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