Abel’s Theorem in Problems and Solutions
《Abel’s Theorem in Problems and Solutions》 · 第 263–264 页 · Chapter 10，Section 10.1，Definition of Teichmüller Space；EPUB 原书页码标记 page_263、page_264，数学图像 φ 已核对
Let \(S\) be a compact surface with finitely many (perhaps zero) points removed from the interior. We assume for now that \(\chi(S)<0\). […] By a hyperbolic structure on \(S\) we will mean a diffeomorphism \(\phi:S\to X\), where \(X\) is a surface with a complete, finite-area hyperbolic metric with totally geodesic boundary. […] Two hyperbolic structures \(\phi_1:S\to X_1\) and \(\phi_2:S\to X_2\) on \(S\) are homotopic if there is an isometry \(X_1\to X_2\) so that the markings \(I\circ\phi_1:S\to X_2\) and \(\phi_2:S\to X_2\) are homotopic. […] Here homotopies are allowed to move points in the boundary of \(X_2\). We can then define the Teichmüller space of \(S\) as the set of homotopy classes of hyperbolic structures on \(S\): \(\operatorname{Teich}(S)=\{\text{hyperbolic structures on }S\}/\text{homotopy}\).