A universal Riemannian foliated space

Abstract

It is proved that the isometry classes of pointed connected complete Riemannian n-manifolds form a Polish space X with the topology described by the smooth convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifold. The locally non-periodic manifolds define an open dense subspace Y, which becomes a smooth foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspace Z defined by the non-periodic manifolds. Moreover, the leaves have a natural Riemannian structure so that Y becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-determination. Y is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces.