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Homological $(n-2)$-systole in $n$-manifolds with positive triRic curvature

Jingche Chen, Han Hong

TL;DR

The article proves an optimal homological $(n-2)$-systole inequality for closed oriented $n$-manifolds with positive $\operatorname{triRic}$ curvature in dimensions $n=4,5$ (and $n=6$ under a $\gamma$-range). It combines stable weighted $2$-slicings with a refined Antonelli–Xu type volume comparison and a metric-deformation scheme to bound the $(n-2)$-systole by $|\mathbb{S}^{n-2}|$, and characterizes equality via a two-step isometric splitting, ultimately showing $M$ is covered by $\mathbb{S}^{n-2}\times \mathbb{R}^2$. The topological hypothesis on cohomology classes is essential to avoid counterexamples from products like $\mathbb{S}^{n-2}(r_1)\times \mathbb{S}^2(r_2)$. Overall, the work extends prior two-dimensional and higher-codimension systolic results to codimension two, using a blend of variational, geometric-analytic, and splitting arguments.

Abstract

In this paper, we prove an optimal systolic inequality and characterize the case of equality on closed Riemannian manifolds with positive triRic curvature. This extends prior work of Bray-Brendle-Neves \cite{BrayBrenleNevesrigidity} and Chu-Lee-Zhu \cite{chuleezhu_n_systole} to higher codimensions. The proof relies on the notion of stable weighted $k$-slicing, a weighted volume comparison theorem and metric-deformation.

Homological $(n-2)$-systole in $n$-manifolds with positive triRic curvature

TL;DR

The article proves an optimal homological -systole inequality for closed oriented -manifolds with positive curvature in dimensions (and under a -range). It combines stable weighted -slicings with a refined Antonelli–Xu type volume comparison and a metric-deformation scheme to bound the -systole by , and characterizes equality via a two-step isometric splitting, ultimately showing is covered by . The topological hypothesis on cohomology classes is essential to avoid counterexamples from products like . Overall, the work extends prior two-dimensional and higher-codimension systolic results to codimension two, using a blend of variational, geometric-analytic, and splitting arguments.

Abstract

In this paper, we prove an optimal systolic inequality and characterize the case of equality on closed Riemannian manifolds with positive triRic curvature. This extends prior work of Bray-Brendle-Neves \cite{BrayBrenleNevesrigidity} and Chu-Lee-Zhu \cite{chuleezhu_n_systole} to higher codimensions. The proof relies on the notion of stable weighted -slicing, a weighted volume comparison theorem and metric-deformation.
Paper Structure (11 sections, 15 theorems, 127 equations)

This paper contains 11 sections, 15 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminary
  4. Volume estimate
  5. Weighted slicings
  6. Splitting
  7. Proof of the inequality
  8. Proof of the rigidity
  9. Local conformal change of the metric
  10. Splitting of $\Sigma_1$
  11. Splitting of $M$

Key Result

Theorem 1.3

Let $(M^{n},g)$ be a closed, oriented $n$-dimensional Riemannian manifold with $n=4,5$. If there exist cohomology classes $\alpha_{1},\alpha_{2}\in H^{1}(M)$ satisfying $\alpha_{1}\smile \alpha_{2}\neq 0$, then Moreover, if equality holds, then $M$ is isometrically covered by $\mathbb{S}^{n-2}\times \mathbb{R}^{2}$.

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 22 more