Researcher Database
Last Updated :2024/07/25
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Profile and Settings
Profile and Settings
Name (Japanese)
FuruhataName (Kana)
HitoshiName
200901048049232641
Alternate Names
Achievement
Research Interests
Research Areas
Research Experience
- 2007 - 2022 北海道大学大学院理学研究院数学部門 助教授
- 1999 - 2007 北海道大学大学院理学研究科数学専攻 講師
- 1996 - 1999 東北大学大学院情報科学研究科 助手
- 1995 - 1996 日本学術振興会 (東北大学) 日本学術振興会特別研究員
Published Papers
- Atsushi Fujioka, Hitoshi FuruhataColloquium Mathematicum 172 (2) 173 - 190 0010-1354 2023 [Refereed]
- Hitoshi FuruhataInformation Geometry 7 (S1 (2024)) 99 - 108 2511-2481 2022/11/22 [Refereed][Invited]
- Hitoshi Furuhata, Izumi Hasegawa, Naoto SatohCommunications of the Korean Mathematical Society 37 (3) 851 - 864 2022/07 [Refereed]
- Hitoshi Furuhata, Jun-ichi Inoguchi, Shimpei KobayashiInformation Geometry 4 (1) 177 - 188 2511-2481 2020/10/20 [Refereed][Not invited]
- Naoto Satoh, Hitoshi Furuhata, Izumi Hasegawa, Toshiyuki Nakane, Yukihiko Okuyama, Kimitake Sato, Mohammad Hasan Shahid, Aliya Naaz SiddiquiInformation Geometry 4 189 - 213 2511-2481 2020/09/04 [Refereed][Not invited]
- FUJIOKA Atsushi, FURUHATA HitoshiBull. Braz. Math. Soc., NS. 50 (1) 291 - 313 2019 [Refereed][Not invited]
- FUJIOKA Atsushi, FURUHATA HitoshiAn. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 64 343 - 355 2018 [Refereed][Not invited]
- Hitoshi Furuhata, Izumi Hasegawa, Yukihiko Okuyama, Kimitake SatoJournal of Geometry 108 (3) 1175 - 1191 1420-8997 2017/12/01 [Refereed][Not invited]
A notion of a Kenmotsu statistical manifold is introduced, which is locally obtained as the warped product of a holomorphic statistical manifold and a line. A statistical manifold is a Riemannian manifold equipped with a torsion-free affine connection satisfying the Codazzi equation. It can be considered as being in information geometry, Hessian geometry and various submanifold theory. On the other hand, a Kenmotsu manifold is in a meaningful class of almost contact metric manifolds. In this paper, we construct a suitable statistical structure on it. Although the notion of the warped product of Riemannian manifolds is well known, the one for statistical manifolds is not established. We consider it in general, and study the statistical sectional curvature of the warped product of two statistical manifolds. We show that a Kenmotsu statistical manifold of constant ϕ-sectional curvature is constructed from a special Kähler manifold, which is an important example of holomorphic statistical manifold. A Sasakian statistical manifold is also studied from the viewpoint of the warped product of statistical manifolds. - Hitoshi Furuhata, Izumi Hasegawa, Yukihiko Okuyama, Kimitake Sato, Mohammad Hasan ShahidJOURNAL OF GEOMETRY AND PHYSICS 117 179 - 186 0393-0440 2017/07 [Refereed][Not invited]
A notion of Sasakian statistical structure is introduced. The condition for a real hypersurface in a holomorphic statistical manifold to admit such a structure is given. (C) 2017 Elsevier B.V. All rights reserved. - Hitoshi FuruhataLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 10589 179 - 185 1611-3349 2017 [Refereed][Not invited]
This article is a digest of [2, 3] with additional remarks on invariant submanifolds of Sasakian statistical manifolds. - Hitoshi Furuhata, Izumi HasegawaGeometry of Cauchy-Riemann Submanifolds 179 - 215 2016/01/01 [Refereed][Invited]
