On Certain Rigidity Results of Compact Regular $(\kappa, \mu) $-Manifolds

TL;DR

This paper explores the rigidity of certain geometric structures on compact regular $(ppa,ta)$-manifolds, establishing uniqueness and classification results for metrics and structures that preserve their canonical bi-Legendrian and bi-Lagrangian configurations.

Contribution

It provides new rigidity theorems for Riemannian and semi-Riemannian metrics on these manifolds, including uniqueness of Sasakian structures and classification of para-contact structures.

Findings

Uniqueness of Sasakian structure orthogonalizing the bi-Legendrian structure.

Explicit classification of para-contact structures in semi-Riemannian case.

Unified framework for analyzing rigidity in both Riemannian and semi-Riemannian settings.

Abstract

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 $(\kappa, \mu)$-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 $(\kappa, \mu)$-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 $(\kappa, \mu)$-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 $(\kappa, \mu)$-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.