DEMONSTRATIO MATHEMATICA Vol. XXVII No 2 1994 Formella GEODESIC MAPPINGS OF PSEUDO-RIEMANNIAN MANIFOLDS l. Introduction Let (M, g) and (M, g) be two pseudo-Riemannian manifolds of class Coo. A mapping 'Y : (M,g) (M,g) is said to be geodesic if it maps geodesics of (M,g) onto geodesics of (M,g), Le. if it preserves the geodesics. In the case under consideration the metrics g and g are said to be geodesically corresponding. Assume that two metrics g and g are given on M. Let denote a ring of differentiable functions and let .r(M) denote of differentiable vector fields on M. Each one of the below given conditions is necessary and sufficient so that the manifold (M, g) admits a geodesic mapping onto (M,g): (1) (2) (3) Vx Y = Vx Y + (X1/J)Y + (Y1/J)X, (V Xg)(Y, Z) = 2(X 1/J)g(Y, Z) + (Y1/J)g(X, Z) + (Z1/J)g(X, Y), (V xa)(Y, Z) = (Y<p)g(X, Z) + (Z<p)g(X, Y), for all X, Y, Z E .r(M), where <p,1/J E V and V are the Levi-Civita connections with respect to g and g and a is a symmetric nonsingular bilinear form on M. In a chart (U, x) on M the local components of g, g, a, X <p and X 1/J given by satisfy (4) (5) gij = g(Xi,Xj ), gij = g(Xi,Xj ), aij = a(Xi,Xj) , <P,i = Xi<P, 1/J,i = Xi1/J aij = exp(21/J)9stgsigtj, <P,i = - exp(21/J)gstgsi 1/J,t, where Xi = E .r(U) and gij are components of (gij)-l.