A universal Riemannian foliated space

Abstract

It is proved that the isometry classes of pointed connected complete Riemannian n-manifolds form a Polish space, M⁎∞(n), with the topology described by the C∞ 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 M⁎,lnp∞(n)⊂M⁎∞(n), which becomes a C∞ foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspace M⁎,np∞(n)⊂M⁎,lnp∞(n) defined by the non-periodic manifolds. Moreover the leaves have a natural Riemannian structure so that M⁎,lnp∞(n) becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-determination. M⁎,lnp∞(n) is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces.