Some results on concircular vector fields and their applications to Ricci solitons
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1535-1547
https://doi.org/10.4134/BKMS.2015.52.5.1535
Printed September 30, 2015
Bang-Yen Chen
619 Red Cedar Road
Abstract : A vector field on a Riemannian manifold $(M,g)$ is called concircular if it satisfies $\nabla_X v=\mu X$ for any vector $X$ tangent to $M$, where $\nabla$ 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 {\it Ricci soliton} if it satisfies the following Ricci soliton equation: \begin{equation}\notag\frac{1}{2}{\mathcal L}_{\xi }g+Ric=\lambda g,\end{equation} where ${\mathcal L}_{\xi }g$ is the Lie-derivative of the metric tensor $g$ with respect to $\xi $, $Ric$ 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.
Keywords : concircular vector field, Ricci soliton, submanifolds, Einstein manifold, concircular potential field, concurrent vector field, concircular curvature tensor
MSC numbers : 53C25, 53C40
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