We consider the notion of cosmological symmetry, i.e. spatial homogeneity and isotropy, in the field of teleparallel gravity and geometry, and provide a complete classification of all homogeneous and isotropic teleparallel geometries. We explicitly construct these geometries by independently employing three different methods, and prove that all of them lead to the same class of geometries. Further, we derive their properties, such as the torsion tensor and its irreducible decomposition, as well as the transformation behavior under change of the time coordinate, and derive the most general cosmological field equations for a number of teleparallel gravity theories. In addition to homogeneity and isotropy, we extend the notion of cosmological symmetry to also include spatial reflections, and find that this further restricts the possible teleparallel geometries. This work answers an important question in teleparallel cosmology, in which so far only particular examples of cosmologically symmetric solutions had been known, but it was unknown whether further solutions can be constructed.

中文翻译：

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宇宙学远平行几何的完整分类

我们在远平行引力和几何领域考虑宇宙学对称性的概念，即空间均匀性和各向同性，并提供所有均匀和各向同性远平行几何的完整分类。我们通过独立地使用三种不同的方法明确地构建这些几何，并证明它们都导致同一类几何。进一步推导了它们的性质，如扭转张量及其不可约分解，以及时间坐标变化下的变换行为，并推导出了许多远平行引力理论中最普遍的宇宙场方程。除了同质性和各向同性之外，我们将宇宙对称性的概念扩展到还包括空间反射，并发现这进一步限制了可能的远平行几何形状。这项工作回答了遥平行宇宙学中的一个重要问题，到目前为止，只有宇宙学对称解的特定例子是已知的，但是否可以构建进一步的解是未知的。