# AppendixAList of Symbols

Symbol Description Location
$\mathbb{C}$ the complex plane Paragraph
$\mathbb{S}^1$ the unit circle Example 3.2.1
$i_C(z)$ inversion in the circle $C$ Paragraph
$\infty$ the point at $\infty$ Paragraph
$\mathbb{C}^+$ the etended complex plane Paragraph
$(\mathbb{C},{\cal T})$ translational geometry Example 4.1.6
$(\mathbb{C},{\cal E})$ Euclidean geometry Example 4.1.11
$(\mathbb{C}^+,{\cal M})$ 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 7.5.9
$H_g$ the handlebody surface of genus $g$ Paragraph
$C_g$ the cross-cap surface of genus $g$ Paragraph
$\mathbb{K}^2$ the Klein bottle Example 7.5.15
$\chi(S)$ the Euler characteristic of a surface Definition 7.5.22
$X/G$ 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