Mohammed Labbi | Searcharxiv

arXiv2013

Compact manifolds with positive $Γ_2$-curvature

The Schouten tensor \ $A$ \ of a Riemannian manifold \ $(M,g)$ provides important scalar curvature invariants $σ_k$, that are the symmetric functions on the eigenvalues of $A$, where, in particular, $σ_1$ \ coincides with the standard scalar curvature \ $\Scal(g)$. Our goal here is to study compact manifolds with positive \ $Γ_2$-curvature, \ i.e., when $σ_1(g)>0$ and $σ_2(g)>0$. In particular, we prove that a 3-connected non-string manifold $M$ admits a positive$Γ_2$-curvature metric if and only if it admits a positive scalar curvature metric. Also we show that any finitely presented group $π$ can always be realised as the fundamental group of a closed manifold of positive $Γ_2$-curvature and of arbitrary dimension greater than or equal to six.

math.DG↗

arXiv2013

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 $α$-invariant and Witten genus $ϕ_W$. 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 $\Ca P^2$.

math.DG↗