Rethinking the link between matter and geometry

In the present manuscript, I examine an intriguing relation at the classical level between general relativity and a theory where matter couples uniquely multiplicatively to geometry in the Lagrangian density. Interestingly, the gravitational constant $G$ is replaced by a novel fundamental constant, whose value is not tied to any classical phenomenon; while the value of $G$ itself becomes related to the dynamics of the universe. I concentrate on different aspects of the Equivalence Principle, as the theory is expected to violate all of its different formulations.