Invariant measures on the space of all Riemannian manifolds

Ian Biringer (Boston College)

Abstract: We will discuss `unimodular' measures on the space of all pointed Riemannian manifolds (M,p). These measures can be described in different ways: through a conservation of mass formula, via transverse measures on foliated spaces, or as measures that (when lifted to the space of unit tangent bundles of Riemannian manifolds) are invariant under geodesic flow. Unimodular measures are useful as they serve as limiting objects for the distribution of geometries seen near randomly chosen base points in a very large closed Riemannian manifold M. Applications to sequences of closed Riemannian manifolds (in particular, certain locally symmetric spaces) will be given.