Rigidity theorems for submetries in positive curvature

We derive general structure and rigidity theorems for submetries $f: M \to X$,where$M$isaRiemannianmanifoldwithsectionalcurvature$\sec M \ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diam X \leq \pi/2 $\mathrm{.}Incaseofequality,thereisaRiemanniansubmersion$$\mathbb{S}$ \to M$fromaunitsphere,andasaconsequence,$f$ is known up to metric congruence. A similar rigidity theorem also holds in the general context of Riemannian foliations.