Invent. math. 65, 179-207 (1981) [~l ve~ tiovle$ mathematicae 9 Springer-Verlag 1981 Existence of Metrics With Prescribed Ricci Curvature: Local Theory Dennis M. DeTurck* Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, N.Y. 10012, USA Introduction and Statement of Results In this paper we examine the following question: Given a symmetric tensor Rij defined on a manifold of dimension n > 3, can a metric g be found so that, in some neighborhood of a given point p, R is the Ricci curvature tensor of g? The definition of Ricci curvature transforms this fundamental question into the problem of finding a solution of the following nonlinear system of second-or- der partial differential equations: Ricc(g)_t?F~ OF~] t_F~Ft t , ,_ (o.1) = - Fit Fsj - Rij Ox ~ •x j for a given Rij, where giS are the Christoffel symbols of the metric g. We systematically write the above system as Ricc(g)=R. We prove the following results: Theorem A. If Ri~ is a C m+` (resp. C ~~ analytic) tensor field (m>2) in a neigh- borhood of a point p on a manifold of dimension n> 3, and if R-l(p) exists, then there is a C m+` (C a, analytic) Riemannian metric g such that Ricc(g)=R in a neighborhood of p. Theorem A% If Rii is analytic and if R-l(p) exists, then there exists an analytic metric g of any desired signature such that Ricc(g)= R near p. These theorems were announced in [D1], and the proof of Theorem A was sketched in [D2]. The plan of this paper is as follows. In Sect. 1, we lay some groundwork to motivate the direction the proof will finally takel defining several notations * The research for this paper was supported in part by the National Science Foundation