关于CPE度量、真空静态空间和$σ_2$ -奇异空间之间的相互作用
On the interplay between CPE metrics, vacuum static spaces and $σ_2$-singular spaces
摘要：
我们称CPE度量为限制在酉体积具有常数数量曲率的度量空间中的全数量曲率泛函的临界点。本文给出了在$σ_2$ -奇异空间中CPE度量是Einstein度量的一个充要条件。这一结果从一个新的几何角度加深了我们对CPE度量和Besse猜想的理解。此外，我们证明了CPE条件可以用相关的真空静态空间条件来代替，以$σ_2$ -奇异空间来刻画闭Einstein流形。
英文摘要：
We call CPE metrics the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume. In this short note, we give a necessary and sufficient condition for a CPE metric to be Einstein in therms of $σ_2$-singular spaces. Such a result improves our understanding about CPE metrics and Besse's conjecture with a new geometric point of view. Moreover, we prove that the CPE condition can be replaced by the related vacuum static space condition to characterize closed Einstein manifolds in terms of $σ_2$-singular spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
