Appendix A: List of Symbols
- Page ID
- 80009
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)| Symbol | Description | Location |
|---|---|---|
| \(\mathbb{C}\) | the complex plane | Section 2.1 |
| \(\mathbb{S}^1\) | the unit circle | Example 3.2.1 |
| \(i_C(z)\) | inversion in the circle \(C\) | Section 3.2 |
| \(\infty\) | the point at \(\infty\) | Section 3.3 |
| \(\mathbb{C}^+\) | the etended complex plane | Section 3.3 |
| \((\mathbb{C},{\cal T})\) | translational geometry | Example 4.1.4 |
| \((\mathbb{C},{\cal E})\) | Euclidean geometry | Example 4.1.5 |
| \((\mathbb{C}^+,{\cal M})\) | Möbius geometry | Definition of Möbius Geometry (Section 4.2) |
| \((\mathbb{D},{\cal H})\) | the Poincaré disk model for hyperbolic geometry | Definition of Poincaré disk model for hyperbolic geometry (Section 5.1) |
| \(\mathbb{D}\) | the hyperbolic plane | Definition of Poincaré disk model for hyperbolic geometry (Section 5.1) |
| \(\mathbb{S}^1_\infty\) | the circle at \(\infty\) in \((\mathbb{D},{\cal H})\) | Section 5.1 |
| \(\mathbb{P}^2\) | the projective plane | Definition of Projective Plane (Section 6.2) |
| \((\mathbb{P}^2,{\cal S})\) | the disk model for elliptic geometry | Definition of Disk Model for Elliptic Geometry (Section 6.2) |
| \((\mathbb{P}^2_k,{\cal S}_k)\) | the disk model for elliptic geometry with curvature \(k\) | 0" href="/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/07:_Geometry_on_Surfaces/7.02:_Elliptic_Geometry_with_Curvature__k__0">Section 7.2 |
| \((\mathbb{D}_k,{\cal H}_k)\) | the disk model for hyperbolic geometry with curvature \(k\) | Section 7.3 |
| \((X_k,G_k)\) | \(2\)-dimensional geometry with constant curvature \(k\) | Definition of Geometry \((X_k, G_k)\) (Section 7.4) |
| \(\mathbb{R}^n\) | real \(n\)-dimensional space | Section 7.5 |
| \(X_1 \# X_2\) | the connected sum of two surfaces | Section 7.5 |
| \(\mathbb{T}^2\) | the torus | Example 7.5.3 |
| \(H_g\) | the handlebody surface of genus \(g\) | Section 7.5 |
| \(C_g\) | the cross-cap surface of genus \(g\) | Section 7.5 |
| \(\mathbb{K}^2\) | the Klein bottle | Example 7.5.4 |
| \(\chi(S)\) | the Euler characteristic of a surface | Definition of Euler Characteristic (Section 7.5) |
| \(X/G\) | the quotient set built from geometry \((X,G)\) | Section 7.7 |
| \(\mathbb{H}^3\) | hyperbolic \(3\)-space | Section 8.1 |
| \(\mathbb{S}^3\) | the \(3\)-sphere | Section 8.1 |
| \(\mathbb{T}^3\) | the \(3\)-torus | Example 8.1.1 |