AI 中文总结
该综述从几何和拓扑视角研究平移曲面，给出其等价定义，探索模空间结构，包括自然分层，还研究了周期映射等内容，突出了理论中常被动力学观点掩盖的基础方面。
AI 中文摘要
本综述从几何和拓扑角度审视平移曲面，着重关注常被动力学观点所掩盖的方面。在给出平移曲面的几个等价定义，尤其是基于多边形模型和基本区域的定义后，我们依瑟斯顿的精神探索其模空间的结构，包括自然分层。最后一部分致力于周期映射、将表示实现为周期特征以及全纯性映射的纤维的研究。这种方法突出了该理论一个基础但有时被低估的方面。
英文摘要
This survey offers a geometric and topological perspective on translation surfaces, with a focus on aspects that are often overshadowed by the dynamical viewpoint. After presenting several equivalent definitions of translation surfaces, particularly those based on polygonal models and fundamental domains, we explore the structure of their moduli spaces in the spirit of Thurston, including natural stratifications. The final part is devoted to the study of the period map, the realisation of representations as period characters and the fibres of the holonomy map. This approach highlights a foundational, yet sometimes underrated, facet of the theory.
Comments70 pages, 26 figures. Survey to appear as a chapter in the book "In the Tradition of Thurston IV" (K. Ohshika and A. Papadopoulos ed.)
