Curvature properties of 3-\((\alpha ,\delta )\)-Sasaki manifolds

Abstract

We investigate curvature properties of 3-\((\alpha ,\delta )\)-Sasaki manifolds, a special class of almost 3-contact metric manifolds generalizing 3-Sasaki manifolds (corresponding to \(\alpha =\delta =1\)) that admit a canonical metric connection with skew torsion and define a Riemannian submersion over a quaternionic Kähler manifold with vanishing, positive or negative scalar curvature, according to \(\delta =0\), \(\alpha \delta >0\) or \(\alpha \delta <0\). We shall investigate both the Riemannian curvature and the curvature of the canonical connection, with particular focus on their curvature operators, regarded as symmetric endomorphisms of the space of 2-forms. We describe their spectrum, find distinguished eigenforms, and study the conditions of strongly definite curvature in the sense of Thorpe.