# Constant curvature

curvatureconstantconstant curveconstant positive curvatureconstant sectional curvaturepositive curvatures
In mathematics, constant curvature is a concept from differential geometry.wikipedia
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### Hyperbolic space

hyperbolic 3-spacehyperbolic planehyperbolic 4-space
In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

### Space form

space forms
In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K.

### Mathematics

mathematicalmathmathematician
In mathematics, constant curvature is a concept from differential geometry.

### Differential geometry

differentialdifferential geometerdifferential geometry and topology
In mathematics, constant curvature is a concept from differential geometry.

### Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvature
Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry.

### Manifold

manifoldsboundarymanifold with boundary
Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry.

### Sphere

sphericalhemisphereglobose
For example, a sphere is a surface of constant positive curvature.

### Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
The Riemannian manifolds of constant curvature can be classified into the following three cases:

### Elliptic geometry

ellipticelliptic spaceelliptical geometry

### Euclidean geometry

plane geometryEuclideanEuclidean plane geometry

### Hyperbolic geometry

hyperbolic planehyperbolichyperbolic surface

### Symmetric space

Riemannian symmetric spacesymmetric spaceslocally symmetric space

### Riemann curvature tensor

Riemann tensorcurvature tensorcurvature

### Parallel (geometry)

parallelparallel linesparallelism

### Killing vector field

Killing vectorKilling vector fieldsKilling vectors

### Isometry

isometriesisometricisometrically

### Covering space

universal covercovering mapuniversal covering space

### Flat manifold

flatBieberbach manifoldsflat (n-1)-manifold

### Geodesic manifold

geodesically completegeodesic completeness

### Isomorphism

isomorphicisomorphouscanonical isomorphism

### Metric signature

signatureSignature changeindefinite signature

### Eugenio Beltrami

BeltramiBeltrami, Eugenio
He was the first to prove consistency of non-Euclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, and in the interior of an n-dimensional unit sphere, the so-called Beltrami–Klein model.

### CAT(k) space

CAT(0)CAT(0) spaceCAT(''k'') space
Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature k. In a space, the curvature is bounded from above by k. A notable special case is k=0; complete spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard.

### Uniformization theorem

uniformizationuniformisationUniformisation Theorem
In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature.