The projective transformation on a space with parallel Ricci tensor

Recently we have proved [2] that a complete connected Riemannian space M, 2<dimM", with parallel Ricci tensor does not admit a non-isometric conformal transformation, unless M is isometric either to the Euclidean space or to the sphere. An analogous fact is true for a protective transformation, as the following main theorem of this paper shows. THEOREM 1. Let g and g be two complete Riemannian metrics on a connected manifold M with dimension > 1 whose Ricci tensors R and R ! are parallel. If g and g are protectively related, then 1) Levi-Civita connections of g and g coincide, or 2) g and g are of positive constant curvature.