A new study has explored extensions of Cartan's geometric framework, demonstrating how torsion and non-metricity, two fundamental properties of spacetime, can be naturally described within this formulation. The work builds on Élie Cartan's idea that the metric and intrinsic curvature of a manifold can be determined by rolling a surface of the same dimensionality over it without slipping or twisting. This concept underpins the MacDowell-Mansouri formulation of General Relativity, which describes gravity as a gauge theory.

The researchers considered scenarios where the rolling surface can twist or even evolve in shape as it moves. They found that the act of twisting during rolling directly corresponds to the presence of torsion in spacetime geometry, while the evolution of the surface's shape relates to non-metricity. These findings provide a unified perspective for understanding these geometric properties, which are crucial for describing gravitational interaction beyond standard General Relativity.

The study also discusses Cartan-geometric formulations of teleparallel gravity and symmetric teleparallelism, alternative theories to General Relativity that explain gravity through torsion or non-metricity instead of curvature. Furthermore, other manifestations of non-metricity in first-order formulations of gravity are explored. This geometric approach offers a powerful tool for investigating and comparing different theories of gravity, paving the way for a deeper understanding of the fundamental structure of spacetime.