关于共形向量场的Ricci平均值及应用

The Ricci Mean Value of Conformal Vector Fields and Application

共形向量场是微分几何中的一个重要组成部分,而Ricci平均值刻画了黎曼度量与向量场之间的关系,因此在n维紧定向黎曼流形上,研究了光滑向量场的Ricci平均值.首先运用活动标架法得到共形向量场的性质,然后结合ξ2的拉普拉斯算子得到判断黎曼流形上一个光滑向量场是共形向量场的必要条件,即若向量场ξ是共形向量场,则关于ξ的Ricci平均值δ(ξ)≥0.并且给出了δ(ξ)=0时黎曼流形的分类.

The Conformal vector field is an important part of differential geometry,the Ricci mean value describes the relationship between Riemannian measure and vector field,therefore the Ricci mean value of smooth vector fields on n-dimensional compact oriented Riemannian manifolds was studied.Firstly,the properties of conformal vector fields are obtained by using the method of moving frame,and then we get the necessary condition for a smooth vector field to be a conformal vector field on Riemannian manifold by combining the Laplacian of|ξ|2,that if a vector fieldξis a conformal vector field,then the Ricci mean ofξis greater than or equal to zero,and we give the classification of Riemannian manifolds when it equals to zero.