Development of Hermitian Geometry on Complex Manifolds

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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.