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space form ：ウィキペディア英語版

space form
In mathematics, a space form is a complete Riemannian manifold ''M'' of constant sectional curvature ''K''. The three obvious examples are Euclidean ''n''-space, the ''n''-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.
==Reduction to generalized crystallography==

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^

.
By rescaling the Riemannian metric on

H^n

, we may create a space

M_K

of constant curvature

K

for any

K < 0

. Similarly, by rescaling the Riemannian metric on

S^n

, we may create a space

M_K

of constant curvature

K

for any

K > 0

. Thus the universal cover of a space form

M

with constant curvature

K

is isometric to

M_K

.
This reduces the problem of studying space form to studying discrete groups of isometries

\Gamma

M_K

M

\pi_1(M)

, will be isomorphic to

\Gamma

. Groups acting in this manner on

R^n

are called crystallographic groups. Groups acting in this manner on

H^2

and

H^3

are called Fuchsian groups and Kleinian groups, respectively.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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