Project/Area Number  10640069 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  NIIGATA UNIVERSITY 
Principal Investigator 
SEKIGAWA Kouei Niigata Univ., Dept. Math., Professor, 理学部, 教授 (60018661)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,600,000 (Direct Cost : ¥3,600,000)
Fiscal Year 1999 : ¥1,900,000 (Direct Cost : ¥1,900,000)
Fiscal Year 1998 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  Almost Kaehler manifold / Quasi Kaehler manifold / Einstein manifold / Weakly *Einstein manifold / Integrability / Holomorphic sectional curvature / Cλmanifold / HitchinThorpe type inequality / 概ケーラー多様体 / Quasiケーラー多様体 / アインシュタイン多様体 / 弱^*アインシュタイン多様体 / 積分可能性 / 正則断面曲率 / Ce多様体 / HitchinThorpe型不等式 / 弱*アインシュタイン多様体 / C_l多様体 / 概(擬)エルミート多様体 / (概)ケーラー多様体 / (*)アインシュタイン多様体 / (反)正則断面曲率 / C_L多様体 / シンプレクティック構造 / Maslou指数 
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 Cλmanifolds (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 4dimensional 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 4dimensional 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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skewsymmetric 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 6dimensional quasiKaehler 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 wellknown that the manifold CM of all closed geodesics in Cλmanifold 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 4dimensional pseudoRiemannian manifolds of metric signature (+ +  ) admits a pair of almost complex structures (J,J'). By making use of this property, we obtained HitchinThorpe type inequality for 4dimensional compact pseudoRiemannian manifolds of metric. Further, we gave several examples of 4dimensional double isotropic Kaehler structures on RィイD14ィエD1 equipped with pseudoRiemannian 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
