Development of Hermitian Geometry on Complex Manifolds

复流形上厄米几何的发展

• 批准号: 07640149
• 负责人: MATSUO Koji
• 金额: $ 1.34万
• 依托单位: Ichinoseki 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: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07640149/
• 关键词: Hermitian connection; Hermitian-flatness; locally conformal Hermitian-flatness; pscudo-curvaturc tensors; LCK manifolds; pseudo-Bochner curvature tensor; complex submanifolds; symmetric sccond fundamental form; 第2基本形式; 擬曲率テンソル; エルミート多様体; 第二基本形式

Purpose of this research was to develop differential gemetry with Hermitian connection on Hermitian manifolds. For this purpose, we started with considering Hermitian analogy of various results in the geometry with Levi-Civita connection, that is, Riemannian geometry and in particular, Kahler geometry which is the intersection of Hermitian geometry and Reimannian geometry.We introduced the local conformal Hermitian-flatness as the analogy of the so-called conformal flatness in Riemannian geometry and constructed the tensor corresponding to Weyl conformal curvature tensor. Also, from the viewpoint of Hermitian geometry we gave new geometric meaning of Bochner curvature tensor which was introduced by S.Bochner on a Kahler manifold as the formal analogy of Weyl conformal curvature tensor. Since these tensors are conformal invarinat, we think that there is a possibility that these have the important role in locally confromal aKahler (LCK) geometry.Moreover, in Hermitian submanifold theory, we can give complex submanifolds (which is LCK itself) of LCK manifolds as co***** submanifolds with symmetric second fundamental form of Hermitian manifolds. We obtained Hermitian anyogys of theorem of Chen and Okumura with respect to the pinching for scalar curvature which means the pinching for sectional curvature and theorem of Yamaguchi and Sato with respect to Bochner-flat Kahler hypersurfaces of Kahler manifolds, etc.Considering Hermitian analogy of the so-called differntial equation of Simons, which is an estimation of Laplacian for the length of the second fundamental form, is our subject in the future.

这项研究的目的是通过在Hermitian流形上与Hermitian的联系发展差异性宝石。为此，我们首先考虑了与Levi-Civita连接的几何结果的赫米尔人类比，即Riemannian几何形状，尤其是Kahler几何形状，Kahler几何形状是遗传几何的相交和reimannian几何形状的相交。我们引入了当地的相互构造的遗产和类比，是如此平坦的属性，是如此平坦的属性，既有般的属性，又是众所周知的，善良的是善良的善良的善良的善良，张量对应于Weyl保形曲率张量。同样，从遗传学几何形状的角度来看，我们给出了Bochner曲率张量的新几何含义，S.Bochner在Kahler歧管上引入了Kahler歧管，作为Weyl Sonformal曲率张量的形式类比。 Since these tensors are conformal invarinat, we think that there is a possibility that these have the important role in locally confromal aKahler (LCK) geometry.Moreover, in Hermitian submanifold theory, we can give complex submanifolds (which is LCK itself) of LCK manifolds as co***** submanifolds with symmetric second fundamental form of Hermitian manifolds.我们获得了有关标量曲率的捏合，获得了Chen和Okumura定理定理的Hermitian Anyogys，这意味着要捏住Yamaguchi和Sato的截面曲率和sato定理，而Bochner-flat kahler kahler hypersurface则是Kahler的高度表面，以相比的象征，以众所周知的归因于同样的象征。 Laplacian在第二种基本形式的长度上是我们将来的主题。

项目成果

Koji MATSUO: "On Local Conformal Hermitian‐Flatness of Hermitian Manifolds" Tokyo Journal of Mathematics.

Koji MATSUO：“论局部共形厄米特-厄米流形的平坦性”《东京数学杂志》。

Koji MATSUO: "Conformal invariant tensors on Hermitian manifolds" Bulletin of Korean Mathematical Society. Vol.33,No.3. 455-463 (1996)

Koji MATSUO：“埃尔米特流形上的共形不变张量”韩国数学会通报。

Koji MATSUO: "Examples of locally conformal Kahler strucutres" Note di Matematica. Vol.15,No.2. 147-152 (1998)

Koji MATSUO：“局部共形卡勒结构的示例”Note di Matematica。