Functional measures on the space of n-dimensional Riemannian geometries

By exploiting some recent results in global Riemannian geometry we construct families of probability measures on the path space associated with the set of n-dimensional (n≥3) Riemannian geometries. As an example of such construction we characterize a Gaussian stochastic process which yields a natural notion of Brownian motion on the set of Riemannian manifolds. An ultraviolet cutoff L parametrizes this class of measures. The limit L→0, as well as the probability of finding a random geometry in a given state, is discussed.