Many authors define an isometry of a metric space to be a distance-preserving map of the space onto itself. In this note, we discuss spaces for which surjectivity is a consequence of the distance-preserving property rather than an initial assumption. These spaces include, for example, the three classical (Euclidean, spherical, and hyperbolic) geometries of constant curvature that are usually discussed independently of each other. In this partly expository paper, we explore basic ideas about the isometries of a metric space, and apply these to various familiar metric geometries.

中文翻译：

---

度量几何的立体等距

许多作者将度量空间的等距定义为空间到自身的距离保持映射。在本笔记中，我们讨论了满射性是距离保持特性的结果而不是初始假设的空间。例如，这些空间包括常曲率的三个经典（欧几里得、球面和双曲）几何，它们通常彼此独立讨论。在这篇部分说明性的论文中，我们探索了关于度量空间的等距的基本思想，并将其应用于各种熟悉的度量几何。