Hyperbolic geometry

Remarks

Any two hyperbolic planes in this sense are isomorphic, so hyperbolic geometry is a non-Euclidean geometry with a single model up to isomorphism. In differential geometry, the hyperbolic $n$-space $\mathbb{H}^n$ is the complete simply connected Riemannian manifold of dimension $n$ with constant sectional curvature $-1$. For $n = 2$ this is a surface of Gaussian curvature $-1$, 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 $\set{(x, y) : y > 0}$ 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.