The quaternionic holomorphic differential geometry of totally complex submanifolds

2016 Fiscal Year Final Research Report

• Project/Area Number: 25400065
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Ochanomizu University
• Principal Investigator: TSUKADA KAZUMI お茶の水女子大学, 基幹研究院, 教授 (30163760)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 四元数多様体 / 全複素部分多様体 / 四元数射影空間 / 複素グラスマン多様体 / 横断的複素部分多様体 / ツイスター空間

Outline of Final Research Achievements

We studied complex submanifolds of quaternionic manifolds and quaternionic Kaehler manifolds. We developed the theory of transversally complex submanifolds in a quaternion projective space. We found the (2,0)+(0,2)-part of the second fundamental form and the function S whose values are complex structures of a quaternionic vector space as the invariants of the quaternionic differential geometry. We studied the theory of the construction and the classification of totally complex submanifolds in a complex Grassmann manifold of 2-planes. We showed that the projective cotangent bundle of a complex projective space is a twistor space of the complex Grassmann manifold. Applying this fact, we construct maximal totally complex submanifolds of the complex Grassmann manifold from complex submanifolds of a complex projective space.

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