Highly connected manifolds of positive 𝑝-curvature

Abstract

We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive p p -curvature. The p p -curvature was defined and studied by the second author in earlier papers. It turns out that the positivity of p p -curvature is preserved under surgeries of codimension at least p + 3 p+3 . This gives a key to reducing a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3 3 -connected manifolds with positive 2 2 -curvature in terms of the bordism groups Ω ∗ spin \Omega ^{\operatorname {spin}}_* , Ω ∗ s t r i n g \Omega ^{\mathrm {string}}_* , and by means of an α \alpha -invariant and a Witten genus ϕ W \phi _W . Here we use results from Anand Dessai (2009), which provide appropriate generators of the ring Ω ∗ s t r i n g ⊗ Q \Omega ^{\mathrm {string}}_*\otimes \mathbf {Q} in terms of “geometric C a P 2 {\mathbb C}{\mathbf a}\mathbf {P}^2 -bundles”, where the Cayley projective plane C a P 2 {\mathbb C}{\mathbf a} \mathbf {P}^2 is a fiber and the structure group is F 4 F_4 which is the isometry group of the standard metric on C a P 2 {\mathbb C}{\mathbf a}\mathbf {P}^2 .