Rigidity theorems for submetries in positive curvature

We derive general structure and rigidity theorems for submetries f:M→Xf:M→X, where M   is a complete Riemannian manifold with sectional curvature secM≥1. When applied to a non-trivial Riemannian submersion, it follows that diamX≤π/2. In case of equality, there is a Riemannian submersion S→MS→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.