Intermediate curvatures and highly connected manifolds

TL;DR

This paper proves that certain highly connected manifolds in dimension 4j+1 can be endowed with metrics of 2-positive Ricci curvature after connected sum with a homotopy sphere, extending previous results to a broader class.

Contribution

It establishes the existence of 2-positive Ricci curvature metrics on (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1 after connected sum with a homotopy sphere, broadening curvature existence results.

Findings

All such manifolds admit 2-positive Ricci curvature metrics after connected sum.

The result extends previous work on positive Ricci curvature to a new class of highly connected manifolds.

The condition of 2-positive Ricci curvature involves the sum of the two smallest eigenvalues of the Ricci tensor.

Abstract

We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues of the Ricci tensor is positive at every point. This result is a counterpart to a previous result of the authors concerning the existence of positive Ricci curvature on highly connected manifolds in dimensions 4j-1 for j > 1, and in dimensions 4j+1 for j > 0 with torsion-free cohomology.