We show that a Riemannian foliation on a topological $n$-sphere has leaf dimension 1 or 3 unless n=15 and the Riemannian foliation is given by the fibers of a Riemannian submersion to an 8-dimensional sphere. This allows us to classify Riemannian foliations on round spheres up to metric congruence.
We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.
We prove that a contractible orbifold is a manifold.
We describe all affine maps from a Riemannian manifold to a metric space and all possible image spaces.
We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several
classes of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of singular Riemannian foliations.
We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.
We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.
