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

A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies del xv = mu X for any vector X tangent to M, where del is the Levi-Civita connection and mu is a non-trivial function on M. A smooth vector field xi on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: 1/2L(xi)g + Ric = lambda g, where L(xi)g is the Lie-derivative of the metric tensor g with respect to Ric is the Ricci tensor of (M, g) and lambda is a constant. A Ricci soliton (M, g, xi, lambda) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field xi is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.