Twistor theory in Submanifold geometry

• Project/Area Number: 16K05119
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Ibaraki University
• Principal Investigator: Kimura Makoto 茨城大学, 理工学研究科(理学野), 教授 (30186332)
• Co-Investigator(Kenkyū-buntansha): 入江 博 茨城大学, 理工学研究科(理学野), 准教授 (30385489) ; 大塚 富美子 茨城大学, 理工学研究科(理学野), 准教授 (90194208)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: ツイスター空間 / ホップ超曲面 / 複素グラスマン多様体 / 四元数ケーラー構造 / パラ四元数ケーラー構造 / 実超曲面 / φ-断面曲率 / スカラー曲率 / リッチ・ソリトン / 線織面 / グラスマン多様体 / 複素射影空間 / 複素双曲空間 / ガウス写像 / ルジャンドル部分多様体 / 部分多様体

Outline of Final Research Achievements

There are a lot of applications of twistor theory in mathematics and geometry. We showd that Hopf hypersurface in a complex projective space is realized as a circle bundle over maximal horizontal submanifold of twistor space of complex 2-plane Grassmann manifold.

Academic Significance and Societal Importance of the Research Achievements

複素双曲空間のホップ超曲面については、部分的な構造定理はいくつか知られていたが統一的に説明できる結果は得られていなかった。我々は、不定値複素グラスマン多様体の「パラ四元数ケーラー構造」に関する３つのツイスター空間を用いて、ホップ超曲面の統一的な構造定理を得た。特に、3つの内の1つのツイスター空間は我々が初めて明らかにしたもので、今後の展開が期待される。

