A vector field on a pseudo-Riemannian manifold $N$ is called concircular if it satisfies $abla_X v=μX$ for any vector $X$ tangent to $N$, where $abla$ is the Levi-Civita connection of $N$. A concircular vector field satisfying $abla_X v=μX$ 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.
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机译：如果伪黎曼流形$ N $上与向量$ X $相切的任何向量$ X $都满足$ nabla_X v =μX$，则将其称为圆弧，其中$ nabla $是$的Levi-Civita连接N $。如果函数$μ$是非恒定的，则满足$ nabla_X v =μX$的圆形向量场称为非平凡的圆形向量场。如果函数$μ$是一个非零常数，则圆弧向量域$ v $被称为并发向量域。在本文中，我们证明了复维$> 1 $的每个伪Kaehler流形都不容许非平凡的圆周矢量场。我们还证明，每当伪Kaehler流形具有一维复数时，该结果都是错误的。在最后一节中，我们对准并行向量场的伪Kaehler流形提供了一些说明。
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