We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M . Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C . With yet another structure on M , the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of ‘parabolic geometry’) and we provide examples when this geometry is ‘flat,’ meaning that its symmetries comprise the split form of the exceptional Lie algebra $${\mathfrak {g}}_2$$ g 2 .

中文翻译：

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飞碟的空气动力学

我们识别飞碟的配置空间 C 上的各种结构，在三维光滑流形 M 中移动。Always C 是一个五维接触流形。如果 M 具有射影结构，则 C 是它的扭曲空间，并配备了一个几乎接触的勒让德结构。相反，如果 M 具有共形结构，则飞碟根据 C 上的 CR 结构移动。对于 M 上的另一种结构，C 中的接触分布在扭曲立方体上配备了一个锥体。这定义了 C 上某种类型的 Cartan 几何（更具体地说，是一种“抛物线几何”），我们提供了当这种几何是“平坦的”时的示例，这意味着它的对称性包括异常李代数 $${ \mathfrak {g}}_2$$ g 2 。