A statistical manifold is a smooth manifold equipped with a pair of a Riemannian metric and a torsion-free affine connection satisfying the Codazzi equation. We naturally have various dualistic geometric objects on it. In this article, the basics for statistical submanifolds in holomorphic statistical manifolds are given. We define the sectional curvature for a statistical structure, and study CR-submanifolds in a holomorphic statistical manifold of constant holomorphic sectional curvature. We prove that this sectional curvature of such a space vanishes if it admits a totally umbilical and a dual-totally umbilical generic submanifolds. Furthermore, we show that a Lagrangian submanifold is of constant sectional curvature if the statistical shape operator and its dual operator commute. Similarly, we generalize several theorems in the classical CR-submanifold theory. - Projective surfaces and pre-normalized Blaschke immersions of codimension twoFUJIOKA Atsushi, FURUHATA Hitoshi, SASAKI TakeshiInt. Electron. J. Geom. 9 100 - 110 2016 [Refereed][Not invited]
- Atsushi Fujioka, Hitoshi Furuhata, Takeshi SasakiJournal of Geometry 105 (1) 87 - 102 0047-2468 2014/04 [Refereed][Not invited]
Centroaffine minimal surfaces are considered as an interesting class of surfaces from the viewpoint of not only variational problems in centroaffine differential geometry but also integrable systems. Typical examples of centroaffine minimal surfaces are proper affine spheres centered at the origin when we regard them as centroaffine surfaces. On the other hand, the study of projective minimal surfaces has a long history in projective differential geometry. Typical examples of projective minimal surfaces are proper affine spheres again, and so-called Demoulin surfaces or Godeaux-Rozet surfaces. In this paper, we shall regard centroaffine surfaces as projective surfaces and study projective minimality of centroaffine minimal surfaces. Using the fact that any centroaffine minimal surfaces have a one-parameter family of deformation known as associated surfaces, we shall give a classification of indefinite centroaffine minimal surfaces whose associated surfaces are all projective minimal, which includes centroaffine surfaces with vanishing Tchebychev operator and those found by the first author before. We shall also show that any indefinite centroaffine minimal surface whose associated surfaces are all Godeaux-Rozet surfaces is a proper affine sphere. © 2013 Springer Basel. - Hitoshi Furuhata, Takashi KuroseTOHOKU MATHEMATICAL JOURNAL 65 (1) 31 - 42 0040-8735 2013/03 [Refereed][Not invited]
We classify the maximal Hessian manifolds of constant Hessaian sectional curvature nonpositive. - FURUHATA Hitoshi, HU Na, Luc VRANCKENJ. 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 Shaker Verlag GmbH 136 - 142 0945-0882 2013 [Refereed][Invited]
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. - Hitoshi FuruhataDIFFERENTIAL GEOMETRY AND ITS APPLICATIONS 29 S86 - S90 0926-2245 2011/08 [Refereed][Not invited]
A rigidity theorem for a statistical hypersurface of Hesse-Einstein type is given. (C) 2011 Elsevier B.V. All rights reserved. - Hitoshi FuruhataDIFFERENTIAL GEOMETRY AND ITS APPLICATIONS 27 (3) 420 - 429 0926-2245 2009/06 [Refereed][Not invited]
The condition for the curvature of a statistical manifold to admit a kind of standard hypersurface is given as a first step of the statistical submanifold theory. A complex version of the notion of statistical structures is also introduced. (C) 2008 Elsevier B.V. All rights reserved. - Hitoshi Furuhata, Luc VranckenResults in Mathematics 49 (3-4) 201 - 217 1422-6383 2006/12 [Refereed][Not invited]
