Title: Intermediate curvatures and highly connected manifolds,
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. When j=1 we
extend the above to certain classes of simply-connected non-spin 5-manifolds.
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.