In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

of a space (more precisely a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

) and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

is a surface of constant positive curvature.

Classification

The

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

s of constant curvature can be classified into the following three cases: *

elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...

– constant positive sectional curvature *

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

– constant vanishing sectional curvature *

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...

– constant negative sectional curvature.

Properties

* Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel

\nabla \mathrm=0

. * Every space of constant curvature is locally maximally symmetric, i.e. it has

\frac n (n+1)

number of local isometries, where

n

is its dimension. * Conversely, there exists a similar but stronger statement: every maximally symmetric space, i.e. a space which has

\frac n (n+1)

(global) isometries, has constant curvature. * ( Killing–Hopf theorem) The

universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete ...

of a manifold of constant sectional curvature is one of the model spaces: **

(sectional curvature positive) ** plane (sectional curvature zero) **

hyperbolic manifold In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, re ...

(sectional curvature negative) * A space of constant curvature which is geodesically complete is called space form and the study of space forms is intimately related to generalized crystallography (see the article on space form for more details). * Two space forms are isomorphic if and only if they have the same dimension, their metrics possess the same

signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...

and their sectional curvatures are equal.

References

* Moritz Epple (2003
From Quaternions to Cosmology: Spaces of Constant Curvature ca. 1873 — 1925
invited address to International Congress of Mathematicians * {{DEFAULTSORT:Constant Curvature Differential geometry of surfaces Riemannian geometry Curvature (mathematics)