A generalization of Geroch's conjecture

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 $n\le 7$ the manifolds $N^n=M^{n-m}\times {\mathbb{T}}^m$ do not admit a metric of positive $m$-intermediate curvature.

final version; to appear in Comm. Pure. Appl. Math