Curvatures and Non-metricities in the Non-Relativistic Limit of Bosonic Supergravity

We construct a metric-like formulation of the non-relativistic (NR) limit of bosonic supergravity at the Lagrangian level. This formulation is particularly useful for decomposing relativistic tensors, such as powers of the Riemann tensor, in a manifest covariant form with respect to infinitesimal diffeomorphisms. The construction is purely geometrical and is based on a torsionless connection, mimicking the construction of the relativistic theory. The formulation contains non-vanishing non-metricities, which are associated with the gravitational fields of the theory ($τ_{μν}$, $h_{μν}$, $τ^{μν}$, $h^{μν}$). The non-metricities are fixed by requiring compatibility with the relativistic metric, before taking the NR expansion. We provide a fully covariant decomposition of the relativistic Riemann tensor, Ricci tensor, and scalar curvature. Our results establish an equivalence between the vielbein approach of string Newton--Cartan geometry at the level of the Lagrangian and the proposed construction. We also discuss potential applications, including a pure metric rewriting of the two-derivative finite bosonic supergravity Lagrangian under the NR limit, a powerful simplification in deriving NR bosonic $α'$-corrections and extensions to more general $f(R,Q)$ Newton--Cartan geometries.