Asian Journal of Mathematics

Volume 26 (2022)

Number 3

Intermediate curvatures and highly connected manifolds

Pages: 407 – 454



Diarmuid Crowley (School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, Australia)

David J. Wraith (Department of Mathematics and Statistics, National University of Ireland, Maynooth, County Kildare, Ireland)


We show that after forming a connected sum with a homotopy sphere, all $(2j-1)$-connected $2j$-parallelisable manifolds in dimension $4j+1, j \geq 2$, 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 $j \geq 2$, and in dimensions $4j+1$ for $j \geq 1$ with torsion-free cohomology. A key technical innovation involves performing surgery on links of spheres within $2$-positive Ricci curvature.


$k$-positive Ricci curvature, intermediate curvatures, highly connected manifolds

2010 Mathematics Subject Classification

53C20, 57R65

Received 16 September 2021

Accepted 15 February 2022

Published 6 March 2023