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 Xᵛ=X for any vector X tangent to M, where is the Levi-Civita connection and is a non-trivial function on M. A smooth vector field on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: 12L_g+Ric=g, where L_g is the Lie-derivative of the metric tensor g with respect to, Ric is the Ricci tensor of (M, g) and is a constant. A Ricci soliton (M, g, , ) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field 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.