mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, constant curvature is a concept from

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

. 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 p ...

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 n ...

) 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 geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

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 space ...

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 ...

– 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 Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...

\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 In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuou ...

) 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 spa ...

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, r ...

(sectional curvature negative) * A space of constant curvature which is

geodesically complete In mathematics, a complete manifold (or geodesically complete manifold) is a ( pseudo-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map ...

is called

space form Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consid ...

and the study of space forms is intimately related to generalized crystallography (see the article on

for more details). * Two space forms are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

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 Moritz Epple (born 7 May 1960, in Stuttgart) is a German mathematician and historian of science. Biography Epple studied mathematics, philosophy and physics in Copenhagen, London, and at the University of Tübingen, where he received in 1987 his ...

(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)