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.

24 pages