A universal Riemannian foliated space

TL;DR

This paper constructs a universal Riemannian foliated space that encapsulates the topology and geometry of pointed connected complete Riemannian manifolds, providing a framework for their classification and realization as dense leaves.

Contribution

It introduces a universal Riemannian foliated space with a canonical partition, characterizing manifolds via $C^ abla$ convergence and covering-continuity, and establishes its universality among such spaces.

Findings

The space $ ext{M}_*^ abla(n)$ is Polish with a topology from $C^ abla$ convergence.

The space admits a canonical partition into leaves, forming a Riemannian foliated space.

Complete connected Riemannian manifolds can be realized as dense leaves within this space.

Abstract

It is proved that the isometry classes of pointed connected complete Riemannian $n$-manifolds form a Polish space, $\mathcal{M}_*^\infty(n)$, with the topology described by the $C^\infty$ 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 $\mathcal{M}_{*,\text{lnp}}^\infty(n)\subset\mathcal{M}_*^\infty(n)$, which becomes a $C^\infty$ foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspace $\mathcal{M}_{*,\text{np}}^\infty(n)\subset\mathcal{M}_{*,\text{lnp}}^\infty(n)$ defined by the non-periodic manifolds. Moreover the leaves have a natural Riemannian structure so that $\mathcal{M}_{*,\text{lnp}}^\infty(n)$ becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-continuity. $\mathcal{M}_{*,\text{lnp}}^\infty(n)$ is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-continuous compact sequential Riemannian foliated spaces.