On a metric symplectization of a contact metric manifold

In this article, we investigate metric structures on the symplectization of a contact metric manifold and prove that there is a unique metric structure, which we call the metric symplectization, for which each slice of the symplectization has a natural induced contact metric structure. We then study the curvature properties of this metric structure and use it to establish equivalent formulations of the $(\kappa ,\mu )$-nullity condition in terms of the metric symplectization. We also prove that isomorphisms of the metric symplectizations of $(\kappa ,\mu )$-manifolds determine $(\kappa ,\mu )$-manifolds up to D-homothetic transformations. These classification results show that the metric symplectization provides a unified framework to classify Sasakian manifolds, K-contact manifolds and $(\kappa ,\mu )$-manifolds in terms of their symplectizations.