2025/05/02 by Caramello, Francisco C., Neubauer, Francisco A.
#53C12 #57R30 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.01378
We prove two transverse analogs of classical theorems involving the rigidity of the sphere: the Grove--Shiohama diameter sphere theorem and the Berger--Klingenberg quarter-pinched sphere theorem. We obtain, more precisely, that if a Killing foliation of a compact, connected manifold has transverse sectional curvature greater than 1 and transverse diameter greater than π/2, then the space of its leaf closures is homeomorphic to the orbit space of a continuous action on a sphere. 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. This, in turn, enables us to prove that a transversely quarter-pinched, complete Riemannian foliation of a connected manifold, with codimension greater than or equal to 3, develops on the universal cover to a fibration over a sphere.