Geometric structures related to neutral metrics

2015 Fiscal Year Final Research Report

• 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
• 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

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.

Free Research Field

微分幾何学