SOME RESULTS ON CONCIRCULAR VECTOR FIELDS AND THEIR APPLICATIONS TO RICCI SOLITONS

Abstract

Abstract. A vector field on a Riemannian manifold (M,g) is called con-circular if it satisfies ∇ X v = µX for any vector X tangent to M, where∇is the Levi-Civita connection and µ is a non-trivial function on M. Asmooth vector field ξ on a Riemannian manifold (M,g) is said to definea Ricci soliton if it satisfies the following Ricci soliton equation:12L ξ g + Ric = λg,where L ξ g is the Lie-derivative of the metric tensor g with respect to ξ,Ric is the Ricci tensor of (M,g) and λ is a constant. A Ricci soliton(M,g,ξ,λ) on a Riemannian manifold (M,g) is said to have concircularpotential field if its potential field ξ is a concircular vector field.In the first part of this paper we determine Riemannian manifoldswhich admit a concircular vector field. In the second part we classify Riccisolitons with concircular potential field. In the last part we prove someimportant properties of Ricci solitons on submanifolds of a Riemannianmanifold equipped with a concircular vector field. 1. IntroductionA. Fialkow introduced in [13] the notion of concircular vector fields on aRiemannian manifold M as vector fields which satisfy(1) ∇