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 $0< 2\text{Ric}_N<\text{scal}_N$ is area-rigid in the sense that any area non-increasing spin map $f\colon M\to N$ from a closed connected oriented Riemannian manifold $M$ with non-vanishing $\hat{A}$-degree and $\text{scal}_M\geq \text{scal}_N \circ f$ is a Riemannian submersion with $\text{scal}_M=\text{scal}_N \circ f$. 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 $f\colon M\to N$ by replacing the topological condition on the $\hat{A}$-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. $\text{pr}_1\colon S^{2n}\times T^k \to S^{2n}$. We develop a technique to extract from a non-vanishing higher index a geometrically useful family of almost $\mathcal{D}$-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.