Compact manifolds with positive Γ2Γ2-curvature

The Schouten tensor A   of a Riemannian manifold (M,g)(M,g) provides the important σkσk-scalar curvature invariants, that are the symmetric functions in the eigenvalues of A  , where, in particular, σ1σ1 coincides with the standard scalar curvature Scal(g)Scal(g). Our goal here is to study compact manifolds with positive Γ2Γ2-curvature, i.e., when σ1(g)>0σ1(g)>0 and σ2(g)>0σ2(g)>0. In particular, we prove that a 3-connected non-string manifold M   admits a positive Γ2Γ2-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 Γ2Γ2-curvature and of arbitrary dimension greater than or equal to six.