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Appendix A: List of Symbols

  • Page ID
    80009
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    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
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