Geometric structures related to neutral metrics

• Project/Area Number: 24540062
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Miyagi University of Education
• Principal Investigator: Kamada Hiroyuki 宮城教育大学, 教育学部, 教授 (00249799)
• Co-Investigator(Renkei-kenkyūsha): AIHARA YOSHIHIRO 福島大学, 人間発達文化学類, 教授 (60175718) ; NAYATANI SHIN 名古屋大学, 大学院多元数理科学研究科, 教授 (70222180) ; NAKAGAWA YASUHIRO 佐賀大学, 大学院工学系研究科, 教授 (90250662) ; IZEKI HIROYASU 慶應義塾大学, 理工学部, 教授 (90244409) ; NAKATA FUMINORI 福島大学, 人間発達文化学類, 准教授 (80467034)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥5,200,000 (Direct Cost: ¥4,000,000、Indirect Cost: ¥1,200,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: ニュートラル計量 / ニュートラル構造 / パラ超複素構造 / 四元数CR構造 / 強積分可能性 / ツイスター空間 / ツイスター概CR構造 / 四元数ＣＲ構造

Outline of Final Research Achievements

A pseudo-Riemannian metric on a manifold is called a neutral metric if it has neutral signature, and a family of local neutral metrics that conicide, except for multiplication by -1, on the overlaps, is called a neutral structure. Davidov et al. obtained examples of compact complex surfaces with a quaternion-like structure (parahypercomplex structure) that admit compatible neutral structures, but never admit any compatible neutral metric. Then we show that their example of a hyperelliptic surface can be deformed to a compatible neutral structure, which is not locally conformal parahyperkahler. Also, ｗe introduce the notion of strong integrability for a quaternionic CR manifold (of dimension greater than 7), and show that, under ultra pseudoconvexity and strong integrability, a partially integrable almost CR structure (called the twistor almost CR structure) is defined naturally on its twistor space.

Research Products

• [Journal Article] Quaternionic CR geometry (2013)
  • Author(s): Hiroyuki Kamada and Shin Nayatani
  • Journal Title: Hokkaido Mathematical Journal Volume: 印刷中（in press）
  • Peer Reviewed
• [Presentation] Quaternionic CR structure: a geometric structure modeled on a real hypersurface in a quaternionic manifold (2015)
  • Author(s): Hiroyuki Kamada
  • Organizer: Workshop on almost Hermitian and contact geometry
  • Place of Presentation: Castle Bedlewo, Bedlewo, Poland
  • Year and Date: 2015-10-18
  • Int'l Joint Research / Invited
• [Presentation] Geometric structures modeled on a real hypersurface in a quaternionic manifold (2014)
  • Author(s): Hiroyuki Kamada
  • Organizer: Fourth International Colloquim on Differential Geometry and its Related Fields
  • Place of Presentation: St. Cyril and St. Methodius University of Veliko Tarnovo (Veliko Tarnovo, Bulgaria)
  • Year and Date: 2014-09-08 – 2014-09-11
• [Book] Current Developments in Differential Geometry and its Related Fields (Almost CR Structure on the Twistor Space of a Quaternionic CR Manifold) (2015)
  • Author(s): Hiroyuki Kamada and Shin Nayatani; A. Arvanitoyeorgos, Y. Sakane and M. Statha; T. Adachi; F. Nakata; N. Ando; N.A. Ivanov; M. Ivanova and H. Manev; N. Ejiri and T. Shoda; Y. Tursun; H. Hashimoto and K. Suzuki; G. Nakova; H. Matsuzoe; M. Ohashi; Q. Shi; M.J. Hristov
  • Total Pages: 241
  • Publisher: World Scientific
• [Book] Current Developments in Differential Geometry and its Related Fields (Almost CR structure on the twistor space of a quaternionic CR manifold） (2015)
  • Author(s): Hiroyuki Kamada and Shin Nayatani 他
  • Total Pages: 234
  • Publisher: World Scientific