Space form

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

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

Constant curvature

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

Killing–Hopf theorem

The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form M^n with curvature K = -1 is isometric to H^n, hyperbolic space, with curvature K = 0 is isometric to R^n, Euclidean n-space, and with curvature K = +1 is isometric to S^n, the n-dimensional sphere of points distance 1 from the origin in R^{n+1}.
These manifolds are called space forms.

Mathematics

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

Complete metric space

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

Euclidean space

EuclideanspaceEuclidean vector space
The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form M^n with curvature K = -1 is isometric to H^n, hyperbolic space, with curvature K = 0 is isometric to R^n, Euclidean n-space, and with curvature K = +1 is isometric to S^n, the n-dimensional sphere of points distance 1 from the origin in R^{n+1}. The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

N-sphere

n''-spheresphere4-sphere
The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form M^n with curvature K = -1 is isometric to H^n, hyperbolic space, with curvature K = 0 is isometric to R^n, Euclidean n-space, and with curvature K = +1 is isometric to S^n, the n-dimensional sphere of points distance 1 from the origin in R^{n+1}. The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

Hyperbolic space

hyperbolic 3-spacehyperbolic planehyperbolic 4-space
The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form M^n with curvature K = -1 is isometric to H^n, hyperbolic space, with curvature K = 0 is isometric to R^n, Euclidean n-space, and with curvature K = +1 is isometric to S^n, the n-dimensional sphere of points distance 1 from the origin in R^{n+1}. The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

Simply connected space

simply connectedsimply-connectedmultiply connected
The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

Covering space

universal covercovering mapuniversal covering space
The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form M^n with curvature K = -1 is isometric to H^n, hyperbolic space, with curvature K = 0 is isometric to R^n, Euclidean n-space, and with curvature K = +1 is isometric to S^n, the n-dimensional sphere of points distance 1 from the origin in R^{n+1}.

Riemannian manifold

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

Discrete space

discrete topologydiscretediscrete metric
This reduces the problem of studying space forms to studying discrete groups of isometries \Gamma of M_K which act properly discontinuously.

Group (mathematics)

groupgroupsgroup operation
This reduces the problem of studying space forms to studying discrete groups of isometries \Gamma of M_K which act properly discontinuously.

Isometry

isometriesisometricisometrically
This reduces the problem of studying space forms to studying discrete groups of isometries \Gamma of M_K which act properly discontinuously.

Group action (mathematics)

group actionactionorbit
This reduces the problem of studying space forms to studying discrete groups of isometries \Gamma of M_K which act properly discontinuously.

Fundamental group

fundamental groupoidfundamental groupsfirst homotopy group
Note that the fundamental group of M, \pi_1(M), will be isomorphic to \Gamma. The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic fundamental groups are homeomorphic.

Space group

crystallographic groupspace groupsList of the 230 crystallographic 3D space groups
Groups acting in this manner on R^n are called crystallographic groups.

Fuchsian group

Fuchsian symmetry groups
Groups acting in this manner on H^2 and H^3 are called Fuchsian groups and Kleinian groups, respectively.

Kleinian group

Kleinian
Groups acting in this manner on H^2 and H^3 are called Fuchsian groups and Kleinian groups, respectively.

Compact space

compactcompact setcompactness
The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic fundamental groups are homeomorphic.

Aspherical space

asphericalaspherical manifoldasphericity
The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic fundamental groups are homeomorphic.

Isomorphism

isomorphicisomorphouscanonical isomorphism
The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic fundamental groups are homeomorphic.

Homeomorphism

homeomorphichomeomorphismstopologically equivalent
The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic fundamental groups are homeomorphic.

Diffeomorphism

diffeomorphicdiffeomorphismsdiffeomorphism group
One might also wish to conjecture that the manifolds are diffeomorphic, but John Milnor's exotic spheres are all homeomorphic and hence have isomorphic fundamental group, showing this to be false.

John Milnor

MilnorJohn W. MilnorJohn Willard Milnor
One might also wish to conjecture that the manifolds are diffeomorphic, but John Milnor's exotic spheres are all homeomorphic and hence have isomorphic fundamental group, showing this to be false.