Critical Metrics Of A Class Of Functional

In recent years the critical metric of functional on the space of Riemannian metrics is one of the popular research fields in differential geometry.The main purpose of such issues is to find the best or more typical metrics by studying the critical point of the functional on the space of Riemannian metrics.Let M be a compact smooth n-manifold for n>3,and the space of all Rieman-nian metrics on M is denoted by M(M).In this paper,we consider a class of tensors defined byAg = Ricg-Sg/an+bg,(0.2)where Ricg,Sg represent the Ricci tensor and scalar curvature on M respectively.Ac-cording to the normalized L2-norm of the tensors Ag,we can get a class of functional A:M(M)? R on M(M).We calculate the first variation of the functional and ob-tain the Euler-Lagrange equation.Critical metrics of the functional A are the solutions to the Euler-Lagrange equation.And then we can find that Einstein metric is critical metric of the functional for a compact smooth manifold M of arbitrary dimension.And the other cases of critical metrics are discussed simply.