arXiv subjects

Thomas Tony

Publications and source records attributed to Thomas Tony.

3 recordsLinked to original sources

arXiv2026

Scalar-rigid submersions are Riemannian products

Scalar-rigid maps are Riemannian submersions by works of Llarull, Goette--Semmelmann, and the second named author. In this article we show that they are essentially Riemannian products of the base manifold with a Ricci-flat fiber. As an application we obtain a Llarull-type theorem for non-zero degree maps onto products of manifolds of non-negative curvature operator and positive Ricci curvature with some enlargeable manifold. The proof is based on spin geometry for Dirac operators and an analysis connecting Clifford multiplication with the representation theory of the curvature operator.

math.DG↗

arXiv2024

Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index

Every closed connected Riemannian spin manifold of non-zero $\hat{A}$-genus or non-zero Hitchin invariant with non-negative scalar curvature admits a parallel spinor, in particular is Ricci-flat. In this note, we generalize this result to closed connected spin manifolds of non-vanishing Rosenberg index. This provides a criterion for the existence of a parallel spinor on a finite covering and yields that every closed connected Ricci-flat spin manifold of dimension $\geq 2$ with non-vanishing Rosenberg index has special holonomy.

math.DG↗

arXiv2024

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.

math.DG↗