In this paper, we mainly study variational formulae of functionals determined by the k-curvature of the modified Schouten tensor which is defined on the space \({\mathcal {G}}(M)\) of Riemannian metrics on a compact manifold M. Firstly, we consider the functional \({\mathcal {F}}_k^{\tau }\) given by

where \(\sigma _k(P^{\tau })\) is the k-curvature of the modified Schouten tensor \(P^{\tau }\) with \(\tau \in {\mathbb {R}}\), and then prove that, for a critical metric g of the functional \({\mathcal {F}}_k^{\tau }\), \(k\ge 3\), the k-curvature \(\sigma _k(P^{\tau })\) is constant if the corresponding Cotton tensor vanishes. Secondly, we study a more general functional \({\mathcal {F}}_3^{\tau ,\theta }\) also given by the modified Schouten tensor \(P^{\tau }\). As the result, we establish the corresponding Euler–Lagrange equation and then compute the second variational formulae for the functional \({\mathcal {F}}_3^{\tau ,\theta }\) restricted to a given conformal metric class.

在本文中，我们主要研究由修正Schouten张量的k曲率确定的泛函变分公式，该张量定义在紧流形M上的黎曼度量空间\({\mathcal {G}}(M)\)上。首先，我们考虑由下式给出的函数\({\mathcal {F}}_k^{\tau }\)

其中\(\sigma _k(P^{\tau })\)是修改后的 Schouten 张量\(P^{\tau }\)的k曲率，其中\(\tau \in {\mathbb {R}} \)，然后证明，对于函数\({\mathcal {F}}_k^{\tau }\)、\(k\ge 3\)的临界度量g，k曲率\(\如果相应的 Cotton 张量消失，则sigma _k(P^{\tau })\)为常数。其次，我们研究一个更通用的函数\({\mathcal {F}}_3^{\tau ,\theta }\)也由修改后的 Schouten 张量\(P^{\tau }\)给出。结果，我们建立了相应的欧拉-拉格朗日方程，然后计算限制于给定共角度量类的函数 \({\mathcal {F}}_3^{\tau ,\theta }\) 的二阶变分公式。