We study the center map of an equiaffine immersion which is introduced using the equiaffine support function. The center map is a constant map if and only if the hypersurface is an equiaffine sphere. We investigate those immersions for which the center map is affine congruent with the original hypersurface. In terms of centroaffine geometry, we show that such hypersurfaces provide examples of hypersurfaces with vanishing centroaffine Tchebychev operator. We also characterize them in equiaffine differential geometry using a curvature condition involving the covariant derivative of the shape operator. From both approaches, assuming the dimension is 2 and the surface is definite, a complete classification follows. © 2006 Birkhäuser-Verlag Basel. - H Furuhata, T KuroseBULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN 9 (4) 573 - 587 1370-1444 2002/10 [Refereed][Not invited]
We explicitly determine the self-dual centroaffine surfaces of codimension two with constant affine mean curvature and indefinite affine fundamental form by giving representation formulas. - Codazzi structures induced by minimal affine immersionsFURUHATA HitoshiBanach Center Publ. 57 17 - 19 2002 [Refereed][Not invited]
- H FuruhataBULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN 7 (1) 125 - 134 1370-1444 2000/01 [Refereed][Not invited]
Minimal equi-centroaffine immersions of codimension two are characterized as solutions of a certain variational problem. We determine the moduli space of such immersions of R-2 into R-4 whose induced connection and affine fundamental form coincide with the ones of the Clifford torus. - ICHIYAMA Toshiyuki, FURUHATA Hitoshi, URAKAWA HajimeTsukuba Math. J. Institute of Mathematics, University of Tsukuba 23 (3) 551 - 564 0387-4982 1999 [Refereed][Not invited]
- Hitoshi FuruhataJournal of Geometry 65 (1-2) 111 - 116 1420-8997 1999 [Refereed][Not invited]
We give an intrinsic characterization of isometric pluriharmonic immersions of Kahler manifolds into semi-Euclidean spaces with real codimension one which is a generalization of the Ricci-Curbastro theorem. © Birkhäuser Verlag, 1999. - Holomorphic centroaffine immersions and the Lelieuvre correspondenceFURUHATA Hitoshi, MATSUZOE HiroshiResult. Math. 33 294 - 305 1998 [Refereed][Not invited]
- FURUHATA Hitoshi, MATSUZOE Hiroshi, URAKAWA HajimeInterdiscip. Inform. Sci. 東北大学 4 (2) 125 - 127 1340-9050 1998 [Refereed][Not invited]
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. - H FuruhataGEOMETRIAE DEDICATA 66 (3) 303 - 311 0046-5755 1997/07 [Refereed][Not invited]
We prove a cylinder theorem for isometric pluriharmonic immersions of complete Kahler manifolds into semi-Euclidean spaces under an assumption concerning the index of relative nullity. - H FuruhataPACIFIC JOURNAL OF MATHEMATICS 176 (1) 1 - 14 0030-8730 1996/11 [Refereed][Not invited]
We classify isometric pluriharmonic immersions of a Kahler manifold into an indefinite Euclidean space. The moduli space of such immersions is explicitly constructed in terms of complex matrices. Some examples of these immersions are also given. - FURUHATA HitoshiTohoku Math. Publ. 東北大学 1 1 - 70 1343-9499 1995 [Refereed][Not invited]
- H FURUHATABULLETIN OF THE LONDON MATHEMATICAL SOCIETY 26 487 - 496 0024-6093 1994/09 [Refereed][Not invited]
A classification of isometric minimal immersions of Kahler manifolds into Euclidean spaces is given, which is a generalization of the Calabi-Lawson theory concerning minimal surfaces. Moreover, we explicitly construct a nonholomorphic isometric minimal immersion of a complete Kahler manifold, biholomorphic to C-2, into R(6).
