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 (${\tau }_{\mu \nu }$, $h_{\mu \nu }$, ${\tau }^{\mu \nu }$, $h^{\mu \nu }$). 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 ${\alpha }^{\mathrm{\prime }}$-corrections and extensions to more general $f(R,Q)$ Newton–Cartan geometries.