Scalar curvature rigidity and the higher mapping degree

Abstract

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 $f:M\to N$ from a closed connected oriented Riemannian manifold $M$ with non-vanishing $\hat{A}$-degree and ${\text{scal}}_M\ge {\text{scal}}_N\circ f$ 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 $f:M\to N$ 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. ${\text{pr}}_1:S^{2n}\times T^k\to S^{2n}$. 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.

36 pages, 3 figures; v2: minor improvements; To appear in J. Funct. Anal