Conformal connections and conformal transformations

Abstract

Introduction. In the previous paper [5], we have studied the groups of projective transformations of affinely connected manifolds by the application of the theory of normal projective connections. The main purpose of the present paper is to study the conformal properties of complete Riemannian spaces of some special type by the application of the theory of normal conformal connections. We shall introduce a family of Riemannian spaces, the elements of which are characterized by conditions on the Ricci tensor fields and will be called of type e (Definition 4). Einstein spaces are of type e and, in general, a Riemannian space of type e is locally isomorphic with the direct product of two Einstein spaces of different signs. Our main result (Theorem 1) states that two conformally equivalent Riemannian metrics on a manifold which are complete and of type e necessarily coincide, except the case where either of the two metrics is an Einstein metric with the vanishing or negative Ricci tensor field. Some results are also obtained in the exceptional case. We see from this theorem that the group of all conformal transformations of a complete Riemannian space of type e5 coincides with the group of all isometries, except the case of an Einstein space with the vanishing or negative Ricci tensor field (Theorem 2).