Model geometries in the space of Riemannian structures and Hamilton's flow

Hamilton's theorem states that, under suitable conditions, a given (three- or four-dimensional) Riemannian manifold can be smoothly deformed, via an heat-type equation, into a space of constant sectional curvature. This result is examined here in detail. On using some recently proven compactness properties of the space of Riemannian structures, we provide a natural setting for understanding the geometrical rationale behind Hamilton's results. This allows us to simplify the existing proof of the global nature of Hamilton's initial-value problem and to discuss it as a distinguished dynamical system on the space of Riemannian metrics. Finally, we briefly argue about the possible applications of these results either to general relativity or to the quantum field theory of extended objects. (author)