Riemannian foliations and quasifolds (Q6623334)

It is well known that, by the Reeb stability theorem, the leaf space of a Riemannian foliation with compact leaves is an orbifold. This paper aims to show that the leaf spaces of Killing Riemannian foliations are diffeological quasifolds. Some examples of Killing Riemannian foliations are\N\begin{itemize}\N\item[(1)] Riemannian foliations on simply-connected manifolds are Killing,\N\item[(2)] foliations of compact Riemannian manifolds whose leaves are orbits of connected Lie subgroups of isometries are Killing, and\N\item[(3)] every toric symplectic quasifold, which is equivariantly symplectomorphic to the leaf space of a Killing foliation [\textit{F. Battaglia} and \textit{D. Zaffran}, Springer INdAM Ser. 23, 1--21 (2017; Zbl 1406.14039)].\N\end{itemize}\NThe main result of the paper goes as follows (Theorem 6.2).\N\NTheorem. If \(\left( M,\mathcal{F}\right) \)\ is a complete Killing foliation of a connected manifold \(M\), with complete transverse action of its structure algebra, then \(M\mathcal{F}/\)\ is a diffeological quasifold.\N\NThe theorem is proved by showing that any complete Killing foliation is locally developable and then applying a description of leaf spaces of developable foliations worked out in the paper. To demonstrate local developability, the authors use \textit{P. Molino}'s [Riemannian foliations. With appendices by G. Cairns, Y. Carrière, E. Ghys, E. Salem, V. Sergiescu. Transl. from the French by Grant Cairns. Boston, MA etc.: Birkhäuser Verlag (1988; Zbl 0633.53001), \S 4] structure theory for Riemannian foliations and specifically \textit{E. Fedida}'s [C. R. Acad. Sci., Paris, Sér. A 272, 999--1001 (1971; Zbl 0218.57014)] theorem for compact Lie-\(\mathfrak{g}\) foliations.