Concircular Vector Fields and Pseudo-Kaehler Manifolds

Bang-Yen Chen

A vector field on a pseudo-Riemannian manifold $N$ is called concircular if it satisfies $\nabla_X v=µX$ for any vector $X$ tangent to $N$, where $\nabla$ is the Levi-Civita connection of $N$. A concircular vector field satisfying $\nabla_X v=µX$ is called a non-trivial concircular vector field if the function $µ$ is non-constant. A concircular vector field $v$ is called a concurrent vector field if the function $µ$ is a non-zero constant. In this article we prove that every pseudo-Kaehler manifold of complex dimension $>1$ does not admit a non-trivial concircular vector field. We also prove that this result is false whenever the pseudo-Kaehler manifold is of complex dimension one. In the last section we provide some remarks on pseudo-Kaehler manifolds which admit a concurrent vector field.