Type: Article
Publication Date: 1967-07-01
Citations: 19
DOI: https://doi.org/10.2969/jmsj/01930328
This paper is a direct continuation of a previous one [7], which we shall refer to for terminologies and notations.In the previous paper, one of the present authors determined the structure of complete Riemannian manifolds admitting a concircular scalar field, and furthermore the structure of complete product Riemannian manifolds admitting a nonisometric conformal vector field under an assumption relative to dimension of manifolds.Here and hereafter we say a vector field to be isometric or conformal if it generates a one- parameter group of isometric or conformal transformations, respectively.A vector field is said to be complete if it generates a global one-parameter group of transformations.After preliminaries are stated in \S 1, we shall study in \S 2 the structure of manifolds, named pseudo-hyperbolic spaces in [7], in more details.In \S 3, the expression of a concircular scalar field in a space form will be obtained.In \S 4, we shall consider complete product Riemannian manifolds admitting a conformal vector field and obtain the equations satisfied by the associated scalar field in a simpler way than that in [7].As a consequence, the structure of manifolds having such properties is determined without assumption relative to dimension.The purpose of \S 5 and of the present paper is to prove the following MAIN THEOREM.If a complete reducible Riemannian manifold admits a complete nonisometric conformal vector field, then the manifold is locally Euclidean and the vector field is homothetic.This is a generalization of Tanaka's theorem [5] for manifolds with par- allel Ricci tensor and of Tachibana's [4] for compact manifolds.Our method of proof is elementary and different from theirs.\S 1. Preliminaries.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 $C^{\infty}$ .