Extend the classification beyond Levi-Civita geometry

Generalize the classification of FLRW-compatible tensor fields and the resulting perfect-fluid universality from Levi-Civita connections to spacetimes with non-metricity and non-vanishing torsion under appropriate compatibility conditions.

Background

The monomial decomposition and its perfect-fluid corollary are established specifically for FLRW spacetimes equipped with a Levi-Civita connection. The proof relies on properties of metric-compatible, torsion-free covariant differentiation and on the resulting behavior of the metric and foliation vector.

The authors identify a broader geometric direction: determining whether analogous tensor classifications and universality results survive when the connection has non-metricity or torsion. Such an extension would require specifying the compatibility conditions under which the symmetry-based construction remains valid.