nLab
hyperbolic geometry

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Riemannian geometry of Riemannian manifolds with constant negative sectional curvature.

Related concepts

synthetic geometrykind of space
Euclidean geometryEuclidean space
hyperbolic geometryhyperbolic space
elliptic geometry

References

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

  • 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

Last revised on April 25, 2018 at 03:19:42. See the history of this page for a list of all contributions to it.