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Asian Journal of Mathematics
Volume 26 (2022)
Number 3
Intermediate curvatures and highly connected manifolds
Pages: 407 – 454
DOI: https://dx.doi.org/10.4310/AJM.2022.v26.n3.a3
Authors
Abstract
We show that after forming a connected sum with a homotopy sphere, all $(2j-1)$-connected $2j$-parallelisable manifolds in dimension $4j+1, j \geq 2$, 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.
Keywords
$k$-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