), 06 Sep. 2009, Japanese, Oral presentation
[Domestic Conference]
- 大学院共通授業科目（一般科目）：自然科学・応用科学, 2024年, 修士課程, 大学院共通科目
- 大学院共通授業科目（一般科目）：自然科学・応用科学, 2024年, 修士課程, 大学院共通科目
- 幾何学特論Ａ, 2024年, 修士課程, 理学院
- 幾何学特論Ｂ, 2024年, 修士課程, 理学院
- 幾何学続論, 2024年, 学士課程, 理学部
- 微分積分学Ⅰ, 2024年, 学士課程, 全学教育
- 微分積分学Ⅱ, 2024年, 学士課程, 全学教育
■ Research Themes
- Geometry of statistical structures for homogeneous spaces and submanifolds
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Apr. 2022 - Mar. 2027
古畑 仁
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, Principal investigator, 22K03279 - 諸分野に現れるアファインはめ込みの研究
Apr. 2014 - Mar. 2018
古畑 仁
Principal investigator, Competitive research funding - Research on classical differential geometry from modern view points and its applications
Grants-in-Aid for Scientific Research
01 Apr. 2010 - 31 Mar. 2015
KUROSE Takashi; SUYAMA Yoshihiko; HAMADA Tatsuyoshi; KAWAKUBO Satoshi; MATSUURA Nozomu; INOGUCHI Junichi; FURUHATA Hitoshi; FUJIOKA Atsushi
In this research program, classical differential geometry, geometry of curves, surfaces and hypersurfaces in various spaces, have been studied, mainly with the method of the theory of integrable systems. Many results on classical differential geoemtry and its application have been achieved; for instance, through the observation that certain sorts of changes with time of curves yield equations dealt with in the theory of integrable systems, geometric descriptions and/or interpretations of several accomplishments of the theory have been given. Moreover, by applying geometry of hypersurfraces in affine spaces, new properties of statistical manifolds, which appear in informtion geometry, the study of mathematical statistics and information theory with differential geometric tools and methods, have been obtained and the statistical manifolds satisfying some curvature condition have been explicitely constructed and classified.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 22540107 - 部分多様体にあらわれる統計構造の幾何学
Apr. 2009 - Mar. 2013
古畑 仁
Principal investigator, Competitive research funding - Research on the differential sysytems and geometric structures associated with simple graded Lie algebras
Grants-in-Aid for Scientific Research
2007 - 2010
YAMAGUCHI Keizo; IZUMIYA Shuichi; ONO Kaoru; ISHIKAWA Goo; MATSUMOTO Keiji; HURUHATA Hitoshi; YATSUI Tomoaki; NAKAI Isao; OZAWA Tetasuya; SASAKI Takeshi
We determined the class of differential equations of finite type, which admits extraordinarily rich infinitesimal symmetries, among the classes of differential equations of finite type obtained from the Se-ashi's principle. Moreover we write up explicitly the model equations of this class of finite type equations. Furthermore we investigated the fundamental Reduction procedure in the fields of Contact Geometry of second Order and formulated two fundamental Reduction Theorems.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Hokkaido University, 19340012 - Classical differential geometry from the modern viewpoint and its application
Grants-in-Aid for Scientific Research
2006 - 2009
KUROSE Takashi; SUAYMA Yoshihiko; HAMADA Tatsuyoshi; KAWAKUBO Satoshi; MATSUURA Nozomu; YAMADA Kotaro; INOGUCHI Junichi; FURUHATA Hitoshi
In this research, we studied classical differential geometry from modern viewpoints, such as of the theory of integral systems and of the theory of singularities ; we obtained results on various fields of classical differential geometry and their applications, in particular, the motions of curves associated with integrable systems, explicit construction and the classification of conformally flat hypersurfaces of four-dimensional space forms, real hypersurfaces of complex space forms, surfaces of three-dimensional spaces, affine differential geometry and its applications to Hessian geometry and information geometry, and so on.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Fukuoka University, 18540103 - アファインはめ込みの大域的理論と情報幾何学の基礎的研究
