The Milnor Frame And Einstein Equation On A Kind Of Lie Group

Given a manifold M with a Riemannian structure gij.Riemannian curvature tensor Rijkl,so Ricci tensor can be defined by Rij=∑kRkikj,then M is an Einstein space if this equation Rij=λgij is satisfied for some scalar function λon M,in fact if we calculate the traces of the matrices on both sides,we getλ=R/n,with R=∑iRii is the scalar curvatue of M and n=dimM.It is a general problem for Riemannnian Geometry to determine all Einstein spaces.In order to simplify the problem,we consider the homogeneous space here.In this article,we prove that for H2n+1×Rm with a left-invariant lorentzian metrics.the equation Rij=λgij.There is no solution to this equation when n≥2,and when n=1,the space is Ricci flat,that is to say A=0 is the only solution.