Scalar-rigid submersions are Riemannian products

TL;DR

This paper proves that scalar-rigid Riemannian submersions are essentially products of the base with a Ricci-flat fiber, and applies this to a Llarull-type theorem involving non-zero degree maps onto certain product manifolds.

Contribution

It establishes that scalar-rigid submersions are Riemannian products with Ricci-flat fibers, extending the understanding of their geometric structure.

Findings

Scalar-rigid submersions are Riemannian products of base and Ricci-flat fiber.

A Llarull-type theorem is proved for maps onto products of manifolds with specific curvature conditions.

The proof uses spin geometry, Dirac operators, and Clifford multiplication analysis.

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.