Scalar curvature rigidity and the higher mapping degree

A closed connected oriented Riemannian manifold N with non-vanishing Euler characteristic, non-negative curvature operator and 0<2RicN<scalN is area-rigid in the sense that any area non-increasing spin map f M→ N from a closed connected oriented Riemannian manifold M with non-vanishing Â-degree and scalM≥ scalN∘ f is a Riemannian submersion. This is due to Goette and Semmelmann and generalizes a result by Llarull. In this article, we show area-rigidity for not necessarily orientable manifolds with respect to a larger class of maps f M→ N by replacing the topological condition on the Â-degree by a less restrictive condition involving the so-called higher mapping degree. This includes fiber bundles over even dimensional spheres with enlargeable fibers, e.g. pr₁ S²ⁿ× Tᵏ → S²ⁿ. We develop a technique to extract from a non-vanishing higher index a geometrically useful family of almost D-harmonic sections. This also leads to a new proof of the fact that any closed connected spin manifold with non-negative scalar curvature and non-trivial Rosenberg index is Ricci flat.