Let \(A\) be a linear endomorphism of an \(n\)-dimensional real vector space. The \(k\)-th elementary symmetric polynomial \(\sigma_k(A)\) of \(A\) is, up to a nonzero constant factor, the coefficient of \(\lambda^{n-k}\) in the determinant of \(\lambda\operatorname{Id}-A\). Let \((M,g)\) be a Riemannian manifold of dimension \(n\geq 3\) and \(R\) its Riemannian curvature tensor field. Then \(R=W+P\odot g\), where \(W\) is the Weyl conformal curvature tensor field, \(P:=\frac{1}{n-2}[\text{Ric}-\frac{1}{2(n-1)}\operatorname{Scal}g]\) is the Schouten tensor field, Ric the Ricci tensor field, Scal the scalar curvature and \(\odot\) denotes the Kulkarni-Nomizu product. \(P\) yields, via the metric, a section of \(\text{End}(TM)\) and so one may speak of the scalar invariants \(\sigma_k(P)\). For example, \(\sigma_1(P)=\frac{\text{Scal}}{2(n-1)}\) and \(\sigma_2(P)=\frac{1}{2}[\frac{\text{Scal}^2}{4(n-1)^2}-| P| ^2]\). For a given fixed \(k\) \((1\leq k\leq n)\), the \(\sigma_k\)-Yamabe problem is to find, within a conformal class of metrics on \(M\), a metric for which \(\sigma_k(P)\) is constant.
Recently there has been significant interest and progress in the study of this problem and related curvature prescription problems [see for example, J. Viaclovsky, Duke Math. J. 101, 283–316 (2000; Zbl 0990.53035); S.-Y. A. Chang, M. Gursky and P. Yang, J. Anal. Math. 87, 151–186 (2002; Zbl 1067.58028); M. Gursky and J. Viaclovsky, Ann. Math. (2) 166, No. 2, 475–531 (2007; Zbl 1142.53027); A. Li and Y. Y. Li, Acta Math. 195, 117–154 (2005; Zbl 1216.35038); P. Guan and G. Wang, J. Reine Angew. Math. 557, 219–238 (2003; Zbl 1033.53058)].
For \(k=1\), this is the classical Yamabe problem of prescribing constant scalar curvature, whose solution by Schoen, Aubin, Trudinger, and Yamabe [see J. M. Lee and T. H. Parker, Bull. Am. Math. Soc. 17, 37–91 (1987; Zbl 0633.53062)] was a milestone in global Riemannian geometry. W. Sheng, N. Trudinger and X.-J. Wang [J. Differ. Geom. 77, No. 3, 515–553 (2007; Zbl 1133.53035)] have shown that the equation \(\sigma_k(P)= \text{const}\) has a solution in settings where it is variational, i.e., the Euler-Lagrange equation of some functional (called an action or Lagrangian). This class of settings includes the cases \(k=2\) and when \((M,g)\) is conformally flat [see also J. Viaclovsky, op. cit.].
For \(3\leq k\leq n\), the authors prove that on an \(n\)-dimensional compact Riemannian manifold, the quantity \(\sigma_k(P)\) is variational in a conformal class of metrics \(\mathcal{C}\) if and only if \(\mathcal{C}\) is locally flat. In the cases where \(\sigma_{\frac{n}{2}}(P)\) (\(n\) even) is variational, they show that the problem of explicitly finding an action is related to the problem of finding an action for the \(Q\)-curvature [for the definition of this concept see the first author [The functional determinant, Lecture Notes Series, Seoul. 4. Seoul: Seoul National University (1993; Zbl 0827.58057)], the case of dimensions \(4\) and \(6\) being treated explicitly.