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Connected complete Riemannian spaces of constant curvature (cf. [[Complete Riemannian space|Complete Riemannian space]]). The problem of classifying the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861901.png" />-dimensional Riemannian spaces of arbitrary constant curvature was formulated by W. Killing (1891), who called it the Clifford–Klein problem of space forms. The contemporary formulation of this problem is due to H. Hopf (1925).
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Connected complete Riemannian spaces of constant curvature (cf. [[Complete Riemannian space|Complete Riemannian space]]). The problem of classifying the $  n $-
 +
dimensional Riemannian spaces of arbitrary constant curvature was formulated by W. Killing (1891), who called it the Clifford–Klein problem of space forms. The contemporary formulation of this problem is due to H. Hopf (1925).
  
 
===Examples of space forms.===
 
===Examples of space forms.===
The Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861902.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861903.png" /> is a space form of zero curvature (a so-called flat space); the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861904.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861905.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861906.png" /> is a space form of positive curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861907.png" />; the Lobachevskii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861908.png" /> (a hyperbolic space) is a space form of negative curvature; the flat torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s0861909.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619010.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619011.png" />-dimensional lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619012.png" />, is a space form of zero curvature (a flat space).
+
The Euclidean space $  E  ^ {n} $
 +
of dimension $  n $
 +
is a space form of zero curvature (a so-called flat space); the sphere $  S  ^ {n} $
 +
in $  E  ^ {n+} 1 $
 +
of radius $  r > 0 $
 +
is a space form of positive curvature $  1 / r  ^ {2} $;  
 +
the Lobachevskii space $  \Lambda  ^ {n} $(
 +
a hyperbolic space) is a space form of negative curvature; the flat torus $  T  ^ {n} = E  ^ {n} / \Gamma $,  
 +
where $  \Gamma $
 +
is an $  n $-
 +
dimensional lattice in $  E  ^ {n} $,  
 +
is a space form of zero curvature (a flat space).
  
