On a metric symplectization of a contact metric manifold

TL;DR

This paper introduces a unique metric structure on the symplectization of contact metric manifolds, linking curvature properties and manifold classifications within a unified framework.

Contribution

It establishes the existence and uniqueness of the metric symplectization and connects it to the classification of various contact metric manifolds.

Findings

Unique metric structure on symplectization proven

Curvature properties linked to $(ppa, ta)$-nullity condition

Isomorphisms classify manifolds up to D-homothetic transformations

Abstract

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.