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hyperbolic plane

Contents

Contents

Idea

The hyperbolic plane is the analog of the ordinary Euclidean plane as one passes from Euclidean geometry to hyperbolic geometry. It is the 2-dimensional example of a hyperbolic space.

Definition

As a Riemannian manifold, a hyperbolic plane is a complete contractible 2-dimensional smooth manifold of constant negative Riemann curvature (equivalently, Gaussian curvature).

Properties

The intrinsic geometry of a hyperbolic plane is completely determined by its curvature; any two hyperbolic planes are related by a conformal transformation, and a self-conformality of a hyperbolic plane is necessarily an isometry.

Hyperbolic planes are also homogeneous spaces: there is a unique oriented isometry relating any two geodesic rays.

The incidence geometry of geodesics and points in the hyperbolic plane is equivalent to that of an open circular disc in the Euclidean plane (and several other obviously-equivalent things) — this is the Beltrami disc model.

Last revised on May 22, 2019 at 11:24:47. See the history of this page for a list of all contributions to it.