---
tag: RBZPZTSPZE
kind: definition
name: Hyperbolic geometry
msc: [51M10, 53A35]
defines: GN4N94GQ0Z
uses: [SPCYMFN5V2, VKFPCSRNDH, 7CBT4SM83Y, SQVQ5DQT6W]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_geometry
---

## Definition

A *hyperbolic plane* is a model of the axioms of [[absolute geometry|VKFPCSRNDH]], including Dedekind's axiom of continuity, in which the hyperbolic parallel axiom holds.
That axiom says: for each line $\ell$ and each point $P$ not on $\ell$, at least two distinct lines through $P$ do not meet $\ell$.
*Hyperbolic geometry* is the study of the hyperbolic plane and of its higher-dimensional analogues.

## Remarks

Any two hyperbolic planes in this sense are isomorphic, so hyperbolic geometry is a [[non-Euclidean geometry|SPCYMFN5V2]] with a single model up to isomorphism.
In differential geometry, the hyperbolic $n$-space $\mathbb{H}^n$ is the complete simply connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with constant sectional curvature $-1$.
For $n = 2$ this is a surface of [[Gaussian curvature|SQVQ5DQT6W]] $-1$, and it is a hyperbolic plane in the axiomatic sense.
In it, a triangle with angles $\alpha, \beta, \gamma$ has area $\pi - \alpha - \beta - \gamma$.

Example: the [[Poincaré half-plane model|RHKSMPBSQ1]] $\set{(x, y) : y > 0}$ with the metric $(dx^2 + dy^2)/y^2$ satisfies the axioms.
Its lines are the vertical half-lines and the semicircles centred on the $x$-axis.

Non-example: the Euclidean plane satisfies the axioms of absolute geometry.
It fails the hyperbolic parallel axiom, because each point off a line lies on exactly one parallel.