1999 Fiscal Year Final Research Report Summary
Geometry of Almost Hermitian Manifolds
| Project/Area Number |
10640069
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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 | NIIGATA UNIVERSITY |
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Principal Investigator |
SEKIGAWA Kouei Niigata Univ., Dept. Math., Professor, 理学部, 教授 (60018661)
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| Co-Investigator(Kenkyū-buntansha) |
INNAMI Nobuhiro Niigata Univ., Dept. Math. and Inform. Professor, 大学院・自然科学研究科, 教授 (20160145)
WATABE Tsuyoshi Niigata Univ., Dept. Math., Professor, 理学部, 教授 (60018257)
MATSUSHITA Yasuo Shiga Prefectural Univ., Dept. Math., Professor, 工学部, 教授 (90144336)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Almost Kaehler manifold / Quasi Kaehler manifold / Einstein manifold / Weakly *-Einstein manifold / Integrability / Holomorphic sectional curvature / Cλ-manifold / Hitchin-Thorpe type inequality |
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| Research Abstract |
The main purpose of our research project is to investigate the following subjects in Geometry of Almost Hermitian Manifolds :
(1) The Goldberg Conjecture (Compact almost Kaehler Einstein manifold is integrable)
(2) Structure of almost Hermitian manifolds of pointwise constant holomorphic sectional curvature
(3) Symplectic structures on the manifolds of closed geodesics of Cλ-manifolds
(4) Other related topics
Subject(1). It is known that the conjecture is true in the case where the scalar curvature is nonnegative. In the terms of the project, we obtained further some partial positive answers to the conjecture. For example, we showed that a 4-dimensional strictly almost Kaehler Einstein and weakly *-Einstein manifold is a Ricci flat space of pointwise constant holomorphic sectional curvature τ */8, and also that a 4-dimensional compact almost Kaehler Einstein and weakly *-Einstein manifold is Kaehler. Quite recently, we generalized the last result slightly to the case where the length of the
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skew-symmetric part of the Ricci *-tensor is constant.
Subject(2). For arbitrary almost Hermitian manifold, the constancy of holomorphic sectional curvature grantees the integrability of the almost Hermitian manifold. For example, there exists examples of compact locally flat almost Hermitian manifolds which are not integrable. So. It is natural to consider the subject (2) restricting to some special classes of almost Hermitian manifolds. In the terms of the project, we proved that a 6-dimensional quasi-Kaehler manifold of constant sectional curvature is a locally flat Kaehler manifold or a nearly Kaehler manifold of positive constant sectional curvature.
Subject(3). It is well-known that the manifold CM of all closed geodesics in Cλ-manifold M admits a symplectic structure ω and all closed geodesics through a specified point of M is a Lagrangian submanifold of CM. So, it is quaite natural to consider almost Kaehler structure associated to the symplectic structureω. Recently, we determined the manifold CM for each compact rank one symmetric space.
Subject(4) A 4-dimensional pseudo-Riemannian manifolds of metric signature (+ + - -) admits a pair of almost complex structures (J,J'). By making use of this property, we obtained Hitchin-Thorpe type inequality for 4-dimensional compact pseudo-Riemannian manifolds of metric. Further, we gave several examples of 4-dimensional double isotropic Kaehler structures on RィイD14ィエD1 equipped with pseudo-Riemannian metric of metric signature (+ + - -). In quite recent work, we also obtained two kinds of generalizations of the Hitchin's lemma with. respect to the integrability of quaternionic almost Kaehler manifolds. We have also some results for the problems in billiyard dynamics. Less
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