Rigidity theorems for submetries in positive curvature

Abstract

We derive general structure and rigidity theorems for submetries $f:M\to X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M\ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diamX\le \pi /2$. In case of equality, there is a Riemannian submersion $\mathbb{S}\to M$ from a unit sphere, and as a consequence, $f$ is known up to metric congruence. A similar rigidity theorem also holds in the general context of Riemannian foliations.

References in corpus (1)

• On the Structure of Equidistant Foliations of Euclidean Space