MISC
Books etc
- 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 (Contributor)
共立出版 2022/08 (ISBN: 9784320114784) x, 196p - 曲面 ---幾何学基礎講義古畑 仁 (Single work)
数学書房 2013 (ISBN: 9784903342382) - 北大高校生講座 数学の並木道北大数学科, 中村郁 (Contributor第５章 形をはかる話)
日本評論社 2004 (ISBN: 4535784116)
Presentations
- 統計2重調和写像 [Not invited]上野龍, 古畑仁日本数学会２０２４年度年会幾何学分科会 2024/03
- A class of mappings between statistical manifolds [Invited]古畑仁RIMS共同研究(公開型) 部分多様体と群作用の幾何学 2023/06
- 「幾何学と言えば三角形」は本当か？ [Invited]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 2023/06
- 数学をすべての人々へ アウトリーチの試み [Invited]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） 2023/02
- 幾何学者たちのヘレネー古畑仁北大道新アカデミー 2022/07
- 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 2022/05
- 双極小統計部分多様体といくつかの問題 [Not invited]古畑仁Tsudoi KK 2022/03
- 統計部分多様体に対する不等式 [Invited]古畑仁福岡大学微分幾何セミナー 2022/02
- Submanifold theory in statistical manifolds [Invited]古畑仁日本数学会２０２１年度秋季総合分科会 2021/09
- Centroaffine surfaces of cohomogeneity one [Not invited]Atsushi Fujioka, Hitoshi Furuhata日本数学会２０２０年度年会幾何学分科会（開催中止） 2020/03
- Statistical sectional curvature and warped product statistical manifold [Not invited]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） 2020/03
- Statistical submanifolds and warped product spaces [Invited]FURUHATA HitoshiThe 18th International Conference, Graduate School of Mathematics, Nagoya University, Information Geometry and Affine Differential Geometry III 2019/03
- Centroaffine surfaces of cohomogeneity one [Not invited]FURUHATA Hitoshi研究集会「幾何学のスペクトル」 2018/12
- 統計多様体，具体例からの入門 [Not invited]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) 2018/11
- 統計多様体の定曲率空間 [Invited]古畑 仁福岡大学微分幾何研究集会（Geometry and Analysis 2018） 2018/11
- Sasakian statistical manifolds and hypersurfaces [Invited]FURUHATA HitoshiPNU-HU Joint Symposium 2017/12
- Sasakian statistical manifolds and warped product [Invited]FURUHATA HitoshiSeminar on Differential Geometry 2017/11
- Sasakian statistical manifolds II [Invited]FURUHATA Hitoshi3rd conference on Geometric Science of Information 2017/11
- 中心写像から見た曲面の中心アファイン幾何学 [Invited]古畑 仁第６４回幾何学シンポジウム 2017/08
- アファイン空間の曲線論入門 [Invited]古畑 仁第５回水戸幾何セミナー 2017/07
- 剱持統計多様体 [Invited]古畑 仁関大微分幾何研究会 2017/06
- 佐々木構造をもつ統計多様体 [Invited]古畑 仁名城研究集会「多様体上の計量と幾何構造」 2017/03
- もう一度はじめから，本当のアファイン曲線 [Invited]古畑 仁北海道大学幾何学コロキウム 2016/04
- 正則統計多様体のＣＲ部分多様体 [Invited]古畑 仁福岡大学微分幾何学研究集会 2015/10
- Affine differential geometry and surfaces in the projective space [Invited]FURUHATA Hitoshi研究集会「Differential Geoemetry with Dajczer」 2015/10
- 統計多様体の曲率 [Not invited]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) 2015/09
- 正則統計多様体とその曲率 [Invited]古畑 仁RIMS共同研究「統計多様体の諸分野への応用」 2014/11
- ヘッセ多様体とアファイン球面 [Not invited]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) 2014/01
- 中心アファイン極小曲面の射影極小性 [Invited]古畑 仁福岡大学微分幾何学研究集会 2013/11
- 接続の幾何学における部分多様体論にむけて [Not invited]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) 2013/03
- Statistical immersions of spaces of constant Hessian curvature [Invited]FURUHATA HitoshiThe 8th HU and SNU Symposium on Mathematics -Recent developments of Geometry and Topology- 2012/12
- Geometry of statistical submanifolds [Invited]FURUHATA HitoshiPADGE2012 2012/08
- Submanifold theory for statistical manifolds [Invited]FURUHATA HitoshiWorkshop on Differential Geometry 2012/06
- A realization problem for statistical manifolds [Not invited]FURUHATA Hitoshi福岡大学微分幾何学研究集会 2011/11
