Llarull proved that the round n-sphere is extremal, meaning that one cannot simultaneously increase both its scalar curvature and its metric. Goette and Semmelmann generalized this result to spin maps f:M→N of nonzero Â-degree onto certain Riemannian manifolds with nonnegative curvature operator, showing that in the rigidity case, f is already a Riemannian submersion.
In this talk, I present a recent generalization of this rigidity theorem, where the original topological condition on the Â-degree is replaced by a weaker one involving the higher mapping degree. I will also discuss ongoing joint work with Oskar Riedler, in which we strengthen the rigidity statement by showing that the comparison map f: M → N is not only a Riemannian submersion but that M is locally a Riemannian product. The proof uses spinorial methods based on the Dirac operator.