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AppendixAList of Symbols

Symbol Description Location
\(\mathbb{C}\) the complex plane Paragraph
\(i_C(z)\) inversion in the circle \(C\) Paragraph
\(\mathbb{S}^1\) the unit circle Example 3.2.1
\(\infty\) the point at \(\infty\) Paragraph
\(\mathbb{C}^+\) the etended complex plane Paragraph
\((\mathbb{C},{\cal T})\) translational geometry Example 4.1.4
\((\mathbb{C},{\cal E})\) Euclidean geometry Example 4.1.9
\((\mathbb{C}^+,{\cal M})\) Möbius geometry Definition 4.2.1
\((\mathbb{D},{\cal H})\) the Poincaré disk model for hyperbolic geometry Definition 5.1.1
\(\mathbb{D}\) the hyperbolic plane Definition 5.1.1
\(\mathbb{S}^1_\infty\) the circle at \(\infty\) in \((\mathbb{D},{\cal H})\) Paragraph
\(\mathbb{P}^2\) the projective plane Definition 6.2.6
\((\mathbb{P}^2,{\cal S})\) the disk model for elliptic geometry Definition 6.2.8
\((\mathbb{P}^2_k,{\cal S}_k)\) the disk model for elliptic geometry with curvature \(k\) Paragraph
\((\mathbb{D}_k,{\cal H}_k)\) the disk model for hyperbolic geometry with curvature \(k\) Paragraph
\((X_k,G_k)\) 2-dimensional geometry with constant curvature \(k\) Definition 7.4.1
\(\mathbb{R}^n\) real \(n\)-dimensional space Paragraph
\(X_1 \# X_2\) the connected sum of two surfaces Paragraphs
\(\mathbb{T}^2\) the torus Example 7.5.11
\(H_g\) the handlebody surface of genus \(g\) Paragraph
\(C_g\) the cross-cap surface of genus \(g\) Paragraph
\(\mathbb{K}^2\) the Klein bottle Example 7.5.18
\(\chi(S)\) the Euler characteristic of a surface Definition 7.5.30
\(X/G\) the quotient set built from geometry \((X,G)\) Paragraph
\(\mathbb{H}^3\) hyperbolic 3-space Paragraphs
\(\mathbb{S}^3\) the 3-sphere Paragraphs
\(\mathbb{T}^3\) the 3-torus Example 8.1.2