2005 Fiscal Year Final Research Report Summary
Totally complex submanifolds of a quaternion projective space
| Project/Area Number |
15540065
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | OCHANOMIZU UNIVERSITY |
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Principal Investigator |
TSUKADA Kazumi Ochanomizu University, Department of Mathematics, Professor, 理学部, 教授 (30163760)
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| Co-Investigator(Kenkyū-buntansha) |
EJIRI Norio Meijo University, Department of Mathematics, Professor, 理学部, 教授 (80145656)
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| Project Period (FY) |
2003 – 2005
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| Keywords | a quaternion projective space / a quaternionic Kahler manifold / totally complex submanifolds / twistor space / totally real submanifolds / Einstein-Kahler submanifolds / a complex quadric / isotropic Kahler immersions |
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| Research Abstract |
In this research, we investigate totally complex submanifolds of a quaternion projective space (more generally a quaternionic Kahler manifold). It is one aspect of the interplay of quaternionic differential geometry and complex differential geometry and makes so-called "quaternionic complex differential geometry". A submanifold M of a quaternionic Kahler manifold (M^^~, Q, g^^~) is said to be totally complex if there exists a section I^^~ of the bundle Q|_M such that (1)I^^~^2 = -id, (2)I^^~TM = TM (3)KTM ⊥ TM for any K ∈ Q|_M with <I^^~, K> = 0. Typical examples are half dimensional totally complex submanifolds of a quaternion projective space HP^n with parallel second fundamental form, which have been classified by Tsukada(head investigator of this research). The twistor space Z of (M^^~,Q,g^^~) is defined by Z = {I^^~ ∈ Q|I^^~^2 =-id}, which is an S^2-bundle over M^^~. The twistor space Z has a natural complex structure and it admits an Einstein-Kahler metric if M^^~ has positive scalar curvature.
Main results of this research are the following :
1.We show fundamental theorem on the existence and the uniqueness for half dimensional totally complex submanifolds of HP^n or the quaternion hyperbolic space HH^n. This result is an affirmative answer to the conjecture by Alekseevsky and Marchiafava.
2.For a totally complex submanifold M of M^^~, we consider a new natural lift to the twistor space Z of M^^~ and construct a totally real and minimal submanifold of Z.
3.We characterize half dimensional totally complex submanifolds of HP^n with parallel second fundamental form under some curvature condition such as Einstein-Kahler.
4.We investigate fundamental properties of isotropic Kahler submanifolds of a complex quadric, whose theory is analogous to that of totally complex submanifolds.
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