CONTACT黎曼流形上S.TANNO联络的数量曲率

THE SCAIAR CURVATURE OF S. TANNO' CONNECTION ON CONTACT RIEMANNIAN MANIFOLDS

自从陈省身和 R.S.Hamilton(见[1])研究3-维紧致 Contact流形上的与 Webster 挠率相关的 Dirichlet energy 以来,对 Contact 流形上几何学的研究取得了很大的进展,获得不少新的成果。特别 N.Tanak 和S.Tanno 讨论了同一流形上 Contact 黎曼结构与 stongly Pseudo—Con-vex 可积 CR-结构的等价性,把 SPC 可积 CR-结构的某些已知结果自然地推广到 Contact 黎曼流形上,丰富了 Contact 黎曼流形的几何内容。为了在Contact 流形上给出相关于 Contact 形式的标准黎曼度量,S.Tanno 讨论了 Contact 流形上的变分问题,并在 Contact 黎曼流形上定义了联络,它是非退化可积 CR-流形上标准仿射联络的自然推广。本文在 Contact 黎曼流形上讨论了关于联络的截面曲率及相关的几个等价条件,并在此基础上给出了联络的曲率张量与数量曲率公式。

Since S.S.chern and R.S.Hamilton have done the research about a critical metric of the Dirichlet energy Concerning the Webster torsion tensor on three-dimensional Contact manifolds. A lot of new geometric results have been given on Contact ma- nifolds. The Canonical affine Connection on a nondegenrate integr- able CR-manifold Was generalized to the Contact Riemannian manifolds by S.TANND,that richen the geometric Content of Contact manifolds One of the important problems in the study of Contact man- ifolds is to find differential geometric propertieswhich are ind- ependent of the choice of Contact forms. The purpose of this paper is,based on S.TANNO' Connect- ion on Contact Riemannian manifolds,to prove four equivatent Condition Concerning the sectional curvature,and give the for- mulas of the Curvature tensor and sclalar curvature.