On some critical Riemannian metrics and Thorpe-type conditions

We study critical metrics of higher-order curvature functionals on compact Riemannian $n$-manifolds $(M,g)$. For an integer $k$ with $2\le 2k\le n$, let $R^k$ denote the $k$-th exterior power of the Riemann curvature tensor. We investigate the Riemannian functionals $$H_{2k}(g)={\int }_M\mathrm{tr}(R^k){\mathrm{d}\mathrm{v}\mathrm{o}\mathrm{l}}_g\text{and}{G}_{2k}(g)={\int }_M\mathrm{\parallel }{R}^k{\mathrm{\parallel }}^2{\mathrm{d}\mathrm{v}\mathrm{o}\mathrm{l}}_g,$$ which generalize the Hilbert–Einstein functional and the total squared norm curvature, obtained for $k=1$ respectively. Using the formalism of double forms, we develop a systematic variational framework yielding compact first variation formulas for these functionals. Two key lemmas streamline the variational computations. A central technical ingredient is a generalization of the classical Lanczos identity to symmetric double forms of arbitrary even degree, providing explicit algebraic relations between the tensors ${\text{cc}}^{2k-1}(R^k\circ R^k)$ and ${\text{cc}}^{4k-1}(R^{2k})$. As a main geometric application, we introduce $(2k)$-Thorpe and $(2k)$-anti-Thorpe metrics, defined by self-duality and anti-self-duality conditions on $g^{r-2k}{R}^k$ in even dimensions $n=2r$. In the critical dimension $n=4k$, these metrics are absolute minimizers of $G_{2k}$, with the minimum determined by the Euler characteristic. For $n>4k$, they satisfy a harmonicity property leading to rigidity results under suitable curvature positivity assumptions. We further establish equivalences among variational criticality conditions. For hyper-$(2k)$-Einstein metrics, characterized by $\text{cc}{R}^k=\lambda g^{2k-1}$, being critical for $G_{2k}$ is equivalent to being $(4k)$-Einstein and to being weakly $(2k)$-Einstein. In the locally conformally flat setting, we classify all $4$-Thorpe metrics, showing that they are either space forms or Riemannian products ${\mathbb{S}}^r(c)\times {\mathbb{H}}^r(-c)$.