Compact manifolds with positive $\Gamma_2$-curvature

The Schouten tensor \ $A$ \ of a Riemannian manifold \ $(M,g)$ provides important scalar curvature invariants $\sigma_k$, that are the symmetric functions on the eigenvalues of $A$, where, in particular, $\sigma_1$ \ coincides with the standard scalar curvature \ $\Scal(g)$. Our goal here is to study compact manifolds with positive \ $\Gamma_2$-curvature, \ i.e., when $\sigma_1(g)>0$ and $\sigma_2(g)>0$. In particular, we prove that a 3-connected non-string manifold $M$ admits a positive$\Gamma_2$-curvature metric if and only if it admits a positive scalar curvature metric. Also we show that any finitely presented group $\pi$ can always be realised as the fundamental group of a closed manifold of positive $\Gamma_2$-curvature and of arbitrary dimension greater than or equal to six.