Scientists have proposed a new formulation of gravity based on an invariant Weyl-integrable space-time (IWIST) geometry. In this framework, the background geometry is not imposed a priori but is dynamically determined through the Palatini variational principle. The key is a compatibility condition that defines a Weyl-integrable geometry, preserved by a local Weyl transformation of the metric and a geometric scalar field. This approach seeks to offer an alternative perspective on the description of gravity, integrating Weyl symmetry in a fundamental way.

The theory is constructed from an action invariant under both diffeomorphisms and Weyl transformations, using a Weyl-covariant variational procedure based on an invariant extension of the divergence theorem. This allows for the derivation of field equations for the metric, the Weyl scalar, and the Weyl gauge field. Subsequently, the theory is reformulated in the Einstein-Riemann frame, where the effective metric is Riemannian and the scalar field is reinterpreted as a physical degree of freedom of geometric origin, suggesting a deep connection between geometry and fundamental physics.

A novel aspect of this proposal is a geometric mechanism for the explicit breaking of Weyl symmetry. This is achieved by introducing a coupling between the Weyl gauge field and a current constructed from the scalar sector. Although the resulting theory maintains diffeomorphism covariance, it departs from local Weyl invariance. The Einstein-Riemann representation of this broken-symmetry theory reveals an additional generally covariant interaction, implying that the Weyl-symmetric model is a particular limit of a more general gravitational theory, opening avenues for exploring new fundamental interactions.