We analytically derive the covariant form of the Riemann (curvature) tensor for homogeneous metric-affine cosmologies. That is, we present, in a cosmological setting, the most general covariant form of the full Riemann tensor including also its non-Riemannian pieces which are associated to spacetime torsion and non-metricity. Having done so we also compute a list of the curvature tensor by-products such as Ricci tensor, homothetic curvature, Ricci scalar, Einstein tensor etc. Finally we derive the generalized metric-affine version of the usual Gauss–Bonnet density in this background and demonstrate how under certain circumstances the latter represents a total derivative term.

我们通过分析推导出齐次度量仿射宇宙学的黎曼（曲率）张量的协变形式。也就是说，我们在宇宙学环境中展示了完整黎曼张量的最一般协变形式，包括其与时空扭转和非度量相关的非黎曼部分。这样做后，我们还计算了曲率张量副产品的列表，例如 Ricci 张量、同位曲率、Ricci 标量、爱因斯坦张量等。 最后，我们在此背景下推导出通常的 Gauss-Bonnet 密度的广义度量仿射版本和演示在某些情况下后者如何代表总导数项。