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 Unknown characterUnknown character 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 (usually 2-dimensional) manifolds with constant negative scalar curvature.

synthetic geometry
Euclidean geometry
hyperbolic geometry
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

Revised on September 15, 2015 14:33:07 by Urs Schreiber (147.231.89.106)