Positive intermediate Ricci curvature on connected sums

TL;DR

This paper explores conditions under which connected sums of manifolds can admit positive intermediate Ricci curvature, introducing the concept of $k$-core metrics and establishing new plumbing techniques.

Contribution

It introduces $k$-core metrics as a generalization for positive Ricci curvature and develops a new plumbing method to construct manifolds with positive intermediate Ricci curvature.

Findings

Connected sums with $k$-core metrics admit positive $k^{th}$ intermediate Ricci curvature.

Linear sphere bundles over such manifolds also admit positive intermediate Ricci curvature.

Examples include $bH P^n$ with $(4n-3)$-core metrics and $bO P^2$ with a 9-core metric.

Abstract

We consider the problem of performing connected sums in the context of positive $k^{th}$ intermediate Ricci curvature. We show that such connected sums are possible if the manifolds involved possess `$k$-core metrics' for some $k$. Here, a $k$-core metric is a generalization of the notion of core metric introduced by Burdick for positive Ricci curvature. Further, we show that connected sums of linear sphere bundles over bases admitting such metrics admit positive $k^{th}$ intermediate Ricci curvature for $k$ in a particular range. This follows from a plumbing result we establish, which generalizes other recent plumbing results in the literature and is possibly of independent interest. As an example of a manifold admitting a $k$-core metric, we prove that $\mathbb{H} P^n$ admits a $(4n-3)$-core metric and that $\mathbb{O}P^2$ admits a $9$-core metric, and we show that in both cases these are optimal.