# 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.