On some critical Riemannian metrics and Thorpe-type conditions
Mohammed Larbi Labbi
TL;DR
The paper develops a comprehensive variational framework for higher-order curvature functionals on compact Riemannian manifolds using the double-forms formalism. It proves a generalized Lanczos identity linking contractions of powers of the Riemann tensor, characterizes critical metrics for $G_{2k}$ and $H_{2k}$, and introduces $(2k)$-Thorpe and $(2k)$-anti-Thorpe geometries that generalize Einstein and conformally flat structures in even dimensions. In critical dimension $n=4k$ these Thorpe-type metrics minimize $G_{2k}$ and tie the minimum to the Euler characteristic; for larger dimensions they exhibit harmonicity of $R^k$ and rigidity under curvature positivity. The work also establishes equivalences among several generalized Einstein conditions (weakly Einstein, $(2k)$-Einstein, hyper $(2k)$-Einstein) and provides explicit gradient formulas for the functionals, enabling a unified understanding of higher-order curvature variational problems with connections to Lovelock theory and Gauss–Bonnet geometry.
Abstract
We study critical metrics of higher-order curvature functionals on compact Riemannian $n$-manifolds $(M,g)$. For an integer $k$ with $2 \leq 2k \leq n$, let $R^k$ denote the $k$-th exterior power of the Riemann curvature tensor. We investigate the Riemannian functionals \[H_{2k}(g)=\int_M \operatorname{tr}(R^k)\,\mathrm{dvol}_g\quad\text{and}\quad G_{2k}(g)=\int_M \|R^k\|^2\,\mathrm{dvol}_g,\] 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 $\cc^{2k-1}(R^k \circ R^k)$ and $\cc^{4k-1}(R^{2k})$. 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 $\cc R^k=λg^{2k-1}$, 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 $\mathbb{S}^r(c) \times \mathbb{H}^r(-c)$.