Sannidhi Alape | Searcharxiv

arXiv2024

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 $(κ, μ)$-nullity condition in terms of the metric symplectization. We also prove that isomorphisms of the metric symplectizations of $(κ, μ)$-manifolds determine $(κ, μ)$-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 $(κ, μ)$-manifolds in terms of their symplectizations.

math.DG↗

arXiv2023

On Certain Rigidity Results of Compact Regular $(κ, μ) $-Manifolds

In this article, we investigate the Riemannian and semi-Riemannian metrics on the base space of the Boothby-Wang fibration of a closed regular non-Sasakian $(κ, μ)$-manifold. To this end, we study a natural class of deviations of the projection map from being (semi-)Riemannian submersions. We consider deviations that preserve the canonical bi-Legendrian structure on the given $(κ, μ)$-manifold. We present rigidity results for Riemannian and semi-Riemannian metrics on the base space which orthogonalize the natural bi-Lagrangian structure induced by the $(κ, μ)$-structure. This approach gives a unified framework to analyze rigidity results in both categories. More precisely, in the Riemannian category, we obtain uniqueness of Sasakian structure on the given $(κ, μ)$-manifold which orthogonalizes the canonical bi-Legendrian structure. In the semi-Riemannian category, we obtain an explicit description of the finitely many para-contact structures which orthogonalize the canonical bi-Legendrian structure.

math.DG↗