The Gauss-Bonnet inequality beyond aspherical conjecture

TL;DR

This paper proves a Gauss-Bonnet inequality for certain closed Riemannian manifolds with nonnegative scalar curvature, characterizing when they are flat or split as a product involving a 2-sphere, extending previous results beyond aspherical cases.

Contribution

It extends the Gauss-Bonnet inequality to higher dimensions under specific topological conditions and establishes a splitting result when equality holds, unifying earlier work on homotopical systoles.

Findings

Manifolds are either flat or have Gauss-Bonnet quantity ≤ 8π.

Equality case implies the universal cover splits as a product of a 2-sphere and Euclidean space.

Unified results on homotopical 2-systole estimate from previous studies.

Abstract

Up to dimension five, we can prove that given any closed Riemannian manifold with nonnegative scalar curvature, of which the universal covering has vanishing homology group $H_k$ for all $k\geq 3$, either it is flat or it has Gauss-Bonnet quantity (defined by (1.3)) no greater than $8\pi$. In the second case, the equality for Gauss-Bonnet quantity yields that the universal covering splits as the Riemannian product of a $2$-sphere with non-negative sectional curvature and the Euclidean space. We also establish a dominated version of this result and its application to homotopical $2$-systole estimate unifies the results from [BBN10] and [Zhu20].