Hyperbolic Geometry - Core Definitions

Hyperbolic geometry, discovered independently by Bolyai and Lobachevsky in the 1820s, is a non-Euclidean geometry where the parallel postulate fails. Through any point not on a line, infinitely many lines exist that never intersect the given line.

DefinitionHyperbolic Space

Hyperbolic $n$ -space $\mathbb{H}^n$ is a complete, simply-connected Riemannian manifold of constant negative curvature $-1$ . In the Poincaré disk model, $\mathbb{H}^2$ is the open unit disk with metric:

ds^2 = \frac{4(dx^2 + dy^2)}{(1 - x^2 - y^2)^2}

Lines are represented by arcs of circles orthogonal to the boundary, and by diameters.

The defining feature: given a line $\ell$ and point $P \notin \ell$ , infinitely many lines through $P$ don't intersect $\ell$ . This contrasts with Euclidean geometry (exactly one parallel) and spherical geometry (no parallels—all lines intersect).

DefinitionHyperbolic Distance

In the hyperboloid model (upper sheet of $x^2 + y^2 - z^2 = -1$ in Minkowski space), the distance between points $p$ and $q$ is:

d(p,q) = \cosh^{-1}(-\langle p, q \rangle_L)

where $\langle \cdot, \cdot \rangle_L$ is the Lorentzian inner product $x_1x_2 + y_1y_2 - z_1z_2$ .

Triangles in hyperbolic geometry have angle sum strictly less than $\pi$ . The defect $\delta = \pi - (\alpha + \beta + \gamma)$ equals the area of the triangle, providing a remarkable link between angles and area absent in Euclidean geometry.

ExamplePoincaré Half-Plane Model

Another model: the upper half-plane $\{(x,y) : y > 0\}$ with metric:

ds^2 = \frac{dx^2 + dy^2}{y^2}

Lines are vertical rays and semicircles centered on the $x$ -axis. This model is conformally equivalent to the disk model via Cayley transform.

Remark

Historically, hyperbolic geometry resolved a 2000-year question: is the parallel postulate independent of other Euclidean axioms? By constructing a consistent geometry where it fails, Bolyai and Lobachevsky proved independence, revolutionizing mathematics and opening non-Euclidean geometries.

The isometry group of $\mathbb{H}^2$ is $PSL(2,\mathbb{R})$ , acting by Möbius transformations on the disk or half-plane. Hyperbolic geometry has constant curvature $K = -1$ , making it maximally symmetric among negatively curved spaces.