  Several problems in physics, in particular the averaging problem in gravity,
can be described in a formalism derived from the real-space Renormalization
Group (RG) methods. It is shown that the RG flow is provided by the
Ricci-Hamilton equations which are thereby provided with a {\it physical}
interpretation. The connection between a manifold deformation according to
these equations and Thurston's conjecture is exhibited. The significance of
criticality which naturally appears in this framework is briefly discussed.
This article summarizes also recent work with M. Carfora. Moreover, a report on
some work in progress is given and some open issues in the averaging problem
pointed out.
