This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$ -principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

中文翻译：

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松散的恩格尔结构

本文对恩格尔结构及其分类的研究做出了贡献。主要结果引入了 Engel 结构松散族的概念，并表明两个这样的族是 Engel 同伦的当且仅当它们在形式上是同伦的。当辅助数据固定时，这意味着一个完整的 $h$ 原则。作为一个推论，我们证明洛伦兹和可定向嘉当延长被它们的形式数据分类为同伦。