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 K\"ahler 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.