Constant curvature


Constant curvature

In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points. The circle has constant curvature, also, in a natural (but different) sense.

The standard surface geometries of constant curvature are elliptic geometry (or spherical geometry) which has positive curvature, Euclidean geometry which has zero curvature, and hyperbolic geometry (pseudosphere geometry) which has negative curvature. Since Riemann surfaces can be taken to have constant curvature, there is a large supply of other examples, for negative curvature.

For higher dimensional manifolds, constant curvature is usually taken to mean constant sectional curvature, and a complete manifold of this kind is called a space form. As in the case of surfaces, there are three types of geometries (elliptic, flat, or hyperbolic) according to whether the curvature is positive, zero, or negative. The universal cover of a manifold of constant sectional curvature is one of the model spaces (sphere, Euclidean space, hyperbolic space), and the study of space forms is thus generalized crystallography.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Curvature — In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this …   Wikipedia

  • Curvature of Riemannian manifolds — In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous… …   Wikipedia

  • Curvature of a measure — In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure s distribution of mass is curved . It is related to notions of curvature in geometry. In the form presented below, the… …   Wikipedia

  • Ricci curvature — In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci Curbastro, provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n… …   Wikipedia

  • Gaussian curvature — In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ 1 and κ 2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how… …   Wikipedia

  • Sectional curvature — In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two dimensional plane σp in the tangent space at p. It is the Gaussian curvature of… …   Wikipedia

  • Affine curvature — This article is about the curvature of affine plane curves, not to be confused with the curvature of an affine connection. Special affine curvature, also known as the equi affine curvature or affine curvature, is a particular type of curvature… …   Wikipedia

  • Scalar curvature — In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the… …   Wikipedia

  • Mean curvature — In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The… …   Wikipedia

  • Einstein's constant — or Einstein s gravitational constant, noted kappa; (kappa), is the coupling constant appearing in the Einstein field equation which can be written: G^{alpha gamma} = kappa , T^{alpha gamma} where Gα gamma; is the Einstein tensor and Tα gamma; is… …   Wikipedia


We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.