Highly connected manifolds of positive $p$-curvature

We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive $p$-curvature. The $p$-curvature was defined and studied by the second author. It turns out that positivity of $p$-curvature could be preserved under surgeries of codimension at least $p+3$. This gives a key to reduce a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3-connected manifolds with positive 2-curvature in terms of the spin and string bordism groups, and by means of $\alpha$-invariant and Witten genus ${\varphi }_W$. Here we use results of Dessai, which provide appropriate generators of the rational string bordism ring in terms of “geometric $\text{Ca}{P}^2$-bundles”, where the Cayley projective plane $\text{Ca}{P}^2$ is a fiber and the structure group is $F_4$ which is the isometry group of the standard metric on $\text{Ca}{P}^2$.