Schouten tensor and some topological properties

Abstract

The notion of positive curvature plays an important role in differential geometry. The existence of such a metric often implies some topological properties of the underlying manifold. A typical example is the Bochner vanishing theorem on manifolds of positive Ricci curvature. In this paper, we consider Riemannian metrics with certain type of positivity on the Schouten tensor. This notion of curvature was introduced by Viaclovsky [18] which extends the notion of scalar curvature. Let (M,g) be an oriented, compact and manifold of dimension n > 2. And let Sg denote the Schouten tensor of the metric g, i.e.,