Scalar-rigid submersions are Riemannian products

Scalar-rigid maps are Riemannian submersions by works of Llarull, Goette--Semmelmann, and the second named author. In this article we show that they are essentially Riemannian products of the base manifold with a Ricci-flat fiber. As an application we obtain a Llarull-type theorem for non-zero degree maps onto products of manifolds of non-negative curvature operator and positive Ricci curvature with some enlargeable manifold. The proof is based on spin geometry for Dirac operators and an analysis connecting Clifford multiplication with the representation theory of the curvature operator.