Researchers have proposed an explicitly conformally invariant, off-shell definition of the Weyl tensor within the framework of Galilean geometry. This development is significant because the Weyl tensor, fundamental in general relativity for describing tidal forces and the curvature of matter-free spacetime, lacked a robust analogous formulation in Galilean theories. The new definition allows for the analysis of curvature properties in a non-relativistic framework with conformal symmetry, opening new avenues for understanding the geometric structures of these theories.

In addition to defining the Weyl tensor, the study introduces its associated electric and magnetic parts in the Galilean context. The vanishing of the magnetic part necessitates the existence of observers with specific kinematical properties. While this condition is automatically satisfied by the Newton-Cartan equation, it may not hold true for other Galilean invariant theories. Therefore, the authors propose that imposing the existence of such observers, for whom the off-shell magnetic part is zero, should be a necessary condition for a Galilean invariant theory to be termed 'Newtonian,' in the spirit of the 'Newtonian' condition introduced by Trautman in standard Newton-Cartan gravity.

As a side result, the work also demonstrates the existence of a unique Galilean boost-invariant connection that can be constructed from a Galilean structure and a choice of Coriolis field, even when the clock form is not closed. This finding is notable because it does not require the introduction of extra structure, such as a mass gauge field, which is the standard approach for constructing boost-invariant connections. This simplifies the formulation and could have implications for the construction of gravity theories in the non-relativistic limit.