Compact manifolds with positive $Γ_2$-curvature

Abstract

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.