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 Ric N < scal N is area-rigid in the sense that any area non-increasing spin map f : M → N with non-vanishing A ˆ -degree and scal M ≥ scal N ∘ f is a Riemannian submersion with scal M = scal N ∘ 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 : M → N by replacing the topological condition on the A ˆ -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. p r 1 : S 2 n × T k → S 2 n . We develop a technique to extract from a non-vanishing higher index a geometrically useful family of almost -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.