The canonical submersion of $\mathcal{S}$-manifolds and transverse K\"ahler-Einstein structures

Abstract

This paper is devoted to the study of the holonomy properties of $(2n+s)$-dimensional $\mathcal{S}$-manifolds equipped with their characteristic connection. These structures generalize Sasakian geometry to higher CR-codimensions and, when viewed as geometries with parallel skew-torsion, share many holonomy features with the Sasakian case. We show that $\mathcal{S}$-manifolds of arbitrary CR-codimension $s\geq 1$ provide examples of geometries with parallel skew-torsion whose holonomy is reducible, indecomposable, and of special type. We also deduce that any $\mathcal{S}$-manifold admits a locally defined Riemannian submersion over a K\"ahler manifold. We describe the corresponding curvature relations and establish a bijective correspondence between the K\"ahler-Einstein condition on the base space and a generalized $\eta$-Einstein condition on the total space. As every $\mathcal{S}$-manifold comes with a characteristic foliation whose transverse geometry is K\"ahler, it is natural to relate the $\eta$-Einstein condition to the transverse metric, leading to a bijection between $\eta$-Einstein $\mathcal{S}$-manifolds and transverse K\"ahler-Einstein metrics, extending the classical Sasakian correspondence to arbitrary CR-codimensions. As an application to Ricci-flat metric connections with parallel skew-torsion, we prove that an $\mathcal{S}$-manifold is ${\rm Ric}^{\nabla}$-flat if and only if it is transverse K\"ahler-Einstein with Einstein constant $\lambda=4s$. An illustration of this characterization is presented by a Sasakian example.