Non-Euclidean Geometry¶
Job: Contrast Euclidean, spherical, and hyperbolic geometries via parallel postulates, models, and curvature; connect to Riemannian space forms.
Core¶
Euclid's parallel postulate (Playfair form): through a point not on line \(\ell\), exactly one parallel to \(\ell\). Independent of the other incidence/betweenness/congruence axioms (relative consistency of alternatives).
| Geometry | Parallels | Model curvature \(K\) | Angle sum (triangle) |
|---|---|---|---|
| Euclidean | exactly one | \(0\) | \(=\pi\) |
| Spherical / elliptic | none (great circles meet) | \(+1\) | \(>\pi\) |
| Hyperbolic | infinitely many | \(-1\) | \(<\pi\) |
Spherical geometry. Unit sphere \(S^2\): geodesics = great circles. Antipodal identification \(\mathbb{RP}^2\) yields elliptic plane (lines meet once). Area of triangle \(= \mathrm{excess}=(\alpha+\beta+\gamma)-\pi\) (Girard).
Hyperbolic geometry — models (all isometric): - Klein (projective): disk, geodesics are chords (nonconformal). - Poincaré disk: geodesics circular arcs orthogonal to boundary; conformal; hyperbolic isometries \(\cong\mathrm{PSU}(1,1)\). - Upper half-plane \(\mathbb{H}=\{z:\mathrm{Im}z>0\}\) with \(ds=|dz|/y\); \(\mathrm{Isom}^+(\mathbb{H})\cong\mathrm{PSL}(2,\mathbb{R})\) acting by Möbius maps. - Hyperboloid in Minkowski space \(\mathbb{R}^{2,1}\).
Distance in half-plane: \(\cosh d(z,w)=1+\frac{|z-w|^2}{2\,\mathrm{Im}z\,\mathrm{Im}w}\) (equivalent formulas with \(\mathrm{artanh}\)).
Trigonometry: hyperbolic law of cosines $$ \cosh c=\cosh a\cosh b-\sinh a\sinh b\cos\gamma, $$ and dual formulas; exponential divergence of geodesics.
Riemannian synthesis. Complete simply connected Riemannian \(n\)-manifolds of constant sectional curvature \(K\in\{-1,0,1\}\) are unique up to isometry: \(H^n\), \(\mathbb{R}^n\), \(S^n\). Compact quotients \(\Leftrightarrow\) discrete torsion-free subgroups of isometry groups (Bieberbach for \(K=0\); rich theory for hyperbolic 3-manifolds — Thurston).
Absolute geometry: theorems avoiding the parallel postulate (e.g. SACHERI–LEGENDRE angle sum \(\le\pi\) under Archimedes) hold in Euclidean and hyperbolic.
See it¶
Ideal triangle in \(\mathbb{H}^2\): vertices on \(\partial\mathbb{H}\), angles \(0\), area \(\pi\) (finite!) by Gauss–Bonnet \(\int K=-\mathrm{Area}+\sum(\pi-\text{interior jumps})\ldots\). Maximal area among triangles is \(\pi\) for \(K=-1\).
Check¶
Why is spherical geometry not a model of neutral geometry with the usual incidence axioms without modification?
Great circles meet in two antipodes; "unique line through two points" fails unless one works in elliptic geometry (antipodes identified) or restricts to open hemispheres (local geometry only).
Show angle sum \(<\pi\) in hyperbolic geometry using Gauss–Bonnet.
For a geodesic triangle \(T\), \(\int_T K\,dA+\sum(\pi-\text{ext.})=\ldots\) standard form: \(\int_T K+\sum_i\alpha_i=\pi\) for simply connected regions with geodesic sides \(\Rightarrow\) \(K=-1\Rightarrow \mathrm{Area}=\pi-\sum\alpha_i>0\Rightarrow\sum\alpha_i<\pi\).
Pitfalls¶
- "Parallel" means non-intersecting geodesics — in hyperbolic geometry, ultraparallel vs limiting parallel (asymptotic) differ.
- Constant curvature \(\neq\) constant Gaussian curvature only for surfaces — higher D uses sectional curvature.
- Taxicab / other Minkowski planes are non-Euclidean metric geometries but not constant-curvature Riemannian.
Refs¶
Ratcliffe, Foundations of Hyperbolic Manifolds; Anderson, Hyperbolic Geometry; do Carmo (curves/surfaces); Thurston notes; Greenberg, Euclidean and Non-Euclidean Geometries.