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 <TEX>∇Xᵛ=μX</TEX> for any vector X tangent to M, where <TEX>∇</TEX> is the Levi-Civita connection and <TEX>μ</TEX> is a non-trivial function on M. A smooth vector field <TEX>ξ</TEX> on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: <TEX>1/2Lξg+Ric=λg</TEX>, where <TEX>Lξg</TEX> is the Lie-derivative of the metric tensor g with respect to <TEX>ξ</TEX>, Ric is the Ricci tensor of (M, g) and <TEX>λ</TEX> is a constant. A Ricci soliton (M, g, <TEX>ξ</TEX>, <TEX>λ</TEX>) 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.