The Three Geometries as Constant Curvature

This is the capstone of the metric synthesis. Everything in the spine converges here: the synthetic trichotomy of Euclidean, hyperbolic, and elliptic geometry is revealed as the classification of maximally symmetric spaces by the sign of their constant curvature $K$. A single equation subsumes the parallel postulate, the angle-sum theorems, the defect, and the spherical excess:

$$\boxed{\alpha +\beta +\gamma -\pi =K\cdot \mathrm{Area}(△)}$$

and the postulate holds iff $K=0$.

References: do Carmo, Riemannian Geometry, ch. 8; Lee, Riemannian Manifolds, ch. 8 (space forms); Wolf, Spaces of Constant Curvature; Thurston, Three-Dimensional Geometry and Topology, ch. 2.

1. Homogeneity, Isotropy, and Constant Curvature

What singles out the three classical geometries from the infinitude of Riemannian manifolds is maximal symmetry:

• Homogeneous: every point looks like every other — there is an isometry carrying any point to any other. (Space has no special location.)
• Isotropic: at each point every direction looks alike — isometries can rotate any direction into any other. (Space has no special direction.)

Theorem (constant curvature). A Riemannian surface that is homogeneous and isotropic has constant Gaussian curvature $K$. Conversely, a complete, simply-connected surface of constant curvature is one of exactly three geometries, determined by the sign of $K$.

This is the metric-geometry restatement of the synthetic fact that the angle defect is a global constant, not a local accident: constant curvature is what "the geometry is the same everywhere in every direction" means, and by Gauss–Bonnet that constant governs the angle sum.

2. The Three Space Forms

Up to scaling $K$ to $0,+1,-1$, the complete simply-connected surfaces of constant curvature — the 2-dimensional space forms — are:

In dimension $n$ the same trichotomy gives the space forms ${\mathbb{E}}^n$, $S^n$, and ${\mathbb{H}}^n$, with isometry groups $E(n)={\mathbb{R}}^n⋊O(n)$, $O(n+1)$, and $O(n,1)$ respectively — the subject of isometry-groups.md.

3. The Unified Angle-Sum Law

Local Gauss–Bonnet applied to a geodesic triangle in a space of constant curvature $K$ gives, since $K$ pulls out of the integral,

$$\alpha +\beta +\gamma -\pi ={\iint }_{△}KdA=K\cdot \mathrm{Area}(△).$$

Reading off the three signs recovers every synthetic angle-area theorem in the spine as one identity:

$$\begin{array}{rlrl}K=0& :\alpha +\beta +\gamma =\pi & & \text{(Euclidean; angle sum)}\\ K>0& :\alpha +\beta +\gamma -\pi =KA& & \text{(spherical excess, Girard)}\\ K<0& :\pi -(\alpha +\beta +\gamma )=|K|A& & \text{(hyperbolic defect)}\end{array}$$

The two curved cases each carry an absolute length scale $R=1/\sqrt{|K|}$, which is why neither has similar figures; the flat case has $|K|=0$, no scale, and hence similarity and free rescaling. The postulate's web of equivalents is, from here, just the list of things that happen precisely when $K=0$.

4. The Flat Limit Unifies the Trigonometries

The hyperbolic and spherical laws of cosines are the same formula at curvature radius $R=1/\sqrt{|K|}$ and $R=1/\sqrt{K}$ respectively. Writing the side lengths in units of $R$ and expanding as $R\to \infty$ ($K\to 0$):

$$\underset{K<0}{\underbrace{\cosh \frac{c}{R}=\cosh \frac{a}{R}\cosh \frac{b}{R}-\sinh \frac{a}{R}\sinh \frac{b}{R}\cos \gamma }}\underset{R\to \infty }{\to }\underset{K=0}{\underbrace{c^2=a^2+b^2-2ab\cos \gamma }}\underset{R\to \infty }{\leftarrow }\underset{K>0}{\underbrace{\cos \frac{c}{R}=\cos \frac{a}{R}\cos \frac{b}{R}+\sin \frac{a}{R}\sin \frac{b}{R}\cos \gamma }}.$$

Euclidean geometry is the common flat limit of the two curved geometries — the tangent-plane approximation valid on scales small compared with $R$. This is why local, small-scale measurements cannot distinguish the geometries: curvature is a second-order effect, invisible in the first-order (tangent-space) approximation. It is also why the curvature of physical space went unnoticed until it could be measured over astronomical baselines.

5. From Simply-Connected to All Constant-Curvature Surfaces

The three space forms are the simply-connected representatives. General constant-curvature surfaces are obtained as quotients by discrete groups of isometries acting freely — the Killing–Hopf theorem:

Killing–Hopf. Every complete connected surface of constant curvature $K$ is a quotient $\tilde{M}/\Gamma$, where $\tilde{M}\in \{{\mathbb{E}}^2,S^2,>{\mathbb{H}}^2\}$ is the space form of curvature $K$ and $\Gamma$ is a discrete group of isometries acting freely and properly discontinuously.

Examples: the flat torus and Klein bottle are quotients of ${\mathbb{E}}^2$; the projective plane ${\mathbb{RP}}^2$ is $S^2/\{\pm 1\}$; and every compact surface of genus $\ge 2$ carries a hyperbolic metric as a quotient ${\mathbb{H}}^2/\Gamma$ by a Fuchsian group $\Gamma \subset PSL(2,\mathbb{R})$. By Gauss–Bonnet the sign of $K$ a surface can carry is fixed by its Euler characteristic: $\chi >0$ forces spherical, $\chi =0$ flat, $\chi <0$ hyperbolic — so almost all surfaces are hyperbolic.

6. The Payoff

The constant-curvature picture is the mature form of the entire subject:

• Mathematically, it unifies the three geometries the parallel postulate split apart, indexing them by a single scalar $K$ and organizing them by their isometry groups.
• Physically, the $K<0$ hyperboloid is the velocity space of special relativity — its isometry group $O(n,1)$ is the Lorentz group — and variable curvature is the arena of general relativity, where Einstein's equation makes $K$ (via the Ricci tensor) a dynamical field sourced by matter.
• Philosophically, it settles that which geometry describes physical space is a measurable question about a curvature field, not a synthetic a priori truth — the theme of the philosophy of space and time.

The isometry-group page makes the group column of the §2 table precise; the Erlangen program then reorganizes the whole spine around those groups.