In this post, I shall discuss certain relations between Weyl-integrable geometry, General Relativity, and the scalar-tensor duality. The principal motivation is that the gauge group up-lift from the Weyl fiber bundle onto the renormalization Lie algebra can handle central infrared paradoxes that GR faces given that the scalar-tensor duality gives a description of the gravitational field in terms of a space-time metric and a scalar field with an analytically closed Einstein-Riemann frame that geometrically embeds standard inflationary cosmological models as well as relativistic quantum geometric ones. We start with the following action for a scalar-tensor theory of gravity in the Jordan vacuum-frame:

the Ricci scalar, a function of the scalar field and is a scalar potential. The action in terms of the redefined field reduces to:

Recalling that the Weyl-integrable spacetime action in this frame is given by:

with the Weylian connection and Riemannian curvature satisfying:

as well as the Palatini variational properties of and :

thus yielding the affine connection:

which is the characteristic non-metricity condition for a Weyl-integrable geometry

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