We classify complex surfaces \((M,\,J)\) admitting Engel structures \({\mathcal {D}}\) which are complex line bundles. Namely, we prove that this happens if and only if \((M,\,J)\) has trivial Chern classes. We construct examples of such Engel structures by adapting a construction due to Geiges [7]. We also study associated Engel defining forms and define a unique splitting of TM associated with \({\mathcal {D}}\) J-Engel.

中文翻译：

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复杂表面上的恩格尔结构

我们对复杂曲面\（（（M，\，J）\）进行分类，以允许使用Engel结构\ {{\ mathcal {D}} \}，它们是复杂线束。即，我们证明只有当\（（M，\，J）\）具有平凡的Chern类时，才会发生这种情况。我们通过改编归因于Geiges的构造来构造此类Engel结构的示例[7]。我们还研究了相关的恩格尔定义形式，并定义了与\（{\ mathcal {D}} \）J -Engel相关的TM的唯一拆分。