标题： 三维单模度量李群中的等距浸入 显示英文标题

标题： Isometric immersions into three-dimensional unimodular metric Lie groups

摘要： 我们研究将曲面等距浸入到具有黎曼或洛伦兹左不变度量的单连通三维单模李群，假设在洛伦兹情况下米尔诺算子是可对角化的。 我们提供了所有这些度量李群的全局坐标模型，这些模型以结构常数为参数进行解析依赖，并建立了某些基本定理来表征这样的浸入。 从这个意义上讲，我们研究在多大程度上可以从（a）自然左不变环境框架的切向投影、（b）左不变的高斯映射以及（c）形状算子中恢复浸入。 作为应用，我们证明了一个等距浸入由其左不变高斯映射确定，最多相差某些良好控制的角伴随。 我们还分类了全测地曲面，并在两个具有4维等距群的洛伦兹齐次3-流形族中引入了四个洛伦兹类比的丹尼尔对应关系。 我们还分类了在$\mathbb{R}^3$或$\mathbb{S}^3$中的等距浸入，其左不变高斯映射相差$\mathbb{S}^2$的直接等距。 最后，我们表明丹尼尔对应关系是黎曼单模度量李群中常平均曲率曲面的经典劳森对应关系的最远扩展。 显示英文摘要

摘要： We study isometric immersions of surfaces into simply connected 3-dimensional unimodular Lie groups endowed with either Riemannian or Lorentzian left-invariant metrics, assuming that Milnor's operator is diagonalizable in the Lorentzian case. We provide global models in coordinates for all these metric Lie groups that depend analytically on the structure constants and establish some fundamental theorems characterizing such immersions. In this sense, we study up to what extent we can recover the immersion from (a) the tangent projections of the natural left-invariant ambient frame, (b) the left-invariant Gauss map, and (c) the shape operator. As an application, we prove that an isometric immersion is determined by its left-invariant Gauss map up to certain well controlled angular companions. We also we classify totally geodesic surfaces and introduce four Lorentzian analogues of the Daniel correspondence within two families of Lorentzian homogeneous 3-manifolds with 4-dimensional isometry group. We also classify isometric immersions in $\mathbb{R}^3$ or $\mathbb{S}^3$ whose left-invariant Gauss maps differ by a direct isometry of $\mathbb{S}^2$. Finally, we show that Daniel's is the furthest extension of the classical Lawson correspondence for constant mean curvature surfaces within Riemannian unimodular metric Lie groups.