Space form
In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental 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 [math]\displaystyle{ M^n }[/math] with curvature [math]\displaystyle{ K = -1 }[/math] is isometric to [math]\displaystyle{ H^n }[/math], hyperbolic space, with curvature [math]\displaystyle{ K = 0 }[/math] is isometric to [math]\displaystyle{ R^n }[/math], Euclidean n-space, and with curvature [math]\displaystyle{ K = +1 }[/math] is isometric to [math]\displaystyle{ S^n }[/math], the n-dimensional sphere of points distance 1 from the origin in [math]\displaystyle{ R^{n+1} }[/math].
By rescaling the Riemannian metric on [math]\displaystyle{ H^n }[/math], we may create a space [math]\displaystyle{ M_K }[/math] of constant curvature [math]\displaystyle{ K }[/math] for any [math]\displaystyle{ K \lt 0 }[/math]. Similarly, by rescaling the Riemannian metric on [math]\displaystyle{ S^n }[/math], we may create a space [math]\displaystyle{ M_K }[/math] of constant curvature [math]\displaystyle{ K }[/math] for any [math]\displaystyle{ K \gt 0 }[/math]. Thus the universal cover of a space form [math]\displaystyle{ M }[/math] with constant curvature [math]\displaystyle{ K }[/math] is isometric to [math]\displaystyle{ M_K }[/math].
This reduces the problem of studying space forms to studying discrete groups of isometries [math]\displaystyle{ \Gamma }[/math] of [math]\displaystyle{ M_K }[/math] which act properly discontinuously. Note that the fundamental group of [math]\displaystyle{ M }[/math], [math]\displaystyle{ \pi_1(M) }[/math], will be isomorphic to [math]\displaystyle{ \Gamma }[/math]. Groups acting in this manner on [math]\displaystyle{ R^n }[/math] are called crystallographic groups. Groups acting in this manner on [math]\displaystyle{ H^2 }[/math] and [math]\displaystyle{ H^3 }[/math] are called Fuchsian groups and Kleinian groups, respectively.
See also
References
- Goldberg, Samuel I. (1998), Curvature and Homology, Dover Publications, ISBN 978-0-486-40207-9
- Lee, John M. (1997), Riemannian manifolds: an introduction to curvature, Springer
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