mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, constant curvature is a concept from

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

. 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 po ...

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 (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, 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 is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

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 mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

– constant vanishing sectional curvature *

Hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

– 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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

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

universal cover In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...

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

(sectional curvature positive) ** plane (sectional curvature zero) ** hyperbolic manifold (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 In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...

if and only if they have the same dimension, their metrics possess the same

signature A signature (; from , "to sign") is a 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 intent. Signatures are often, but not always, Handwriting, handwritt ...

and their sectional curvatures are equal.

Classification

Properties

References

Further reading