We introduce the notion of $\varepsilon\eta\,$-Einstein $\varepsilon\,$-contact metric three-manifold, which includes as particular cases $\eta\,$-Einstein Riemannian and Lorentzian (para) contact metric three-manifolds, but which in addition allows for the Reeb vector field to be null. We prove that the product of an $\varepsilon\eta\,$-Einstein Lorentzian $\varepsilon\,$-contact metric three-manifold with an $\varepsilon\eta\,$-Einstein Riemannian contact metric three-manifold carries a bi-parametric family of Ricci-flat Lorentzian metric-compatible connections with isotropic, totally skew-symmetric, closed and co-closed torsion, which in turn yields a bi-parametric family of solutions of six-dimensional minimal supergravity coupled to a tensor multiplet. This result allows for the systematic construction of families of Lorentzian solutions of six-dimensional supergravity from pairs of $\varepsilon\eta\,$-Einstein contact metric three-manifolds. We classify all left-invariant $\varepsilon\eta\,$-Einstein structures on simply connected Lie groups, paying special attention to the case in which the Reeb vector field is null. In particular, we show that the Sasaki and K-contact notions extend to $\varepsilon\,$-contact structures with null Reeb vector field but are however not equivalent conditions, in contrast to the situation occurring when the Reeb vector field is not light-like. Furthermore, we pose the Cauchy initial-value problem of an $\varepsilon\,$-contact $\varepsilon\eta\,$-Einstein structure, briefly studying the associated constraint equations in a particularly simple decoupling limit. Altogether, we use these results to obtain novel families of six-dimensional supergravity solutions, some of which can be interpreted as continuous deformations of the maximally supersymmetric solution on $\widetilde{\mathrm{Sl}}(2,\mathbb{R})\times S^3$.

中文翻译：

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接触度量三流形和洛伦兹几何与六维超重力中的扭转

我们引入了$\varepsilon\eta\,$-Einstein $\varepsilon\,$-contact metric 三流形的概念，其中包括作为特殊情况的$\eta\,$-Einstein Riemannian 和 Lorentzian (para) contact metric 三流形-流形，但另外允许 Reeb 向量场为空。我们证明了 $\varepsilon\eta\,$-Einstein Lorentzian $\varepsilon\,$-contact 三流形与 $\varepsilon\eta\,$-Einstein Riemannian 三流形的乘积带有具有各向同性、完全斜对称、闭合和共闭合扭转的 Ricci-flat Lorentzian 度量兼容连接的双参数族，这反过来产生耦合到张量多重态的六维最小超重力的双参数族解. 该结果允许从 $\varepsilon\eta\,$-Einstein 接触度量三流形对中系统地构建六维超引力的洛伦兹解族。我们将所有左不变的 $\varepsilon\eta\,$-Einstein 结构分类到单连通李群上，特别注意 Reeb 向量场为空的情况。特别是，我们表明 Sasaki 和 K-contact 概念扩展到 $\varepsilon\,$-contact 结构，具有空 Reeb 矢量场，但不是等效条件，与 Reeb 矢量场不轻时发生的情况相反-喜欢。此外，我们提出了 $\varepsilon\,$-contact $\varepsilon\eta\,$-Einstein 结构的柯西初值问题，在一个特别简单的解耦极限中简要研究了相关的约束方程。