Intermediate curvature and splitting theorem

TL;DR

This paper establishes rigidity results for complete noncompact manifolds with nonnegative intermediate curvature, characterizing their geometric structure and constructing metrics with positive intermediate curvature, advancing understanding of curvature conditions.

Contribution

It introduces a new recursion theorem for spectral intermediate curvatures and provides novel splitting theorems and metric constructions, sharpening previous algebraic conditions.

Findings

Manifolds of certain topological types are isometrically covered by product spaces.

Constructed metrics with positive intermediate curvature on specific manifolds.

Reproved existing results using the new recursion theorem.

Abstract

In this paper, we prove several rigidity results for complete noncompact manifolds with nonnegative intermediate curvatures. We show that when either $3\leq n\leq 5$, $1\leq m\leq n-1$, or $6\leq n\leq 7$, $m\in \{1,n-1,n-2\}$, any manifold of the topological type $M^{n-m}\times \mathbb{T}^{m-1}\times \mathbb{R}$ with nonnegative $m$-intermediate curvature is isometrically covered by the canonical product $M\times \mathbb{R}^m$. We also construct smooth metrics on $M^{n-m}\times \mathbb{T}^{m-1}\times \mathbb{R}$ with uniformly positive $m$-intermediate curvature for $6\leq n\leq 7$, $2\leq m\leq n-3$. This proves that the algebraic condition $m^2-mn+m+n>0$ from \cite{chenshuli_end} is sharp. The proof is based on a new recursion theorem for spectral intermediate curvatures and cylindrical splitting theorems. In particular, when $m=n-1$, this provides a new proof of some results by Chodosh--Li \cite{chodoshlisoapbubble} and Zhu \cite{zhu-splitting}. Moreover, the recursion theorem can be used to reprove the result of Brendle--Hirsch--Johne \cite{brendlegeroch'sconjecture}.