This paper is about geometric and Riemannian properties of Engel structures, i.e. maximally non-integrable $2$-plane fields on $4$-manifolds. Two $1$-forms $\alpha$ and $\beta$ are called Engel defining forms if $\mathcal{D}=\ker\alpha\cap\ker\beta$ is an Engel structure and $\mathcal{E}=\ker\alpha$ is its associated even contact structure, i.e. $\mathcal{E}=[\mathcal{D},\mathcal{D}]$. A choice of Engel defining forms determines a distribution $\mathcal{R}$ transverse to $\mathcal{D}$ called the Reeb distribution. We study conditions that ensure integrability of $\mathcal{R}$. For example if we have a metric $g$ which makes the splitting $TM=\mathcal{D}\oplus\mathcal{R}$ orthogonal and such that $\mathcal{D}$ is totally geodesic then there exists an integrable Reeb distribution $\tilde{\mathcal{R}}$. It turns out that integrabilty of $\mathcal{R}$ is related to the existence of vector fields $Z$ whose flow preserves $\mathcal{D}$, so called Engel vector fields. A K-Engel structure is a triple $(\mathcal{D},\,g,\,Z)$ where $\mathcal{D}$ is an Engel structure, $g$ is a Riemannian metric, and $Z$ is a vector field which is Engel, Killing, and orthogonal to $\mathcal{E}$. In this case we can construct Engel defining forms with very nice properties and such that $\mathcal{R}$ is integrable. Moreover we can classify the topology of K-Engel manifolds studying the action of the flow of $Z$. As natural consequences of these methods we provide a construction which is the analogue of the Boothby-Wang construction in the contact setting and we give a notion of contact filling for an Engel structure.

中文翻译：

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恩格尔结构的黎曼性质

这篇论文是关于恩格尔结构的几何和黎曼性质，即最大不可积的$2$-平面场在$4$-流形上。如果 $\mathcal{D}=\ker\alpha\cap\ker\beta$ 是恩格尔结构且 $\mathcal{E}= \ker\alpha$ 是其关联偶数接触结构，即 $\mathcal{E}=[\mathcal{D},\mathcal{D}]$。恩格尔定义形式的选择决定了与 $\mathcal{D}$ 横向的分布 $\mathcal{R}$，称为 Reeb 分布。我们研究确保 $\mathcal{R}$ 可积性的条件。例如，如果我们有一个度量 $g$ 使分裂 $TM=\mathcal{D}\oplus\mathcal{R}$ 正交，并且 $\mathcal{D}$ 是完全测地线，那么存在可积 Reeb分布 $\tilde{\mathcal{R}}$。事实证明，$\mathcal{R}$ 的可积性与矢量场 $Z$ 的存在有关，该矢量场 $Z$ 的流动保留了 $\mathcal{D}$，即所谓的恩格尔矢量场。K-Engel 结构是三元组 $(\mathcal{D},\,g,\,Z)$，其中 $\mathcal{D}$ 是恩格尔结构，$g$ 是黎曼度量，$Z$是 Engel、Killing 和正交于 $\mathcal{E}$ 的向量场。在这种情况下，我们可以构造具有非常好的属性的恩格尔定义形式，并且 $\mathcal{R}$ 是可积的。此外，我们可以对研究 $Z$ 流的作用的 K-Engel 流形的拓扑结构进行分类。作为这些方法的自然结果，我们提供了一种构造，它类似于接触设置中的 Boothby-Wang 构造，并且我们给出了 Engel 结构的接触填充概念。