Twistor theoretic approach for the geometry of non-Riemannian type connection

• Project/Area Number: 24740050
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Fukushima University
• Principal Investigator: NAKATA Fuminori 福島大学, 人間発達文化学類, 准教授 (80467034)
• Research Collaborator: KAMADA Hiroyuki 宮城教育大学, 教育学部, 教授 (00249799) ; HASHIMOTO Hideya 名城大学, 理工学部, 教授 (60218419) ; OHASHI Misa 名古屋工業大学, 工学部, 准教授 (70710359) ; MATSUSHITA Yasuo 滋賀県立大学, 名誉教授
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: ツイスター理論 / 不定値計量 / 複素幾何学 / 波動方程式 / 微分幾何学

Outline of Final Research Achievements

The twistor correspondence for circle invariant indefinite Einstein-Weyl structure is established, and its relation with the theory of integral transforms and wave equations are found. Investigation for the twistor correspondence for the twisted Tod-Kamada metric is also progressing.
On the other hand, by the collaboration with several specialists, we start an investigation for G2 geometry, and obtained a remarkable result. That is, we succeeded to characterize a geometric structure of the associative Grassmannian via twistor theory in the G2 geometry. This theory is strongly related to the theory of isoparametric hypersurfaces. Moreover, we succeeded to write down a map clearly which is important in the investigation of the associative Grassmannian.

Research Products

• [Journal Article] Circle invariant indefinite Einstein-Weyl structures and the twistor correspondence (2016)
  • Author(s): Fuminori Nakata
  • Journal Title: Current Developments in Differential Geometry and its Related Fields Volume: 1 Pages: 45-56
  • Peer Reviewed
• [Journal Article] Wave equations, Integral transforms, and twistor theory on indefinite geometry (2014)
  • Author(s): Fuminori Nakata
  • Journal Title: Prospects of Differential Geometry and its Related Fields(Proceedings) Volume: 1 Pages: 83-97
  • Peer Reviewed
• [Journal Article] An integral transform on a cylinder and the twistor correspondence (2012)
  • Author(s): Fuminori Nakata
  • Journal Title: Differential Geometry and its Applications Volume: 30 Issue: 5 Pages: 428-437
  • DOI: 10.1016/j.difgeo.2012.06.004
  • Peer Reviewed
• [Presentation] 結合的グラスマン多様体に関するツイスター対応について (2016)
  • Author(s): 中田文憲
  • Organizer: 名城大学幾何学研究集会
  • Place of Presentation: 名城大学
  • Year and Date: 2016-03-02
  • Invited
• [Presentation] 対称空間G2/SO(4)のイソトロピー表現について (2016)
  • Author(s): 中田文憲
  • Organizer: 淡路島幾何学研究集会
  • Place of Presentation: 国民宿舎慶野松原荘
  • Year and Date: 2016-01-23
• [Presentation] 自己双対計量とツイスター対応 I, II (2015)
  • Author(s): 中田文憲
  • Organizer: 幾何学阿蘇研究集会
  • Place of Presentation: 休暇村南阿蘇
  • Year and Date: 2015-09-24
  • Invited
• [Presentation] 不定値のツイスター対応と積分変換 (2015)
  • Author(s): 中田文憲
  • Organizer: 関大微分幾何研究集会
  • Place of Presentation: 関西大学
  • Year and Date: 2015-06-21
  • Invited
• [Presentation] twistor correspondence for Associative Grassmannian (2015)
  • Author(s): 中田文憲
  • Organizer: G2幾何小研究集会
  • Place of Presentation: 鶴城コミュニティセンター
  • Year and Date: 2015-05-31
• [Presentation] 正則円板の族によるEinstein-Weyl構造の構成について (2015)
  • Author(s): 中田文憲
  • Organizer: 名城大学幾何学研究集会
  • Place of Presentation: 名城大学
  • Year and Date: 2015-03-11
  • Invited
• [Presentation] 複素・正定値・ニュートラルのツイスター対応 (2015)
  • Author(s): 中田文憲
  • Organizer: GAP Project Spring School
  • Place of Presentation: 立命館大学
  • Year and Date: 2015-03-09 – 2015-03-10
  • Invited
• [Presentation] 正則円板の族によるEinstein-Weyl構造の構成について (2015)
  • Author(s): 中田文憲
  • Organizer: 淡路島幾何学研究集会2015
  • Place of Presentation: 国民宿舎慶野松原荘
  • Year and Date: 2015-01-24
  • Invited
• [Presentation] Circle invariant indefinite Einstein-Weyl structures and the twistor correspondence (2014)
  • Author(s): 中田文憲
  • Organizer: ICDG2014
  • Place of Presentation: Veliko Tarnnovo (ブルガリア)
  • Year and Date: 2014-09-08
  • Invited
• [Presentation] 自己双対性とツイスター対応1,2 (2013)
  • Author(s): 中田文憲
  • Organizer: RIMS共同研究「凝集現象を内包する非線形偏微分方程式に対する幾何学的視点と解析学的視点の協同」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 自己双対計量と波動方程式I・II (2012)
  • Author(s): 中田文憲
  • Organizer: RIMS研究集会「逆問題への応用を意図した解析学の研究」
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Twsitor correspondence for R-invariant indefinite self-dual metric on R^4
  • Author(s): 中田文憲
  • Organizer: 多変数複素解析冬セミナー
  • Place of Presentation: コラッセ福島
• [Presentation] 八元数とツイスター対応
  • Author(s): 中田文憲
  • Organizer: 淡路島幾何学研究集会
  • Place of Presentation: 国民宿舎慶野松原荘
• [Presentation] 不定値のツイスター対応について
  • Author(s): 中田文憲
  • Organizer: 日本数学会東北支部会
  • Place of Presentation: 宮城教育大学
• [Presentation] Twistor correspondence for non-compact indefinite self-dual manifolds
  • Author(s): 中田文憲
  • Organizer: Progress of geometric structures on manifolds
  • Place of Presentation: 名城大学
• [Presentation] 積分変換・波動方程式と不定値のツイスター理論
  • Author(s): 中田文憲
  • Organizer: 福島応用数学研究集会
  • Place of Presentation: コラッセ福島
  • Invited
• [Presentation] ツイスター理論
  • Author(s): 中田文憲・本多宣博
  • Organizer: 秋葉原微分幾何セミナー
  • Place of Presentation: 東京理科大学
  • Invited
• [Book] ４次元微分幾何学への招待（ＳＧＣライブラリ113） (2014)
  • Author(s): 松下泰雄・鎌田博行・中田文憲
  • Total Pages: 187
  • Publisher: サイエンス社