- Statistical Immersions [Invited]FURUHATA HitoshiRIKEN Workshop on Information Geometry 2011/08
- 部分多様体の微分幾何学と統計多様体 [Invited]古畑 仁第１３回ＤＣＳセミナー 2011/05
- 定曲率 Hesse 多様体のモデルについて [Invited]古畑 仁研究集会「擬リーマン幾何学の展開III」 2010/12
- Statistical manifolds --- Hypersurfaces and complex structures [Not invited]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) 2010/10
- Statistical submanifolds and spaces of constant Hessian sectional curvature [Invited]FURUHATA HitoshiDifferential Geometry and its Applications 2010/08
- Centroaffine Surfaces with Special Center [Invited]古畑 仁松江微分幾何学研究会２００９ 2009/12
- 統計多様体の微分幾何学入門 [Invited]古畑 仁北海道大学特異点論セミナー 2009/11
- Spaces of non-positive constant Hessian curvature [Invited]古畑 仁福岡大学微分幾何学研究集会 2009/11
- Examples of Statistical Manifolds [Not invited]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) 2009/09
Association Memberships
Research Projects
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)Date (from‐to) : 2022/04 -2027/03Author : 古畑 仁
- 諸分野に現れるアファインはめ込みの研究Date (from‐to) : 2014/04 -2018/03Author : 古畑 仁
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2010/04 -2015/03Author : KUROSE Takashi, SUYAMA Yoshihiko, HAMADA Tatsuyoshi, KAWAKUBO Satoshi, MATSUURA Nozomu, INOGUCHI Junichi, FURUHATA Hitoshi, FUJIOKA AtsushiIn 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.
- 部分多様体にあらわれる統計構造の幾何学Date (from‐to) : 2009/04 -2013/03Author : 古畑 仁
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2007 -2010Author : YAMAGUCHI Keizo, IZUMIYA Shuichi, ONO Kaoru, ISHIKAWA Goo, MATSUMOTO Keiji, HURUHATA Hitoshi, YATSUI Tomoaki, NAKAI Isao, OZAWA Tetasuya, SASAKI TakeshiWe 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：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2006 -2009Author : KUROSE Takashi, SUAYMA Yoshihiko, HAMADA Tatsuyoshi, KAWAKUBO Satoshi, MATSUURA Nozomu, YAMADA Kotaro, INOGUCHI Junichi, FURUHATA HitoshiIn 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.
- 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Date (from‐to) : 2003/04 -2006/03Author : 古畑 仁
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2003 -2005Author : KUROSE Takashi, SUYAMA Yoshihiko, HAMADA Tatsuyoshi, YAMADA Kotaro, INOGUCHI Jun-ichi, FURUHATA HitoshiIn 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.
- 極小アファイン曲面の計量微分幾何学的研究Date (from‐to) : 2001/04 -2003/03Author : 古畑 仁
- アファインはめ込みの情報幾何学的研究稲盛財団：Date (from‐to) : 1999/04 -2002/03Author : 古畑 仁
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2001 -2002Author : KIYOHARA Kazuyoshi, FURUHATA Hitoshi, ISHIKAWA Goo, IZUMIYA Shuichi, IGARASHI Masayuki, SHIMADA IchirouWe 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：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2000 -2002Author : KUROSE Takashi, YAMADA Kotaro, HAMADA Tatsuyoshi, SUYAMA Yoshihiko, FURUHATA Hitoshi, INOGUCHI Jun-ichiIn 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.
- 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Date (from‐to) : 1999/04 -2001/03Author : 古畑 仁
- Affine Immersions and Statistical StructuresDate (from‐to) : 2000
- 一般余次元のアファインはめ込みの分類および構成Date (from‐to) : 1997/04 -1999/03Author : 古畑 仁
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 1998 -1999Author : AIDA Shigeki, KUWAE Kazuhiro, SEKIME Jun, NAGAI Hideo, TOSHIDA Noluo, KAZUM TetsuyaThe 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.
- 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Date (from‐to) : 1995/04 -1996/03Author : 古畑 仁