# 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**

### 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**

### 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.