The research of quaternionic manifolds by using subamnifold geometry

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03272
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Kanazawa University
• Principal Investigator: Kazuyuki Hasegawa 金沢大学, 学校教育系, 教授 (50349825)
• Co-Investigator(Kenkyū-buntansha): 守屋 克洋 筑波大学, 数理物質系, 助教 (50322011)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 四元数多様体 / 超複素多様体 / 複素部分多様体 / c-射影構造

Outline of Final Research Achievements

We give a construction of a hypercomplex manifold from a quaternion manifold (Q/H-correspondence). We also find a construction of a quaternion manifold from a hypercomplex manifold (H/Q-correspondence). As a concrete example, we obtain a hypercomplex Hopf manifold, which does not allow any hyperkahler structures. In particular, as an application of H/Q-correspondence, it is possible to construct a quaternion manifold from a certain type of complex manifold, which follow from the fact that the tangent bundle of this complex manifold has the structure of a quaternion manifold.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

四元数多様体や超複素多様体は重要な研究対象であり，それらの構成は基本的なテーマである．本研究では，超複素運動量写像を用いて四元数多様体から超複素多様体を構成したが，超複素運動量写像の有用性を示したことや古典的な題材であるc-射影構造を活用して点でも学術的意義があると思われる．また，今後は物理学への応用もふまえ，更なる展開を考えていきたい．