Research Products

• [Int'l Joint Research] 全南大学校/慶北大学校(韓国)
• [Int'l Joint Research] グラナダ大学(スペイン)
• [Int'l Joint Research] グラナダ大学(スペイン)
• [Int'l Joint Research] 全南大学校/慶北大学校(韓国)
• [Int'l Joint Research] 全南大学校/慶北大学校(韓国)
• [Int'l Joint Research] グラナダ大学(スペイン)
• [Journal Article] Ruled Real Hypersurfaces in the Complex Quadric (2021)
  • Author(s): Kimura Makoto、Lee Hyunjin、Perez Juan de Dios、Suh Young Jin
  • Journal Title: The Journal of Geometric Analysis Volume: To appear Issue: 8 Pages: 7989-8012
  • DOI: 10.1007/s12220-020-00564-2
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Real hypersurfaces with constant Phi-sectional curvature in complex projective space (2020)
  • Author(s): Cho Jong Taek、Kimura Makoto
  • Journal Title: Differential Geometry and its Applications Volume: 68 Pages: 101573-101573
  • DOI: 10.1016/j.difgeo.2019.101573
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Transversal Jacobi Operators in Almost Contact Manifolds (2020)
  • Author(s): Cho Jong Taek、Kimura Makoto
  • Journal Title: Mathematics Volume: 9 Issue: 1 Pages: 31-31
  • DOI: 10.3390/math9010031
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] New construction of ruled real hypersurfaces in a complex hyperbolic space and its applications (2019)
  • Author(s): MAKOTO KIMURA, SADAHIRO MAEDA AND HIROMASA TANABE
  • Journal Title: Geometriae Dedicata Volume: Online
  • Peer Reviewed
• [Journal Article] Levi-umbilical real hypersurfaces in Hermitian symmetric spaces (2019)
  • Author(s): Cho Jong Taek、Kimura Makoto
  • Journal Title: Topology and its Applications Volume: 264 Pages: 145-157
  • DOI: 10.1016/j.topol.2019.06.009
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Gradient Ricci soliton on $O(n)$-invariant $n$-dimensional submanifold in $S^n(1)\times S^n(1)$ (2019)
  • Author(s): Jong Taek Cho and Makoto Kimura
  • Journal Title: Journal of the Korean Mathematical Society Volume: 56 Pages: 81-90
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Hopf Real Hypersurfaces in the Indefinite Complex Projective Space (2019)
  • Author(s): Kimura Makoto、Ortega Miguel
  • Journal Title: Mediterranean Journal of Mathematics Volume: 16 Issue: 2
  • DOI: 10.1007/s00009-019-1299-9
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Levi-umbilical real hypersurfaces in Hermitian symmetric spaces (2019)
  • Author(s): Jong Taek Cho and Makoto Kimura
  • Journal Title: Topology and Its Applications Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Integral curves of the characteristic vector field of minimal ruled real hypersurfaces in non-flat complex space forms (2019)
  • Author(s): Makoto Kimura，Sadahiro Maeda and Hiromasa Tanabe
  • Journal Title: Hokkaido Mathematical Journal Volume: 印刷中
  • Peer Reviewed
• [Journal Article] The homogeneous ruled real hypersurface in a complex hyperbolic space (2018)
  • Author(s): Kimura Makoto、Maeda Sadahiro、Tanabe Hiromasa
  • Journal Title: Journal of Geometry Volume: 109 Issue: 1
  • DOI: 10.1007/s00022-018-0421-7
  • Peer Reviewed
• [Journal Article] Characterizations of three homogeneous real hypersurfaces in a complex projective space (2018)
  • Author(s): Makoto Kimura and Sadahiro Maeda
  • Journal Title: Hokkaido Mathematical Journal Volume: 47 Pages: 291-316
  • Peer Reviewed
• [Journal Article] Gradient Ricci soliton on $O(n)$-invariant $n$-dimensional submanifold in $S^n(1)\times S^n(1)$ (2018)
  • Author(s): Jong Taek Cho and Makoto Kimura
  • Journal Title: Journal of the Korean Mathematical Society Volume: 未定
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Integral curves of the characteristic vector field of minimal ruled real hypersurfaces in non-flat complex space forms (2018)
  • Author(s): Makoto Kimura，Sadahiro Maeda and Hiromasa Tanabe
  • Journal Title: Hokkaido Mathematical Journal Volume: 未定
  • Peer Reviewed
• [Journal Article] Hopf hypersurfaces in complex projective space and half-dimensional totally complex submanifolds in complex $2$-plane Grassmannians II (2017)
  • Author(s): Makoto Kimura
  • Journal Title: Differential Geometry and its Applications Volume: 印刷中 Pages: 44-52
  • DOI: 10.1016/j.difgeo.2016.12.007
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Levi-umbilical real hypersurfaces in a complex space form (2017)
  • Author(s): Jong Taek Cho and Makoto Kimura
  • Journal Title: Nagoya Math. J. Volume: 印刷中 Pages: 99-112
  • DOI: 10.1017/nmj.2016.63
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Characterizations of three homogeneous real hypersurfaces in a complex projective space (2017)
  • Author(s): Makoto Kimura and Sadahiro Maeda
  • Journal Title: Hokkaido Mathematical Journal Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Classification of regular complex 2-polyhedra of positive constant excess (2017)
  • Author(s): Jin-ichi Itoh, Fumiko Ohtsuka
  • Journal Title: Topology and its applications Volume: 219 Pages: 43-54
  • DOI: 10.1016/j.topol.2017.01.004
  • Peer Reviewed / Open Access
• [Presentation] Lagrangian submanifolds in complex projective space and quaternionic Kahler geometry (2021)
  • Author(s): Makoto Kimura
  • Organizer: Submanifolds of Symmetric spaces and thier time evolutions
  • Int'l Joint Research / Invited
• [Presentation] Gauss map of real hypersurfaces in non-flat complex space forms and twistor space of complex 2-plane Grassmannian (2019)
  • Author(s): Makoto Kimura
  • Organizer: Workshop on the isoparametric theory, Beijing
  • Int'l Joint Research / Invited
• [Presentation] Real hypersurfaces in non-flat complex space forms and twistor spaces of complex $2$-plane Grassmannian (2019)
  • Author(s): Makoto Kimura
  • Organizer: The 22nd International Workshop on Differential Geometry of Submanifolds in Symmetric Spaces, Daegu
  • Int'l Joint Research / Invited
• [Presentation] Real hypersurfaces in non-flat complex space forms and twistor spaces of complex 2-plane Grassmannian (2019)
  • Author(s): Makoto Kimura
  • Organizer: International workshop on Geometry of submanifolds 2019, Istanbul
  • Int'l Joint Research / Invited
• [Presentation] 非平坦複素空間形の実超曲面と複素2-平面グラスマン多様体のツイスター空間 (2019)
  • Author(s): 木村真琴
  • Organizer: 2019 日本数学会秋季総合分科会幾何学分科会特別講演
  • Invited
• [Presentation] Ruled submanifolds in complex space forms (2019)
  • Author(s): MakotoKimura
  • Organizer: Workshop on Differential Geometry, Topology and Mathematical Physics, Gwangju-2019
  • Int'l Joint Research / Invited
• [Presentation] Twister spaces of complex 2-plane Grassmannian and Hopf hypersurfaces in non-flat complex space forms (2018)
  • Author(s): Makoto Kimura
  • Organizer: Mini Workshop on Submanifolds in Hermitian Symmetric Spaces and Related Topics, Daegu-2018
  • Int'l Joint Research / Invited
• [Presentation] Polar hypersurfaces of curves and surfaces in non-flat complex space forms (2018)
  • Author(s): Makoto Kimura
  • Organizer: Mini Workshop on Submanifolds in Hermitian Symmetric Spaces and Related Topics, Daegu-2018
  • Int'l Joint Research / Invited
• [Presentation] Twistor space of complex 2-plane Grassmannian and Hopf hypersurfaces in non-flat complex space forms (2017)
  • Author(s): Makoto Kimura
  • Organizer: Workshop on Differential Geometry, Gwangju-2017
  • Int'l Joint Research / Invited
• [Presentation] 線織部分多様体について (2017)
  • Author(s): 木村真琴
  • Organizer: 東京理科大学理工学部数学科談話会
  • Invited
• [Presentation] Real hypersurfaces in non-flat complex space forms and (para-)quaternionic geometry of complex 2-plane Grassmannian (2016)
  • Author(s): Makoto Kimura
  • Organizer: Submanifold theory, Yuzawa 2016
  • Place of Presentation: 湯沢グランドホテル
  • Year and Date: 2016-12-02
  • Invited
• [Presentation] Submanifolds of complex space forms and twistor space of complex 2-plane Grassmannian (2016)
  • Author(s): Makoto Kimura
  • Organizer: Quaternionic Differential Geometry and its Related Topics
  • Place of Presentation: お茶の水女子大
  • Year and Date: 2016-09-09
  • Int'l Joint Research / Invited
• [Presentation] Twistor space of complex 2-plane Grassmannian and submanifolds in complex space forms (2016)
  • Author(s): Makoto Kimura
  • Organizer: DGA 2016
  • Place of Presentation: Masaryk Univ., Brno, Czech
  • Year and Date: 2016-07-19
  • Int'l Joint Research
• [Remarks] Makoto Kimura
  • URL: https://kmakoto.sci.ibaraki.ac.jp
• [Remarks] Makoto Kimura
  • URL: http://kmakoto.sci.ibaraki.ac.jp/