(Some) scalar curvature rigid submersions are Riemannian products

30.10.2025, 16:15  –  Building 9, Room 1.22, Golm
Forschungsseminar Differentialgeometrie

Oskar Riedler

A well known result of Sebastian Goette and Uwe Semmelmann states that a spin map \(M\to N\) to a space of non-negative curvature operator is a Riemannian submersion, provided the map satisfies certain Llarull type rigidity assumptions related to scalar curvature. However the only examples known are covered by Riemannian products \(\widetilde M=\widetilde N\times F\) for \(F\) a Ricci flat space.

In this talk I present joint work with Thomas Tony. We show that, provided \(N\) has no “degenerate sphere” factors, in above setting one can indeed conclude that \(M\) is a locally Riemannian product with a Ricci flat factor.

Furthermore we relax the index conditions by making use of the higher mapping degree, so that e.g. also the projection \(\pi_1:(S^n \times T^k, g^*) \to (S^n, g_{round})\) is covered by the theorem, and \(g^*=g_{round}+g_{flat}\).