  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 $\phi_W$.
Here we use results of Dessai, which provide appropriate generators of the
rational string bordism ring in terms of "geometric $\Ca P^2$-bundles", where
the Cayley projective plane $\Ca P^2$ is a fiber and the structure group is
$F_4$ which is the isometry group of the standard metric on $\Ca P^2$.
