On some critical Riemannian metrics and Thorpe-type conditions

Abstract: Let ((M, g)) be a compact, closed Riemannian manifold of dimension (n), and let (k) be an integer such that (2 \leq 2k \leq n). Denote by (R^k) the (k)-th Thorpe tensor, defined as the (k)-fold exterior product of the Riemann curvature tensor (R). We consider the geometric functionals $ H_{2k}(g) := \int_M \operatorname{tr}(R^k) \, \mathrm{dvol}g$ and $ G{2k}(g) :=\int_M |R^k|^2 \, \mathrm{dvol}g,$ where (\operatorname{tr}(R^k)) denotes the full contraction (trace) of (R^k). For (k = 1), (H_2(g)) coincides with the Hilbert--Einstein functional, while (G_2(g)) is the (L^2)-norm of the Riemann curvature tensor. We derive the first variation formulas for (H{2k}) and (G_{2k}), and study some properties of their critical points. In particular, we prove that an Einstein metric is critical for (G_2) if and only if it is critical for (H_4). More generally, we show that a hyper-((2k))-Einstein metric is critical for (G_{2k}) if and only if it is critical for (H_{4k}). We further introduce and study two new classes of Riemannian metrics: the ((2k))-Thorpe and ((2k))-anti-Thorpe metrics, defined via self-duality and anti-self-duality conditions on the Thorpe tensor (R^k), respectively. In dimension (n = 4k), we show that both types are critical for (G_{2k}). In higher dimensions ((n > 4k)), the same duality conditions imply that $R^k$ is harmonic. Under additional positivity assumptions on the Riemann tensor, we further deduce rigidity results for such metrics.