Some properties of the Schouten tensor and applications to conformal geometry

Guan, Pengfei; Viaclovsky, Jeff; Wang, Guofang

doi:10.1090/S0002-9947-02-03132-X

Some properties of the Schouten tensor and applications to conformal geometry
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by Pengfei Guan, Jeff Viaclovsky and Guofang Wang PDF

Trans. Amer. Math. Soc. 355 (2003), 925-933 Request permission

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 $k$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, $\Gamma _k^+$. We prove that this eigenvalue condition for $k \geq n/2$ implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of $\sigma _k$-curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.

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