Generalizations of 3-Sasakian manifolds and skew torsion

Abstract

<jats:title>Abstract</jats:title> <jats:p>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 new classes should admit ‘good’ metric connections with skew torsion. In particular, we introduce the <jats:italic>Reeb commutator function</jats:italic> and the <jats:italic>Reeb Killing function</jats:italic>, we define the new classes of <jats:italic>canonical almost 3-contact metric manifolds</jats:italic> and of 3-(<jats:italic>α</jats:italic>, <jats:italic>δ</jats:italic>)-<jats:italic>Sasaki manifolds</jats:italic> (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 of Kashiwada. We study their behaviour under a new class of deformations, called 𝓗-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-(<jats:italic>α</jats:italic>, <jats:italic>δ</jats:italic>)-Sasaki manifold is Einstein either if <jats:italic>α</jats:italic> = <jats:italic>δ</jats:italic> (the 3-<jats:italic>α</jats:italic>-Sasaki case) or if <jats:italic>δ</jats:italic> = (2<jats:italic>n</jats:italic> + 3)<jats:italic>α</jats:italic>, where dim <jats:italic>M</jats:italic> = 4<jats:italic>n</jats:italic> + 3.</jats:p> <jats:p>In the second part we find these adapted connections. We start with a very general notion of <jats:italic>φ</jats:italic>-<jats:italic>compatible connections</jats:italic>, where <jats:italic>φ</jats:italic> 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 <jats:italic>canonical connection</jats:italic> (they exist exactly on canonical manifolds, hence the name). For 3-(<jats:italic>α</jats:italic>, <jats:italic>δ</jats:italic>)-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the ∇-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated <jats:italic>G</jats:italic><jats:sub>2</jats:sub>-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.</jats:p>

Correction to: Affine connections on 3-Sasakian homogeneous manifolds

“…On one hand, as already stated, any 3-Sasakian manifold (M 4n+3 , g) is Einstein with Ric g = 2(2n + 1)g. Therefore, by (5), the symmetrized of Ric ∇ is a multiple of the metric g if, and only if, the corresponding tensor S is so. On the other hand, the scalar curvature s ∇ is a constant for every affine connection in (39).…”

“…Hence, given ∇ an invariant Einstein with skew torsion connection on the homogeneous 3-Sasakian manifold M = G/H, there should exist λ ∈ R such that S = λg. Now, the identities (3) and (5) imply that the tensor S corresponding with such connection ∇ satisfies S = 4 2(2n + 1) − s ∇ 4n + 3 g.…”

“…The corresponding affine connection was generalized to arbitrary dimension in [5], where is called the canonical connection of the 3-Sasakian manifold (see Remark 5.17). We will denote it by ∇ c .…”

“…We study the symmetry of the Ricci tensors corresponding to our families of affine connections with skew-symmetric torsion. In fact, from (5), the Ricci tensor of ∇ is symmetric just when the divergence of the torsion tensor of ∇ vanishes.…”

“…Finally, in the study of the control system of a n-dimensional Riemannian manifold M rolling on the sphere S n , without twisting or slipping, and under certain assumption, the manifold M can be endowed with a 3-Sasakian structure, [17]. Several generalizations of 3-Sasakian structures have been recently studied, as the 3-(α, δ)-Sasaki manifolds, [5], which are 3-Sasakian manifolds when α = δ = 1, or the 3-quasi-Sasakian manifolds [14]. In general lines, they deal with geometric structures which are less rigid than the 3-Sasakian ones, which permits to clarify some geometrical properties of the 3-Sasakian manifolds.…”