The aim of this series of papers is to generalise the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this first part, we propose a generalisation of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure of Duval et al. is extended to gravitational waves endowed with a lightlike hypersurface-orthogonal Killing vector field. This extension is shown to provide the natural geometric framework underlying the generalisation by Lichnerowicz of the Eisenhart lift. On the Carrollian side, a new class of gravitational waves - dubbed Dodgson waves - is introduced and geometrically characterised. Dodgson waves are shown to admit a lightlike foliation by Carrollian manifolds and furthermore to be the largest subclass of gravitational waves satisfying this property. This extended class allows to generalise the embedding procedure to a larger class of Carrollian manifolds that we explicitly identify. As an application of the general formalism, (Anti) de Sitter spacetime is shown to admit a lightlike foliation by codimension one (A)dS Carroll manifolds.

中文翻译：

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嵌入伽利略和卡罗尔几何。一、引力波

本系列论文的目的是概括 Duval 等人的环境方法。关于伽利略和卡罗尔几何嵌入具有平行射线的引力波。在第一部分中，我们提出了将无扭伽利略和卡罗尔流形嵌入更大类别的引力波中的概括。在伽利略方面，Duval 等人的商程序。被扩展到引力波，其具有类似光的超曲面正交杀伤向量场。该扩展被证明为 Lichnerowicz 对艾森哈特电梯的概括提供了自然的几何框架。在卡罗尔方面，引入了一种新的引力波——被称为道奇森波——并对其进行了几何表征。道奇森波被证明可以通过卡罗尔流形允许类似光的叶理，而且是满足这一特性的引力波的最大子类。这个扩展类允许将嵌入过程推广到我们明确识别的更大的卡罗尔流形类。作为一般形式主义的应用，(Anti) de Sitter 时空被证明允许通过 codimension one (A)dS Carroll 流形产生类似光的叶理。