一类具有特殊曲率性质的(α,β)-度量

A Class of(α,β)-Metric with Special Curvature Properties

在一般的(α,β)-度量F=αφ(s)与Riemann度量α的Ricci曲率之间的关系基础上,证明了一类特殊的(α,β)-度量F=α2/α+β,在维数n≥3的流形上,如果F具有迷向的Ricci曲率,且β是闭的1-形式,则其Ricci曲率等于零.从而得到如果F=αα+2β具有常Ricci曲率,并且β是闭的1-形式,则其Ricci曲率等于零.

We studied an important class of（α,β）-metric on the basis of the relationship of Ricci curvature between Gi and α Gi.We verified that if F=α2α＋β on an n dimension manifold M（n≥3） is of istropic Ricci curvature,i.e.Rmm=（n-1）c（x）F2,where c（x） is a scalar function on M and β is closed 1-form,then c（x）=0.Hence,we obtained that if F=α2/α＋β is of constant Ricci curvature and β is closed,then c（x）=0.