Apr. 2003 - Mar. 2006
古畑 仁
Principal investigator, Competitive research funding - Classical differential geometry from the modern viewpoint and its applications
Grants-in-Aid for Scientific Research
2003 - 2005
KUROSE Takashi; SUYAMA Yoshihiko; HAMADA Tatsuyoshi; YAMADA Kotaro; INOGUCHI Jun-ichi; FURUHATA Hitoshi
In this research, we planned to give a now development of the theories of classical differential geometry by restructuring them from the modern viewpoint, particularly, of the theories of integrable systems and of singularities. Our main results are the following :
1.(1)In affine differential geometry, one of the core theories of classical differential geometry, we mainly studied the geometry of affine hyperspheres and their representation formulae, and showed a relationship with the geometry of holomorphic statistical manifolds and the several properties of the center maps. We also studied the discretization of affine or centroaffine plane curves and gave a description of their time-evolution following discrete soliton equations ; (2)we characterized the classical examples of conformally flat hypersurfaces in 4-dimensional Euclidean space and constructed new examples ; (3)for real hypersurfaces in complex space forms, we introduced a new geometric invariant and classified Hopf real hypersurfaces using the invariant.
2.We studied the geometric properties of surfaces with singularities and obtained the following results : (1)We constructed the theory of flat fronts, the flat surfaces with singularities of a certain kind in 3-dimensional hyperbolic space. In particular, we defined (weak) completeness of flat fronts and showed their global properties ; (2)investigating the properties of the singularities of maximal surfaces in 3-dimensional Minkowski space, we constructed the theory of maxfaces, the spacelike maximal surfaces allowing singularities of a certain kind.
3.We studied transformations of surfaces and showed that the transformations given by the sphere congruences in Moebius geometry are obtained by the complexified line congruences in Euclidean space. We also investigated biharmonic curves in 3-dimensional homogeneous spaces and determined such curves when the homogeneous spaces are irreducible and reductive.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Fukuoka University, 15540100 - 極小アファイン曲面の計量微分幾何学的研究
Apr. 2001 - Mar. 2003
古畑 仁
Principal investigator, Competitive research funding - アファインはめ込みの情報幾何学的研究
Apr. 1999 - Mar. 2002
古畑 仁
稲盛財団, Principal investigator, Competitive research funding - Integrable geodesic flows and Masloy's quantization condition
Grants-in-Aid for Scientific Research
2001 - 2002
KIYOHARA Kazuyoshi; FURUHATA Hitoshi; ISHIKAWA Goo; IZUMIYA Shuichi; IGARASHI Masayuki; SHIMADA Ichirou
We constructed a continuous family of riemanninan metrics on 2-sphere whose geodesic flows possess first integrals of fiber-degree k, for every k greater than 2. They are the first examples, exect the cases where k=3,4, due to Bolsinov and Fomenko. Moreover, the constructed manifolds have the property that every geodesic is closed. Therefore they are conrete examples of the manifolds that Guillemin showed their existence in an abstract manner.
We also investigated the structures of Kahler-Liouville manifolds of general type, I.e., not necessarlly of type (A). As a consequence, we showed that every compact, proper Kahler-Liouville manifold has a bundle structure such that the fiber is a Kahler-Liouville manifold whose geodesic flow is integrable, and the base is (locally) a product of one-dimensional Kahler manifolds. Also, we obtain another class, called of type (B), of Kahler-Liouville manifolds whose geodesic flows are integrable. This class had already appeared in the study of fiber bundle structure of type (A) manifolds, but we now obtained its intrinsic definition.
