A generalization of Geroch's conjecture

Abstract

The Theorem of Bonnet–Myers implies that manifolds with topology $M^{n-1}\times S^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $T^n$ 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 $n\le 7$ and $1\le m\le n-1$ the manifolds $N^n=M^{n-m}\times T^m$ do not admit a metric of positive m-intermediate curvature.