Space form

Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvature

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

Space form Hyperbolic space Exponential map (Riemannian geometry)Gaussian curvature Cartan–Hadamard theorem

Constant curvature

curvatureconstantconstant curve

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

Hyperbolic space Space form Mathematics Differential geometry Sectional curvature

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.

Space form Riemannian manifold Isometry Quotient space (topology)Sphere

Mathematics

mathematicalmathmathematician

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

Algebra Geometry Calculus Mathematical structure Conjecture

Complete metric space

completecompletioncompleteness

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

Cauchy sequence Metric space Mathematical analysis Banach space Unit interval

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.

Three-dimensional space Dot product Reflection (mathematics)Non-Euclidean geometry Platonic solid

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.

3-sphere Simply connected space Spherical coordinate system Hypersphere Ball (mathematics)

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.

Homogeneous space Hyperbolic geometry Sectional curvature Constant curvature Hyperbolic manifold

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.

Fundamental group Connected space Unit disk N-sphere Riemann sphere

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}.

Local homeomorphism Ramification (mathematics)Fiber bundle Spin group Semi-locally simply connected

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds

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

Manifold Differential geometry Metric tensor Riemannian geometry Differential geometry of surfaces

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.

Topological space Comparison of topologies Open set Isolated point Uniform continuity

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.

Lie group Symmetry group Point group Poincaré group Closure (mathematics)

Isometry

isometriesisometricisometrically

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

Euclidean group Congruence (geometry)Reflection (mathematics)Translation (geometry)Geometric transformation

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.

Euclidean space Permutation Principal homogeneous space Polyhedron 2-transitive group

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.

Algebraic topology Homotopy group Monodromy Group (mathematics)Henri Poincaré

Space group

crystallographic groupspace groupsList of the 230 crystallographic 3D space groups

Groups acting in this manner on R^n are called crystallographic groups.

Symmetry group Wallpaper group Crystal structure Bravais lattice Leonhard Sohncke

Fuchsian group

Fuchsian symmetry groups

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

Fuchsian model Lazarus Fuchs Möbius transformation Discrete group Kleinian group

Kleinian group

Kleinian

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

Schottky group Möbius transformation Hyperbolic 3-manifold Discrete group Kleinian model

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.

Unit interval Smoothness Extreme value theorem Distribution (mathematics)Bounded set

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.

Eilenberg–MacLane space Homotopy group Whitehead conjecture Cartan–Hadamard theorem Essential manifold

Isomorphism

isomorphicisomorphouscanonical isomorphism

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

Automorphism Homomorphism Diffeomorphism Homeomorphism Bijection

Homeomorphism

homeomorphichomeomorphismstopologically equivalent

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

Topology Topological property Isomorphism Topological space Torus

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.

Isomorphism Differentiable manifold Bijection Immersion (mathematics)Manifold

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.

Differential topology Fields Medal Jon Folkman Fary–Milnor theorem Tadatoshi Akiba

Space form

Sectional curvature100% (1/1)

Constant curvature100% (1/1)

Killing–Hopf theorem100% (1/1)

Mathematics50% (1/1)

Complete metric space50% (1/1)

Euclidean space50% (1/1)

N-sphere50% (1/1)

Hyperbolic space50% (1/1)

Simply connected space50% (1/1)

Covering space50% (1/1)

Riemannian manifold50% (1/1)

Discrete space50% (1/1)

Group (mathematics)50% (1/1)

Isometry50% (1/1)

Group action (mathematics)50% (1/1)

Fundamental group50% (1/1)

Space group50% (1/1)

Fuchsian group50% (1/1)

Kleinian group50% (1/1)

Compact space50% (1/1)

Aspherical space50% (1/1)

Isomorphism50% (1/1)

Homeomorphism50% (1/1)

Diffeomorphism50% (1/1)

John Milnor50% (1/1)

Sectional curvature

Constant curvature

Killing–Hopf theorem

Mathematics

Complete metric space

Euclidean space

N-sphere

Hyperbolic space

Simply connected space

Covering space

Riemannian manifold

Discrete space

Group (mathematics)

Isometry

Group action (mathematics)

Fundamental group

Space group

Fuchsian group

Kleinian group

Compact space

Aspherical space

Isomorphism

Homeomorphism

Diffeomorphism

John Milnor