page_title: Differential geometry of surfaces
section_title: Surfaces of constant Gaussian curvature
hierarchical_section_title: Differential geometry of surfaces / Examples / Surfaces of constant Gaussian curvature
context_page_description: In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
  Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.
  Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups, namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections.
context_section_description: If a surface has constant Gaussian curvature, it is called a surface of constant curvature.
  The unit sphere in E³ has constant Gaussian curvature +1.
  The Euclidean plane and the cylinder both have constant Gaussian curvature 0.
  The surfaces of revolution with φₜₜ = φ have constant Gaussian curvature –1. Particular cases are obtained by taking φ(t) =C cosh t, C sinh t and C eᵗ. The latter case is the classical pseudosphere generated by rotating a tractrix around a central axis. In 1868 Eugenio Beltrami showed that the geometry of the pseudosphere was directly related to that of the hyperbolic plane, discovered independently by Lobachevsky (1830) and Bolyai (1832). Already in 1840, F. Minding, a student of Gauss, had obtained trigonometric formulas for the pseudosphere identical to those for the hyperbolic plane. The intrinsic geometry of this surface is now better understood in terms of the Poincaré metric on the upper half plane or the unit disc, and has been described by other models such as the Klein model or the hyperboloid model, obtained by considering the two-sheeted hyperboloid q(x, y, z) = −1 in three-dimensional Minkowski space, where q(x, y, z) = x² + y² – z².
  Each of these surfaces of constant curvature has a transitive Lie group of symmetries. This group theoretic fact has far-reaching consequences, all the more remarkable because of the central role these special surfaces play in the geometry of surfaces, due to Poincaré's uniformization theorem (see below).
  Other examples of surfaces with Gaussian curvature 0 include cones, tangent developables, and more generally any developable surface.
caption: Eugenio Beltrami (1835-1899)
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