Homological $k$-systole in $n$-manifolds with positive intermediate curvature

Abstract

In this paper, we prove optimal $k$-systolic inequalities and characterize the case of equality on closed $n$-dimensional Riemannian manifolds with positive intermediate curvature for $3\le n\le 7$. This unifies prior works of Bray-Brendle-Neves \cite{BrayBrenleNevesrigidity} and Chu-Lee-Zhu \cite{chuleezhu_n_systole}, and extends them to higher codimensions. The proof is inspired by our recent work on splitting theorems under intermediate curvature \cite{chenhong2026}.

24 pages; extend $(n-2)$-systolic inequality to all $k$-systolic inequalities, proof is rewritten. Comments are welcome