  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.
