Concircular vector fields and pseudo-Kaehler manifolds

Abstract

A vector field on a pseudo-Riemannian manifold N is called concircular if it satisfies ΔXv = μX for any vector X tangent to N, where ∆ is the Levi-Civita connection of N. A concircular vector field satisfying ∆Xv = µX is called a nontrivial concircular vector field if the function µ is non-constant. A concircular vector field ν 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.