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 new 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 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-(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. In the second part we find 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 ∇-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated G2-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.

中文翻译：

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3-Sasakian 流形和偏扭的推广

摘要在第一部分中，我们定义和研究了近 3 接触度量流形的新类别，牢记两个指导思想：首先，哪些几何对象最适合捕获近 3 接触度量流形的关键属性，其次，新类应该承认与偏扭的“良好”度量连接。特别地，我们引入了 Reeb 交换子函数和 Reeb Killing 函数，我们定义了新的典型几乎 3-接触度量流形和 3-(α, δ)-Sasaki 流形（包括特殊情况下的 3-Sasaki 流形，四元数海森堡群等）并证明后者是超正态的，从而概括了柏田的开创性结果。我们研究了它们在一类新的变形下的行为，称为 𝓗-同位变形，并证明他们承认潜在的四元数接触结构，我们从中推导出 Ricci 曲率。例如，如果 α = δ（3-α-Sasaki 情况）或如果 δ = (2n + 3)α，其中dim M = 4n + 3，则 3-(α, δ)-Sasaki 流形是爱因斯坦流形。第二部分我们找到了这些适应的连接。我们从一个非常普遍的 φ 兼容连接的概念开始，其中 φ 表示几乎接触结构的相关球体的任何元素，并通过某个额外条件使它们唯一，从而产生规范连接的概念（它们完全存在于规范连接上）流形，因此得名）。对于 3-(α, δ)-Sasaki 流形，我们明确地计算了这个连接的扭转，我们证明了它是平行的，我们描述了完整度，∇-Ricci 曲率，我们证明了公制锥是一个 HKT-多方面的。在第 7 维中，