Riemannian foliations and quasifolds

Mathematische Zeitschrift (2024) 308:49 https://doi.org/10.1007/s00209-024-03595-5 Mathematische Zeitschrift Riemannian foliations and quasifolds Yi Lin1 · David Miyamoto2 Received: 8 December 2023 / Accepted: 12 August 2024 © The Author(s) 2024 Abstract It is well known that, by the Reeb stability theorem, the leaf space of a Riemannian foliation with compact leaves is an orbifold. We prove that, under mild completeness conditions, the leaf space of a Killing Riemannian foliation is a diffeological quasifold: as a diffeological space, it is locally modelled by quotients of Cartesian space by countable groups acting affinely. Furthermore, we prove that the holonomy groupoid of the foliation is, locally, Morita equivalent to the action groupoid of a countable group acting affinely on Cartesian space. Keywords Riemannian foliation · Quasifolds · Molino theory · Diffeology Mathematics Subject Classification 57S25 · 57R91 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quasifolds as diffeological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Transverse geometric structures on foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Review of Riemannian foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The leaf space of a developable foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The leaf space of a Killing foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Diffeological quasifolds as leaf spaces of foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction A regular foliation of a manifold M is a partition F of M into submanifolds of a fixed dimension, called leaves, which fit together smoothly. To study the differential geometry of the leaf space M/F , an immediate obstacle is whereas M and F are smooth, the leaf space M/F is rarely a manifold. For instance, consider the foliation (M, F ) of an open Möbius strip, where M is formed by gluing the vertical edges of the square [−1, 1] × (−1, 1) with B David Miyamoto Yi Lin 1 Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA 2 Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany 0123456789().: V,-vol 123 49 Page 2 of 32 Y. Lin, D. Miyamoto a half twist, and where the leaves are the projections of horizontal lines. Then the leaf space is homeomorphic to (−1, 1)/Z2 , which has a singularity at the origin. Nevertheless, the leaf space of a foliation always admits the structure of a diffeology, a generalized smooth structure on a set introduced by Souriau [22]. Indeed, a diffeology propagates nicely to quotients of manifolds, and allows a clean and precise notion of “local model” for singular spaces. For example, this is the case for orbifolds, which are spaces that are locally modelled by quotients of Cartesian space Rn by smooth actions of finite groups. It is not difficult to show the leaf space of our foliation of the Möbius strip is diffeologically diffeomorphic to the orbifold (−1, 1)/Z2 . More generally, the Reeb stability theorem implies: Theorem 1.1 If (M, F ) is a foliation, and every leaf is compact with finite holonomy, then M/F is an orbifold. The requirement of finite holonomy is subtle, but automatic if F has compact leaves and is Riemannian, meaning M admits a Riemannian metric compatible with F . In this case: Corollary 1.2 The leaf space of a Riemannian foliation with compact leaves is an orbifold. The compactness requirement is necessary. As a counterexample, consider a Kronecker foliation induced by irrational flows on a 2-torus. This foliation is Riemannian, but its leaf space is non-Hausdorff and is not an orbifold. Nevertheless, the leaf space of a Kronecker foliation carries the structure of a quasifold, spaces introduced by Prato [18, 19] in order to generalize the Delzant construction in toric geometry to simple non-rational polytopes. Quasifolds are locally modelled by quotients of Cartesian spaces Rn by smooth affine actions of countable groups, and fit naturally in the category of diffeological spaces, cf. [11, 12]. In this article, we show that the leaf spaces of Killing Riemannian foliations are diffeological quasifolds. The definition of a Killing Riemannian foliation is rather complicated (see Sect. 4.1), so here we give three examples: Riemannian foliations on simply-connected manifolds are Killing; foliations of compact Riemannian manifolds whose leaves are orbits of connected Lie subgroups of isometries are Killing; and finally, Battaglia and Zaffran [3] show that every toric symplectic quasifold is equivariantly symplectomorphic to the leaf space of a Killing foliation.1 We prove: Theorem 1.3 If (M, F ) is a complete Killing foliation of a connected manifold M, with complete transverse action of its structure algebra, then M/F is a diffeological quasifold. Lie groupoids give an alternative approach to transverse geometry. Specifically, we have the holonomy groupoid Hol(F ) of a foliation (M, F ). For Killing foliations, we have: Corollary 1.4 In the setting of Theorem 1.3, Hol(F ) is, locally, Morita equivalent to the action groupoids of countable groups acting affinely on a Cartesian space. These are Theorem 6.2 and Corollary 6.3 in this article, respectively. To prove these results, we show that any complete Killing foliation is locally developable, and then apply a description of leaf spaces of developable foliations worked out in Propositions 5.3 and 5.7. To show local develop-ability, we use Molino’s structure theory for Riemannian foliations (see [16] and Sect. 4), and specifically Fedida’s Theorem 5.12 for complete Lie-g foliations. We emphasize that our local models are quotients of Cartesian spaces by countable groups 1 More precisely, they show the relevant foliation is Riemannian, but mention that a careful reading of their proof also shows it is Killing. 123 Riemannian foliations and quasifolds Page 3 of 32 49 acting affinely, and thus we consider our result a form of “affinization”. It is distinct from the usual linearization of proper Lie group actions (or proper Lie groupoids), which are known to be locally isomorphic to the action of the isotropy groups on the normal space to an orbit. For example, in Example 6.4, we apply Theorem 6.2 to conclude that the leaf space of a Kronecker foliation with angle λ is the quotient of R by the subgroup Z + λZ, whereas if we attempt to linearize its holonomy groupoid like a proper groupoid, we get a trivial groupoid. Haefliger [7] gave a model for complete pseudogroups of isometries which, when applied to the holonomy pseudogroup of a complete Killing Riemannian foliation, achieves a result of a (...truncated)