Riemann Tensor and Gauss-Bonnet density in Metric-Affine Cosmology

TL;DR

This paper derives the most general covariant form of the Riemann tensor in homogeneous Metric-Affine Cosmology, including torsion and non-metricity, and extends the Gauss-Bonnet density to this framework.

Contribution

It provides the first analytical derivation of the covariant Riemann tensor and Gauss-Bonnet density in Metric-Affine Cosmology, incorporating non-Riemannian features.

Findings

Derived the covariant Riemann tensor including torsion and non-metricity.

Computed curvature tensor by-products such as Ricci tensor and scalar.

Extended Gauss-Bonnet density to Metric-Affine Cosmology, showing it can be a total derivative.

Abstract

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.