nLab hyperbolic geometry

Idea

Riemannian geometry of Riemannian manifolds with constant negative sectional curvature.

Related concepts

• Riemannian geometry

• Lorentzian geometry

synthetic geometry	kind of space
Euclidean geometry	Euclidean space
hyperbolic geometry	hyperbolic space
elliptic geometry

• hyperbolic manifold

  • hyperbolic 3-manifold

• hyperbolic link

• trigonometry

References

General

• James Cannon, William Floyd, Richard Kenyon, Water Parry, Hyperbolic geometry, in Flavors of geometry, MSRI Publications Volume 31, 1997 (pdf)

• John Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics 149, Springer 2006 (doi:10.1007/978-0-387-47322-2, pdf)

• Igor R. Shafarevich, Alexey O. Remizov: §12 in: Linear Algebra and Geometry (2012) [doi:10.1007/978-3-642-30994-6, MAA-review]

• Brice Loustau, Hyperbolic geometry (arXiv:2003.11180)

See also

• Wikipedia, Hyperbolic geometry

For the special case of hyperbolic plane (but possibly over various fields) see

• Norman J. Wildberger, Universal hyperbolic geometry I: Trigonometry, Geometriae Dedicata 163(1), 2009 arxiv/0909.1377 doi; Universal hyperbolic geometry II: A pictorial overview, arxiv/1012.0880

Relation to AdS3/CFT2

Relation of AdS3/CFT2 to hyperbolic geometry and Arakelov geometry of algebraic curves:

• Yuri Manin, Matilde Marcolli, Holography principle and arithmetic of algebraic curves, Adv. Theor. Math. Phys. 5 (2002) 617-650 (arXiv:hep-th/0201036