The classification of general affine connections in Newton--Cartan geometry: Towards metric-affine Newton--Cartan gravity

TL;DR

The paper addresses the problem of classifying affine connections on Galilei manifolds (the metric structure underlying Newton–Cartan gravity) in the metric-affine setting. It shows a bijection between affine connections and freely specifiable tensor data consisting of the torsion $T$, the Newton–Coriolis form $\Omega$, and the non-metricities $\hat{Q}=\nabla\tau$ and $Q=\nabla h$, subject to two compatibility identities, and provides an explicit formula for the connection coefficients $\Gamma^{\rho}_{\mu\nu}$. This generalizes the familiar metric-compatible (torsionful) Galilei connections and sets a rigorous framework for studying Newtonian limits of general metric-affine gravity, including a principal-bundle viewpoint and a pathway to coordinate-free post-Newtonian expansions. The results enable systematic construction of Newton–Cartan-like limits of metric-affine theories and offer geometric tools for exploring curvature, Bianchi identities, and Newtonian dynamics in a fully covariant setting.

Abstract

We give a full classification of general affine connections on Galilei manifolds in terms of independently specifiable tensor fields. This generalises the well-known case of (torsional) Galilei connections, i.e. connections compatible with the metric structure of the Galilei manifold. Similarly to the well-known pseudo-Riemannian case, the additional freedom for connections that are not metric-compatible lies in the covariant derivatives of the two tensors defining the metric structure (the clock form and the space metric), which however are not fully independent of each other.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• proof : Proof of \ref{['thm:conn_class']}
• Definition 8
• Proposition 9
• Theorem 10
• proof : Proof of \ref{['lem:nablav']}