Transverse sphere theorems for Riemannian foliations

We extend the classical theory of sphere theorems to the transverse geometry of Riemannian foliations. In this setting, we establish transverse analogues of the Grove-Shiohama diameter sphere theorem and of the Berger-Klingenberg quarter-pinched sphere theorem. First, we prove that if a Killing foliation of a compact, connected manifold has transverse sectional curvature greater than 1 and transverse diameter greater than $\pi \mathrm{/}2$, then, after an arbitrarily small deformation, the resulting foliation has leaf space homeomorphic to a good spherical orbifold. Moreover, the space of leaf closures of the original foliation is realized as a further quotient of this spherical model by a torus action. Using this deformation theory we also prove that the space of leaf closures of a Killing foliation of a compact manifold is the Gromov-Hausdorff limit of a sequence of orbifolds. Under transverse quarter-pinching, we prove that this convergence is non-collapsing. As a consequence, we obtain that a complete Riemannian foliation with quarter-pinched transverse sectional curvature and codimension at least 3 develops on the universal cover to a simple foliation given by the fibers of a submersion onto the sphere.