Generalizations of 3-Sasakian manifolds and skew torsion

Abstract

In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the newly defined classes should admit 'good' metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-$(α,δ)$-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result by Kashiwada. We study their behaviour under a new class of deformations, called $\mathcal{H}$-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-$(α,δ)$-Sasaki manifold is Einstein either if $α=δ$ (the 3-$α$-Sasaki case) or if $δ=(2n+3)α$, where $\dim M=4n+3$. The second part is actually devoted to finding these adapted connections. We start with a very general notion of $φ$-compatible connections, where $φ$ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-$(α,δ)$-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the $\nabla$-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated $G_2$-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.