A recurrent curvature-like tensor on semi-definite Kähler manifolds (Q2770057)

\textit{S. Bochner} [Ann. Math. (2) 50, 77-93 (1949; Zbl 0039.17603)] introduced three curvature-like tensors on a Kähler manifold. The first tensor \(W\) is called the Weyl curvature tensor and the third tensor \(B\) is known as the Bochner curvature tensor. In this paper the authors study the second curvature-like tensor \(H\) on a semi-definite Kähler manifold \((M,J, g)\). We recall that a tensor \(T\) on \(M\) is recurrent if there exists and 1-form \(\alpha\) such that \(\nabla T=\alpha\otimes T\), where \(\nabla\) is the Levi-Civita connection on \(M\). In this case \(M\) is said to be \(T\)-recurrent. The main result of the paper is the following:NEWLINENEWLINENEWLINETheorem. Let \(M\) be an \(n\)-dimensional semi-definite Kähler manifold \((n\geq 2)\) with non-null second curvature like tensor \(H\). If it is \(H\)-recurrent, then it is \(R\)-recurrent, where \(R\) is the Riemannian curvature tensor on \(M\).NEWLINENEWLINENEWLINEAlso, they obtain some examples of \(H\)-recurrent indefinite Kähler manifolds.