Complete Riemannian manifolds and some vector fields

Abstract

Introduction and theorems. In this paper we shall always deal with connected Riemannian manifolds with positive definite metric, and suppose that manifolds and quantities are differentiable of class C'. Let M be an n-dimensional Riemannian manifold with metric tensor field (1) g,,. We call a nonconstant scalar field p in M a concircular scalar field, or simply a concircularfield, if it satisfies the equation