Also, we investigated local structures of Hermite-Liouville manifolds and basically clarifled them. Moreover, we construct the structure of Hermite-Liouville manifolds on complex projective spaces. The way of construction is similar to that of a Kahler-Liouvlle manifold, I.e., a complexification of a real Liouville manifold. However, in the Hermite case, plural Liouville manifolds produce one Hermite-Liouville manifold. Therefore, we obtain quite many examples of integrable geodesic flows in this way.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), HOKKAIDO UNIVERSITY, 13640054 - Modern Research of Affine and Projective Geometry and its Applications
Grants-in-Aid for Scientific Research
2000 - 2002
KUROSE Takashi; YAMADA Kotaro; HAMADA Tatsuyoshi; SUYAMA Yoshihiko; FURUHATA Hitoshi; INOGUCHI Jun-ichi
In this research, we studied classical differential geometries, theory of integral systems and information geometry.
1. Classical Differential Geometries (1) We characterized minimal affine hypersurfaces and minimal centroaffine immersions of codimension two. Moreover, we gave an explicit method of constructing self-dual minimal centroaffine surfaces of codimension two.
(2) We studied manifolds with projectively flat torsion-free affine connection whose Ricci curvature is symmetric and definite, and showed fundamental results on the injectivity of the projective developing maps of such manifolds and the convexity of their image.
(3) For conformally flat hypersurfaces of a 4-dimensional sphere, we defined a new conformal invariant. Using the invariant, we characterized the classical examples and constructed new examples.
(4) We developed a very concrete and comprehensive theory on curves and surfaces in 3-dimensional homogeneous spaces.
2. Integrable Systems We investigated various integrable systems appeared in classical differential geometries. We obtained representation formulae for minimal surfaces in 3-dimensional solvable Lie groups and flat surfaces in a 3-dimensional hyperbolic space. We also developed a comprehensive theory of (spacelike) surfaces with harmonic inverse mean curvature in 3-dimensional Riemannian space forms and Lorentzian space forms.
3. Information Geometry and Statistical Manifolds (1) We defined complex statistical manifolds and studied them from the view points of affine differential geometry and of information geometry, especially of quantum estimation theory.
(2) As a generalization of special Kahler manifolds, we defined statistical manifolds with compatible complex structure and investigated their fundamental properties.
(3) On (-1)-conformally flat statistical manifolds, we gave an explicit method of constructing the Volonoi diagrams.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Fukuoka University, 12640097 - ワイル構造および統計構造とアファインはめ込みの幾何学の研究
Apr. 1999 - Mar. 2001
古畑 仁
Principal investigator, Competitive research funding - Affine Immersions and Statistical Structures
2000
Competitive research funding - 一般余次元のアファインはめ込みの分類および構成
Apr. 1997 - Mar. 1999
古畑 仁
Principal investigator, Competitive research funding - Siochastic Analysis on loop spaces
Grants-in-Aid for Scientific Research
1998 - 1999
AIDA Shigeki; KUWAE Kazuhiro; SEKIME Jun; NAGAI Hideo; TOSHIDA Noluo; KAZUM Tetsuya
The results obtained for 1998-1999 are as follows :
1. We proved a Varadhan type asymptotics of diffusion processes in Wiener spaces
2. We gave an irreducibility criterion of diffusion processes on a subset in an Wiener space in terms of a certain connectivity of the subset.
3. We proved a Clark-Ocone formula for pinned Brownian motion.
The following are explanations of the aboves.
1. : This is a generalization of the Fang's results for Ornstein-Uhlenbeck processes to more general diffusion coefficients.
2. : The head investigator proved an irreducibility theorem for diffusion processes on loop spaces using Kusuoka's theorem. In this work, the head investigator proved the generalization.
3. : As an application, the head investigator proved a logarithmic Sobolev inequality on loop space over hyperbolic spaces.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 10640147 - ケーラー多様体からユークリッド空間への等長はめ込みの研究
Apr. 1995 - Mar. 1996
古畑 仁
Principal investigator, Competitive research funding
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