发表机构
Politecnico di Torino; Institute of Mathematics, FSU Jena（都灵理工大学; 耶拿大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完成了Sophus Lie 1882年提出的二维伪黎曼度量的局部等距分类问题，补充了射影对称代数非正则情形的标准形，结合已有正则情形分类实现了完整求解。
AI 中文摘要
我们完成了对二维伪黎曼度量（包括黎曼度量和洛伦兹度量）的局部等距分类，该类度量容许至少二维的射影对称代数。本文的新贡献在于处理非正则情形：我们在射影李对称代数作用非正则的点邻域中，得到了一组完整的、互不等距的标准形，涵盖了所有可能出现的奇异行为。结合已有的正则情形分类，这一工作完成了Sophus Lie于1882年提出的该问题的求解。
英文摘要
We complete the local classification, up to isometry, of $2$-dimensional pseudo-Riemannian metrics (i.e. both Riemannian and Lorentzian), admitting a projective symmetry algebra of dimension at least two. The new contribution is the treatment of the non-regular case: we obtain a complete list of mutually non-isometric normal forms in neighbourhoods of points where the action of the projective Lie symmetry algebra fails to be regular, including all singular behaviours that can occur. Together with the known classification in the regular case, this completes the solution of the problem posed by Sophus Lie in 1882.
