The Obata-Vétois argument and its applications (Q6629501)

A classical result by \textit{M. Obata} [J. Differ. Geom. 6, 247--258 (1971; Zbl 0236.53042)] states that every compact conformally Einstein manifold with constant scalar curvature is itself Einstein. In this paper, similar conclusions are drawn with the scalar curvature replaced by a family of higher-order Riemannian invariants, by a generalized and streamlined version of a recent Obata-type argument by \textit{J. Vétois} [Potential Anal. 61, No. 3, 485--500 (2024; Zbl 07935784)] for the \(Q\)-curvature.\N\NPrecisely, it is shown that if \((M^n, g)\) is a compact conformally Einstein manifold with nonnegative scalar curvature and such that \(Q_4 + a \sigma_2\) is constant, then \((M^n, g)\) is Einstein. Here, \(Q_4\) is the \(Q\)-curvature, \(\sigma_2\) is the \(\sigma_2\)-curvature and \(a \in \mathbb R\) lies in a certain interval \(I_n\) containing zero. The assumption of nonnegative scalar curvature can be weakened to only assuming positive Yamabe constant at the expense of further restricting the range of \(a\).\N\NA second class of results in this paper uses Vétois' result to compute explicitly, for a compact Einstein manifold with nonnegative scalar curvature, the Yamabe-type constant associated to \(I_{a,b} := Q_4 + a \sigma_2 + b|W|^2\), where \(a \in [-4,0]\), \(b \leq 0\), and \(|W|^2\) is the square length of the Weyl tensor \(W\). Through the conformal invariance property of \(I_{a,b}\), this statement can be written equivalently as a sharp Sobolev inequality.