A universal Riemannian foliated space (Q896820)

The authors prove that the isometry classes of pointed connected complete Riemannian \(n\)-manifolds form a Polish space with the topology defined by the \(C^\infty\)-convergence of manifolds: \(\mathcal{M}^\infty_\ast(n)\). This space has a canonical partition into sets defined by varying the distinguished point in each manifold. The set of locally non-periodic manifolds defines an open dense subspace \(\mathcal{M}^\infty_{\ast,\mathrm{lnp}}(n)\) of \(\mathcal{M}^\infty_\ast(n)\). This subspace becomes a \(C^\infty\)-foliated space with the restriction of the canonical partition; its leaves have a natural Riemannian structure. The authors prove that \(\mathcal{M}^\infty_{\ast,\mathrm{lnp}}(n)\) is a Riemannian foliated space universal among all sequential Riemannian foliated spaces verifying a particular property, and they use this fact to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces.