Development of Hermitian Geometry on Complex Manifolds

本研究的目的是发展厄米流形上具有厄米联络的微分方程。为此，我们从考虑Levi-Civita联络几何中各种结果的Hermitian类比，即黎曼几何，特别是Hermitian几何与Reimannian几何的交叉Kahler几何出发，引入局部共形Hermitian平坦性作为黎曼几何中所谓共形平坦性的类比，构造了与Weyl共形曲率张量相对应的张量。从Hermitian几何的观点出发，给出了Bochner在Kahler流形上引入的Bochner曲率张量作为Weyl共形曲率张量的形式类比的新的几何意义.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）几何中有重要作用的可能性，而且在Hermitian子流形理论中，我们可以把LCK流形的复子流形（也就是LCK本身）作为具有对称的Hermitian流形的第二基本形式的余子流形.我们得到了Chen和Okumura关于纯量曲率的Pinching定理即截面曲率的Pinching定理和Yamaguchi和Sato关于Kahler流形的Bochner平坦Kahler超曲面的定理等的Hermitian类比，考虑到所谓的Simons微分方程的Hermitian类比，它是Laplacian对第二基本形式长度的估计，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.