$\mathbb{R}^2$上共形和Killing向量场的Vessiot-Guldberg李代数的几何特征
Geometric features of Vessiot--Guldberg Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$
摘要：
本文对$\mathbb{R}^2$上关于任意伪黎曼度量的共形和Killing向量场的有限维李代数进行了局部分类.详细讨论了它们的几何性质，如它们的不变分布和诱导辛结构。研究结果说明了两个例子的物理性质：米尔恩-平尼方程和投影薛定谔方程的黎曼球。
英文摘要：
This paper locally classifies finite-dimensional Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$ relative to an arbitrary pseudo-Riemannian metric. Several results about their geometric properties are detailed, e.g. their invariant distributions and induced symplectic structures. Findings are illustrated with two examples of physical nature: the Milne--Pinney equation and the projective Schrödinger equation on the Riemann sphere.
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