Some Notes on Metallic K\"{a}hler Manifolds

Abstract

The present paper deals with metallic K\"{a}hler manifolds. Firstly, we
define a tensor $H$ which can be written in terms of the $(0,4)-$Riemannian
curvature tensor and the fundamental $2-$form of a metallic K\"{a}hler
manifold and study its properties and some hybrid tensors. Secondly, we
obtain the conditions under which a metallic Hermitian manifold is conformal
to a metallic K\"{a}hler manifold. Thirdly, we prove that the conformal
recurrency of a metallic K\"{a}hler manifold implies its recurrency and also
obtain the Riemannian curvature tensor form of a conformally recurrent
metallic K\"{a}hler manifold with non-zero scalar curvature. Finally, we
present a result related to the notion of $Z$ recurrent form on a metallic K%
\"{a}hler manifold.