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EINSTEIN-WEYL SPACE AND WEYL SUBMANIFOLD

爱因斯坦韦尔空间和韦尔子流形

基本信息

批准号：
10640098
负责人：
NARITA Fumio
金额：
$ 0.58万
依托单位：
AKITA NATIONAL COLLEGE OF TECHNOLOGY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

Let M be a manifold with a conformal structure [g] and a tortion-free affine connection D.I consider a Gauduchon manifold (M, g, D) with the Gauduchon metric g. Firstly, I obtained that a Gauduchon manifold (M, g, D) (n【greater than or equal】4) is a Gauduchon manifold of constant curvature if and only if (M, g, D) is a Weyl conformally flat Einstein-Weyl manifold. Next, I classified a Gauduchon manifold of constant curvature with Killing vector field. By using this result, I obtained the following result : Let (M, [g], D) be a compact Einstein-Weil manifold and Weyl conformally flat for every g【reverse surface chemistry arrow】[g]. If dimension n of M n【greater than or equal】4 and ω≠0 for every g【reverse surface chemistry arrow】[g], then (M, [g], D) is a Weyl flat manifold. Next, I investigated a Weyl submanifold of a Gauduchon manifold. Let (M, g, D) be a compact Weyl totally umbilical submanifold of a Gauduchon flat manifold which tangent to the Killing vector field B and ω≠0. Then M is a totally geodesic submanifold with Einstein-Weyl structure and the universal covering manifold of M is isometric to the Riemannian product of the sphere and R. If (M, g, D) is a Weyl hypersurface of a Gauduchon flat manifold which orthogonal to the Killing vector field B, then M is Weyl totally umbilical and a totally geodesic submanifold, moreover M is an elliptic space form. Finally, I obtained that (M, [g], J, D) admits an Einstein-Weyl structure if and only if (M, g, J) admits an Einstein-Hermitian structure.

令 M 为具有共形结构 [g] 和无扭转仿射连接 D 的流形。我考虑具有高杜雄度量 g 的高杜雄流形 (M, g, D)。首先，我得到高杜雄流形（M，g，D）（n【大于或等于】4）是常曲率高杜雄流形当且仅当（M，g，D）是Weyl共形平坦的Einstein-Weyl流形。接下来，我对具有 Killing 矢量场的常曲率高杜雄流形进行了分类。通过使用这个结果，我得到了以下结果：设（M，[g]，D）是一个紧凑的爱因斯坦-韦尔流形，并且对于每个g【反向表面化学箭头】[g]，韦尔保角平坦。如果对于每个g【反表面化学箭头】[g]，M n【大于或等于】4的维数n【大于或等于】4且ω≠0，则（M，[g]，D）是Weyl平流形。接下来，我研究了 Gauduchon 流形的 Weyl 子流形。令 (M, g, D) 为高杜雄平坦流形的紧外尔全脐子流形，与 Killing 矢量场 B 相切且 ω≠0。则 M 是具有 Einstein-Weyl 结构的全测地子流形，M 的通用覆盖流形与球体和 R 的黎曼积等距。如果 (M, g, D) 是与 Killing 矢量场 B 正交的高杜雄平坦流形的 Weyl 超曲面，则 M 是全测地线子流形，而且 M 是椭圆空间形式。最后，我得到当且仅当（M，g，J）承认爱因斯坦-埃尔米特结构时（M，[g]，J，D）承认爱因斯坦-外尔结构。