A universal Riemannian foliated space
Résumé
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. Contents Key words and phrases. C ∞ convergence of Riemannian manifolds; locally non-periodic Riemannian manifolds; Riemannian foliated space. * ,c (n) (respectively, M ∞ * ,o (n)) be the F * (n)-saturated subspace of M * (n) consisting of classes [M, x] such that M is compact (respectively, open). Observe that, if [N, y] is close enough to any [M, x] ∈ M ∞ * ,c (n), then N is diffeomorphic to M. Thus M ∞ * ,c (n) is open in M * (n), and therefore M ∞ * ,o (n) is closed. Hence these are Polish subspaces of M * (n), as well as their intersections with any Polish subspace. The intersection of M ∞ * ,c/o (n) and M ∞ * ,(l)np (n) is denoted by M ∞ * ,(l)np,c/o (n). The restrictions of F * (n) to M * ,c/o (n) and M * ,(l)np,c/o (n) are denoted by F * ,c/o (n) and F * ,(l)np,c/o (n), respectively. The second main theorem of the paper is the following. Theorem 1.3. The following properties hold for n ≥ 2: (i) M * ,lnp (n) is Polish and dense in M ∞ * (n). (ii) M ∞ * ,lnp (n) ≡ (M ∞ * ,lnp (n), F * ,lnp (n)) is a foliated space of dimension n. (iii) F * ,lnp,o (n) is transitive. (iv) The foliated space M ∞ * ,lnp (n) has canonical C ∞ and Riemannian structures such that ῑ : Iso(M)\M → ι(M) is an isometry for every locally non-periodic, complete, connected Riemannian manifold M. (v) For any locally non-periodic complete connected Riemannian manifold M , the quotient map M → Iso(M)\M corresponds to the holonomy covering of the leaf ι(M) by ῑ : Iso(M)\M → ι(M). In particular, the set M * ,np (n) is the union of leaves of M ∞ * ,lnp (n) with trivial holonomy groups. The following result states a universal property of M ∞ * ,lnp (n), which involves certain property called covering-determination (Definition 12.1). Theorem 1.4. Let X be a sequential Riemannian foliated space of dimension n ≥ 2 whose leaves are complete. Then X is isometric to a saturated subspace of M ∞ * ,lnp (n) if and only if it is covering-determined.
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