A RIEMANNIAN GENERALIZATION OF A RESULT OF DEWITT CONCERNING RICCI-FLAT LORENTZ METRICS

For any given nonzero real number, alpha, and any curve from some open interval, I, into the space M, of Riemannian metrics on a compact man ifold M, a metric, (g) over tilde, on the manifold I X M can be constr ucted in a natural way the metric is Riemannian for alpha positive and Lorentzian for alpha negative. The aim of this paper is to show that (g) over tilde is an Einstein metric, with constant lambda, if and onl y if the curve of metrics is a solution of a certain Lagrangian on M d efined in terms of the DeWitt metric on M, the total scalar curvature functional and the values alpha and lambda. This result was obtained b y DeWitt [Phys. Rev. 160, 1113 (1967)], for the case where the dimensi on of M is three, alpha is negative and lambda = 0, in the context of a formulation of Einstein equations of evolution as a dynamical system . DeWitt metric is a member of a one parameter family of pseudometrics on M; it is shown here that, no other such a metric can be used to de scribe the relationship of the (n + 1) geometry of I X M with the dyna mics on M, so a characterization of the DeWitt metric is obtained. (C) 1996 American Institute of Physics.