A generalization of Geroch's conjecture

TL;DR

This paper introduces a new curvature concept called $m$-intermediate curvature, bridging Ricci and scalar curvature, and proves non-existence results for positive $m$-intermediate curvature metrics on certain manifolds for dimensions up to 7.

Contribution

It defines $m$-intermediate curvature and demonstrates that manifolds of the form $N^n = M^{n-m} imes ext{Torus}^m$ do not admit positive $m$-intermediate curvature for $n \,\leq\, 7$, extending curvature obstructions.

Findings

Manifolds $N^n = M^{n-m} imes \mathbb{T}^m$ lack positive $m$-intermediate curvature for $n \leq 7.

Introduces $m$-intermediate curvature as a bridge between Ricci and scalar curvature.

Uses stable weighted slicings to establish curvature obstructions.

Abstract

The Theorem of Bonnet--Myers implies that manifolds with topology $M^{n-1} \times \mathbb{S}^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $\mathbb{T}^n$ does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so called $m$-intermediate curvature), and use stable weighted slicings to show that for $n \leq 7$ the manifolds $N^n = M^{n-m} \times \mathbb{T}^m$ do not admit a metric of positive $m$-intermediate curvature.