Some properties of the Schouten tensor and applications to conformal geometry

Abstract

The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the k k th elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, Γ k + \Gamma _k^+ . We prove that this eigenvalue condition for k ≥ n / 2 k \geq n/2 implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of σ k \sigma _k -curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.