Compact manifolds with positive ${\Gamma }_2$-curvature

Abstract

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.

Main change: the initial long proof of theorem 4.1 is replaced by a shorter proof that uses a recent general surgery result by Hoelzel