Scalar-rigid submersions are Riemannian products

TL;DR

The paper proves that scalar-rigid maps are essentially Riemannian products: if a spin map $f:M o N$ satisfies a lower scalar curvature bound, a metric domination, and a nonvanishing higher mapping degree (encoded via the Mishchenko–Fomenko bundle), then $M$ is locally isometric to a product $N imes F$ with $F$ Ricci-flat, and $f$ is the projection to the first factor. The argument fuses spin geometry and Dirac operators with higher index theory to obtain a Riemannian submersion, then uses the Clifford action of the curvature operator to force the O'Neill tensors $A$ and $T$ to vanish, yielding a local product structure. The work yields a Llarull-type rigidity result for maps to products $N imes F$ where $F$ is enlargeable or rationally essential, and clarifies the geometric rigidity landscape for scalar curvature comparison beyond constant-target rigidities. The methods highlight a deep link between curvature, spinorial indices, and submersion geometry, with potential impact on scalar curvature comparison and topological rigidity in higher dimensions.

Abstract

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.

Theorems & Definitions (100)

• Theorem 1
• Example 1.1: Local Riemannian product
• Remark 1.2: Existence of a non-trivial parallel spinor
• Remark 1.3: Non-negative curvature operator
• Corollary 2
• Example 3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Example 1.7: Index theoretic condition in \ref{['MainThm']}