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Hyperbolic Space

Hyperbolic n-space, denoted H^n, is, up to isometry, the simply connected Riemannian manifold of dimension n having constant sectional curvature -1. It is a complete metric space. Rescaling its metric by a factor R^2 gives sectional curvature -R^(-2). The cases n=2 and n=3 are called the hyperbolic plane and hyperbolic 3-space, respectively.

Standard models of hyperbolic space include ball, upper half-space, projective, and hyperboloid models. Although their coordinate descriptions differ, the models are mutually isometric. In dimension 2, familiar examples are the Poincaré hyperbolic disk and the Klein-Beltrami model. Hyperbolic space is also a homogeneous space and a symmetric space.

See also

Hyperbolic Geometry, Hyperbolic Metric, Hyperbolic Plane, Hyperboloid Model, Poincaré Half-Plane Model

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References

Anderson, J. W. Hyperbolic Geometry. New York: Springer-Verlag, 1999.Helgason, S. Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press, 1978.

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Hyperbolic Space

Cite this as:

Weisstein, Eric W. "Hyperbolic Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicSpace.html

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