A universal Riemannian foliated space

摘要

It is proved that the isometry classes of pointed connected complete Riemannian n-manifolds form a Polish space, M-*(infinity)(n), with the topology described by the C-infinity 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(infinity)(n) subset of M-*(infinity)(n), which becomes a C-infinity foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspace M-*,np(infinity)(n) subset of M-*,lnp(infinity)(n) defined by the non-periodic manifolds. Moreover the leaves have a natural Riemannian structure so that M-*,lnp(infinity)(n) becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-determination. M-*,lnp(infinity)(n) is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces. (C) 2015 Elsevier B.V. All rights reserved.