Rigidity of Schouten tensor under conformal deformation (Q6562814)

The main objects of the reviewed paper are the Schouten tensor \(A_g\) and the modified Schouten tensor \(A_g^\tau\) of a complete Riemannian manifold \((M,g)\) of dimension \(n\geq 3,\) defined for simplicity as \[ A_{g}=\mathrm{Ric}-\frac{R}{2(n-1)}g,\quad A_{g}^\tau=\mathrm{Ric}-\frac{\tau R}{2(n-1)}g, \] where \(R\) and \(\mathrm{Ric}\) are the curvature tensor field and the Ricci tensor field, respectively, and \(\tau>0\).\N\NThe authors prove that if after conformal transformations of the standard Euclidean metric \(g_0\) on \(\mathbb{R}^n\) with \(n\geq 3\) one obtaines a complete metric \(g\) whose modified Schouten tensor \(A_{g}^\tau\) is nonnegative for some \(\tau\in\left(0,\frac{2(n-1)}{n}\right)\), then the metrics \(g\) and \(g_0\) must be isometric. Moreover, they show that on a scalar flat complete Riemannian manifold \((M^n,g_0)\) of dimension \(n\geq 3\), any metric \(g\) conformal to \(g_0\) is isometric to \(g_0\) if \(A_g\geq A_{g_0}\).\N\NSince the condition \(A_{g}^\tau\geq 0\) is equivalent to a Ricci pinching condition \( \mathrm{Ric}_g -\epsilon Rg\geq 0 \) for some \(\epsilon >0,\) the main result of the paper can be applied to prove Cheng's theorem, which asserts that a complete, nonflat, locally conformally flat manifold satisfying the above Ricci pinching condition must be compact, see [\textit{L. Cheng}, ``On locally conformally flat manifolds with positive pinched Ricci curvature'', Preprint, \url{arXiv:2303.06648}].