Properties of Modified Riemannian Extensions

Let $M$ be an $n$-dimensional differentiable manifold with a symmetric connection $\nabla $ and $T^{\ast }M$ be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension $\widetilde{g}_{\nabla ,c}$ on $T^{\ast }M$ defined by means of a symmetric $(0,2)$-tensor field $c$ on $M.$ We get the conditions under which $T^{\ast }M $ endowed with the horizontal lift $^{H}J$ of an almost complex structure $J$ and with the metric $\widetilde{g}_{\nabla ,c}$ is a Kähler–Norden manifold. Also curvature properties of the Levi–Civita connection of the metric $\widetilde{g}_{\nabla ,c}$ are presented.