Development of Hermitian Geometry on Complex Manifolds

这项研究的目的是发展厄米特流形上的厄米特连通的微分宝石学。为此，我们首先考虑了具有Levi-Civita联系的几何中的各种结果的厄米相似，即黎曼几何，特别是作为厄米几何和雷曼几何的交集的Kahler几何。我们引入局部共形厄米平坦性作为黎曼几何中所谓的共形平坦度的类比，并构造了对应于Weyl共形曲率张量的张量。从Hermite几何的观点出发，给出了由S.Bochner在Kahler流形上引入的Bochner曲率张量的新的几何意义，作为Weyl共形曲率张量的形式类比.由于这些张量是共形不变的，我们认为这些张量在局部共形AKahler(LCK)几何中有重要作用。此外，在厄米子流形理论中，我们可以将LCK流形的复子流形(即LCK本身)作为具有对称第二基本形式的厄米特流形的余*子流形。我们得到了Chen和Okumura关于截面曲率收缩的Hermite定理，以及Yamaguchi和Sato关于Kahler流形的Bochner平坦Kahler超曲面的定理等。考虑到所谓的Simons微分方程式的埃尔米特类比，这是第二基本形式长度的拉普拉斯估计，是我们未来的主题。

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.