Geodesic
Concircular Vector Fields and Pseudo-Kaehler Manifolds
Kragujevac Journal of Mathematics, Tome 40 (2016) no. 1, p. 7 .

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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.
Classification : 53C55 53B30, 53B35
Keywords: Pseudo-Kaehler manifold, concircular vector field, concurrent vector field 
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Bang-Yen Chen. Concircular Vector Fields and Pseudo-Kaehler Manifolds. Kragujevac Journal of Mathematics, Tome 40 (2016) no. 1, p. 7 . http://geodesic.mathdoc.fr/item/KJM_2016_40_1_a0/