Development of Hermitian Geometry on Complex Manifolds
Project/Area Number  07640149 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Ichinoseki National College of Technology 
Principal Investigator 
MATSUO Koji Ichinoseki National College of Technology, Faculty of General Education, Assistant Professor, 一般教科, 助教授 (80238972)

Project Period (FY) 
1995 – 1997

Project Status 
Completed(Fiscal Year 1997)

Budget Amount *help 
¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1997 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1996 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1995 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  Hermitian connection / Hermitianflatness / locally conformal Hermitianflatness / pscudocurvaturc tensors / LCK manifolds / pseudoBochner 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 / 半対称な接続 / 局所共形ケーラー多様体 / 局所共形エルミート平坦 / 正規概接触リーマン多様体 / リーマン直積 / 佐々木多様体 / 剣持多様体 
Research Abstract 
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 LeviCivita 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 Hermitianflatness as the analogy of the socalled 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 Bochnerflat Kahler hypersurfaces of Kahler manifolds, etc. Considering Hermitian analogy of the socalled differntial equation of Simons, which is an estimation of Laplacian for the length of the second fundamental form, is our subject in the future.

Report
(4results)
Research Output
(9results)