Abstract
The geometry of the sphere is familiar to us, from everyday life as well as from geography. It is a part of the metric geometry of space, yet it represents something of its own in it. The distance of two points on the unit sphere is its angle, measured from the center; thus the angle takes on a whole new meaning: spherical distance. There is a second geometry which is similarly defined, but has exactly opposite properties in many respects: Here the surrounding Euclidean space is replaced by \(\mathbb {R}^{n+1}\) with the Lorentzian scalar product, the spacetime of Special Relativity. The “unit sphere” in this space is a model of the non-Euclidean geometry of Lobachevski and Bolyai, which had caused a great surprise in the early nineteenth century because it contradicted the common belief that Euclidean geometry was the only conceivable geometry.
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Notes
- 1.
Specifically on \({{\mathbb {S}}}^2 = {{\mathbb {C}}}\cup \{\infty \} = \hat {{\mathbb {C}}} = {{\mathbb {C}}}{{\mathbb {P}}}^1\), the group of oriented conformal circle-preserving transformations is the group of fractional-linear functions \(f(z) = \frac {az+b}{cz+d}\) with ad − bc≠0. This in turn is the projective group in complex dimension 1, \(PGL(2,{\mathbb {C}})\). Two-dimensional oriented conformal geometry and one-dimensional complex projective geometry are thus identical,
$$\displaystyle \begin{aligned} PO^+(3,1) = PGL(2,\mathbb{C}) \end{aligned} $$(7.1)(where + stands for orientation-preserving). Therefore, we can also describe the Lorentz transformations of Special Relativity by complex 2 × 2-matrices. Eq. (7.1) is one of the coincidences between low-dimensional Lie groups, of which there are several more (e.g. \(SU(2) = \mathbb {S}^3\subset \mathbb {H}\)).
- 2.
In projective space \(\mathbb {R}\mathbb {P}^n\), the two shells are identified.
- 3.
The subset \(\{x\in \mathbb {R}^{n+1};\, \langle x,x\rangle < 0\}\) of timelike vectors has two connected components: x n+1 > 0 and x n+1 < 0. By O +(n, 1) we denote the subgroup of O(n, 1) preserving each connected component. Apparently this is isomorphic to PO(n, 1) = O(n, 1)∕{±I}.
- 4.
The corresponding argument does not hold when instead x and y lie on a common circle of latitude, because on the northern hemisphere these circles get shorter towards the north. Thus swerving to the north still costs an extra north-south component, but at the same time the length of the east-west component is diminished.
- 5.
These are the two partial derivatives of the map \((\varphi ,\theta ) \mapsto \left ( {\begin {matrix}\sin \theta \cos \varphi \\ \sin \theta \sin \varphi \\ \cos \theta \end {matrix}}\right ) \).
- 6.
Nikolai Ivanovich Lobachevski , 1792 (Nizhny Novgorod)–1856 (Kazan), János Bolyai , 1802 (Clausenburg/Cluj Napoca)–1860 (Neumarkt/Târgu Mureş, now Romania).
- 7.
Jules Henri Poincaré, 1854 (Nancy)–1912 (Paris).
- 8.
For this argument, we need a point where the angles of Euclidean and hyperbolic geometry already coincide; this is the point e n+1 because on the (horizontal) tangential hyperplane \(T_{e_{n+1}}H = {\mathbb {R}}^n\) the Lorentzian scalar product coincides with the Euclidean one. The group PO(n, 1) operates transitively on \({\mathbb {S}}^n_+\) and on H. It preserves the (Euclidean) angles on \({\mathbb {S}}^n_+\) because it operates conformally, and it preserves the angles of hyperbolic geometry on H because it operates isometrically on H (since it preserves the Lorentzian scalar product and thus the hyperbolic distance).
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Eschenburg, JH. (2022). Angular Distance: Spherical and Hyperbolic Geometry. In: Geometry - Intuition and Concepts. Springer, Wiesbaden. https://doi.org/10.1007/978-3-658-38640-5_7
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DOI: https://doi.org/10.1007/978-3-658-38640-5_7
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