Study of topis related to almost complex structures

与几乎复杂结构相关的topis研究

• 批准号: 16540057
• 负责人: SEKGAWA Kouei
• 金额: $ 2.11万
• 依托单位: Niigata University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540057/
• 关键词: Almost complex structure; Kaehler manifold; 6-dimensional sphere; Einstein manifold; Goldberg conjecture; J-holomorphic curve; Solvmanifold; Walker metric; 概超複素多様体; 概ケーラー多様体; Spin(7); Blair-Ianus汎関数; Almost complex structure; Almost Kaehler m

A smooth manifold M admitting a (1,1)-tensor field J satisfying J=・I is called an almost complex manifold and the tensor field J is called the almost complex structure. The concept of almost complex manifold is a generalization of complex manifold.. Almost complex manifold (M, J) is said to be integrable if M admits a complex structure and the derived almost complex structure coincides with the almost complex structure J. Any 2-dimensional almost complex manifold is always integrable. However, this is not true for higher dimensional cases in general. An almost complex manifold (M, J) equipped with a compatible (pseudo) Riemannian metric g is called an almost Hermitian manifold. A Kaehler manifold is the most typical one. In this research project, we study mainly the following topics related to the almost complex structures :(1) Integrability of almost complex structure(2) Submanifolds in almost complex manifolds(3) Intermediate and Related topics to (1),(2)Concerning (1), we study the integrability of almost Kaehler manifolds, for example Goldberg conjecture. Y.Matsushita et al. constructed an 8-dimensional counter example with a neutral Walker metric to the conjecture in the pseuo-Riemannian case. However, the conjecture itself is still remaining open in the case where the scalar curvature is negative. Concerning (2), H.Hashimoto studied several topics related to J-holomorphic curves in the nearly Kaehler 6-sphere S6 from the viewpoint of the Grassmann geometry and obtained interesting results on the deformations of super-minimal J-holomorphic curves and on some tubes around J-holomorphic curves in S6. Recently, the head investigator and H.Hashimoto et al. began to study 6-dimensional oriented submanifolds in the Octonions.and succeed to classify all extrinsic homogeneous almost hermitian 6-manifolds. Concerning (3), for example, K.Hasegawa gave an affirmative answer to the generalized Benson-Gordon conjecture.

接纳(1,1)-张量场J且满足J=·I的光滑流形M称为几乎复流形，张量场J称为几乎复结构。几乎复流形的概念是复流形的推广。如果 M 承认一个复结构并且导出的几乎复结构与几乎复结构 J 一致，则称几乎复流形 (M, J) 是可积的。任何二维几乎复流形总是可积的。然而，对于一般的高维情况来说，情况并非如此。配备兼容（伪）黎曼度量 g 的近复流形 (M, J) 称为近厄米流形。凯勒流形是最典型的一种。在本研究项目中，我们主要研究以下与几乎复杂结构相关的主题：（1）几乎复杂结构的可积性（2）几乎复杂流形中的子流形（3）与（1），（2）相关的中间主题和相关主题（1），我们研究几乎凯勒流形的可积性，例如戈德堡猜想。 Y.松下等人。针对伪黎曼情况下的猜想，构造了一个具有中性沃克度量的 8 维反例。然而，在标量曲率为负的情况下，猜想本身仍然悬而未决。关于(2)，H.Hashimoto从格拉斯曼几何的角度研究了近凯勒6球S6中与J全纯曲线相关的几个课题，并在超最小J全纯曲线和S6中J全纯曲线周围的一些管子的变形方面得到了有趣的结果。最近，首席研究员和 H.Hashimoto 等人。开始研究八元数中的6维定向子流形，并成功对所有外在齐次几乎埃尔米特6-流形进行分类。例如，对于（3），K.Hasekawa 对广义本森-戈登猜想给出了肯定的答案。

项目成果

Notes on 4-dimensional almost hyperhermitian manifolds

关于 4 维几乎超厄米流形的注释

• 发表时间: 2006
• 期刊: Mathematica Balkanica 19
• 作者: Y.Matsushita;S.Haze;P.Law;J.Davidov;K.Seigawa;K.Hasegawa;H.Hashimoto;K.Hasegawa;K.Sekigawa;T.Nihonyanagi
• 通讯作者: T.Nihonyanagi

Almost Kaehler-Einstein structures on 8-dimensional Walker manifolds

8 维沃克流形上的几乎凯勒-爱因斯坦结构

• 发表时间: 2007
• 期刊: Monatshefte fur Mathematik 150
• 作者: Y.Matsushita;S.Haze;P.Law
• 通讯作者: P.Law

A remark on four-dimensional almost Kahler-Einstein manifolds With negative scalar curvature

关于具有负标量曲率的四维几乎卡勒-爱因斯坦流形的评论

• 发表时间: 2004
• 期刊: Intern.J.Math.and Math.Sci. 35
• 作者: Y.Matsushita;S.Haze;P.Law;J.Davidov;K.Seigawa;K.Hasegawa;H.Hashimoto;K.Hasegawa;K.Sekigawa;T.Nihonyanagi;N.Innami;K.Hasegawa;H.Hashimoto;Y.Matsushita;H.Hashimoto;T.Nihonyanagi;M.Chaichi;T.Oguro;R.S.Lemence
• 通讯作者: R.S.Lemence

On the Spin(7) frame fields on S^1 x S^2 x R^3

在 S^1 x S^2 x R^3 上的 Spin(7) 帧字段上

• 发表时间: 2006
• 期刊: J. Research Institute of Meijo University 5
• 作者: Y.Matsushita;S.Haze;P.Law;J.Davidov;K.Seigawa;K.Hasegawa;H.Hashimoto;K.Hasegawa;K.Sekigawa;T.Nihonyanagi;N.Innami;K.Hasegawa;H.Hashimoto
• 通讯作者: H.Hashimoto

Curvature properties of four-dimensional Walker metrics

四维 Walker 度量的曲率性质

• 发表时间: 2005
• 期刊: Classical and Quantum Gravity 22
• 作者: Y.Matsushita;S.Haze;P.Law;J.Davidov;K.Seigawa;K.Hasegawa;H.Hashimoto;K.Hasegawa;K.Sekigawa;T.Nihonyanagi;N.Innami;K.Hasegawa;H.Hashimoto;Y.Matsushita;H.Hashimoto;T.Nihonyanagi;M.Chaichi
• 通讯作者: M.Chaichi