  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 $diam X
\leq \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.