Any space form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619013.png" /> of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619014.png" /> can be obtained from a simply-connected space form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619015.png" /> of the same curvature by factorization with respect to a discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619016.png" /> of freely-acting motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619017.png" /> (i.e. acting fixed-point free). Two spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619019.png" /> are, moreover, isometric if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619021.png" /> are conjugate in the group of all motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619022.png" />. Thus, the problem of classifying space forms reduces to the problem of describing all non-conjugate groups of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619025.png" />, acting freely. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619026.png" /> is called a spherical space form if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619027.png" />, a Euclidean space form if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619028.png" /> and a hyperbolic space form if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619029.png" />; the fundamental group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619030.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619031.png" />. In the study of the classification problem of space forms of non-zero curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619032.png" />, only the sign of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619033.png" /> plays a significant role, so one usually puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619034.png" />.
+
Any space form $  M  ^ {n} $
 +
of curvature $  \sigma $
 +
can be obtained from a simply-connected space form $  \widetilde{M}  {}  ^ {n} $
 +
of the same curvature by factorization with respect to a discrete group $  \Gamma $
 +
of freely-acting motions of $  \widetilde{M}  {}  ^ {n} $(
 +
i.e. acting fixed-point free). Two spaces $  M  ^ {n} = \widetilde{M}  {}  ^ {n} / \Gamma $
 +
and $  M _ {1}  ^ {n} = \widetilde{M}  {}  ^ {n} / \Gamma _ {1} $
 +
are, moreover, isometric if and only if $  \Gamma $
 +
and $  \Gamma _ {1} $
 +
are conjugate in the group of all motions of $  \widetilde{M}  {}  ^ {n} $.  
 +
Thus, the problem of classifying space forms reduces to the problem of describing all non-conjugate groups of motions of $  S  ^ {n} $,  
 +
$  E  ^ {n} $
 +
or $  \Lambda  ^ {n} $,  
 +
acting freely. A space $  M  ^ {n} $
 +
is called a spherical space form if $  M  ^ {n} = S  ^ {n} / \Gamma $,  
 +
a Euclidean space form if $  M  ^ {n} = E  ^ {n} / \Gamma $
 +
and a hyperbolic space form if $  M  ^ {n} = \Lambda  ^ {n} / \Gamma $;  
 +
the fundamental group of $  M  ^ {n} $
 +
is isomorphic to $  \Gamma $.  
 +
In the study of the classification problem of space forms of non-zero curvature $  \sigma $,  
 +
only the sign of $  \sigma $
 +
plays a significant role, so one usually puts $  | \sigma | = 1 $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619035.png" /> is even, then the only motions of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619036.png" /> without fixed points are the central symmetries, mapping each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619037.png" /> into the diametrically-opposite point. The quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619038.png" /> by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619039.png" /> generated by these motions is an elliptic space. Any spherical space form of even dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619040.png" /> is isometric to either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619041.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619042.png" />. The three-dimensional spherical space forms have been classified (cf. [[#References|[2]]]). The next step in the direction of classifying spherical space forms is a general program for solving this problem, as well as its applications to the classification of spherical space forms of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619043.png" /> (cf. [[#References|[4]]]). Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619044.png" /> is compact, and the discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619045.png" /> of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619046.png" /> is finite, in order to classify <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619047.png" />-dimensional spherical space forms it is sufficient to describe all non-conjugate finite subgroups of the orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619048.png" /> acting freely on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619049.png" />. One says that an orthogonal representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619050.png" /> of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619051.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619052.png" /> is fixed-point free if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619053.png" /> the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619054.png" /> of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619055.png" /> is fixed-point free, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619056.png" /> is a [[Faithful representation|faithful representation]]. According to the program developed in [[#References|[4]]], the solution of the Clifford–Klein problem for spherical space forms can be subdivided into several stages. Firstly, one has to find necessary and sufficient conditions on an abstract group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619057.png" /> so that it be the [[Fundamental group|fundamental group]] of a spherical space form, and to classify such groups; one obtains a certain family of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619058.png" />. Secondly, one has to describe all inequivalent irreducible orthogonal representations of each group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619059.png" />, and distinguish among them those representations that are fixed-point free. Finally, one has to determine all automorphisms of the groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619060.png" /> and to clarify which of the representations found are equivalent modulo the automorphisms of the corresponding group. This program has been realized completely in [[#References|[5]]], and has led to an exhaustive classification of spherical space forms. Any finite cyclic group belongs to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619061.png" />; a non-cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619062.png" /> is the fundamental group of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619063.png" />-dimensional spherical space form if (but not only if) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619064.png" /> is relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619065.png" /> and is divisible by the square of an integer.
+
If $  n $
 +
is even, then the only motions of the sphere $  S  ^ {n} $
 +
without fixed points are the central symmetries, mapping each point of $  S  ^ {n} $
 +
into the diametrically-opposite point. The quotient space $  S  ^ {n} / \Gamma $
 +
by the group $  \Gamma $
 +
generated by these motions is an elliptic space. Any spherical space form of even dimension $  n $
 +
is isometric to either $  S  ^ {n} $
 +
or $  P  ^ {n} $.  
 +
The three-dimensional spherical space forms have been classified (cf. [[#References|[2]]]). The next step in the direction of classifying spherical space forms is a general program for solving this problem, as well as its applications to the classification of spherical space forms of dimension $  4 k + 1 $(
 +
cf. [[#References|[4]]]). Since $  S  ^ {n} $
 +
is compact, and the discrete group $  \Gamma $
 +
of motions of $  S  ^ {n} $
 +
is finite, in order to classify $  n $-
 +
dimensional spherical space forms it is sufficient to describe all non-conjugate finite subgroups of the orthogonal group $  O ( n + 1 ) $
 +
acting freely on $  S  ^ {n} $.  
 +
One says that an orthogonal representation $  \pi $
 +
of a finite group $  G $
 +
in $  E  ^ {n+} 1 $
 +
is fixed-point free if for all $  g \in G \setminus  \{ 1 \} $
 +
the transformation $  \pi ( g) $
 +
of the sphere $  S  ^ {n} $
 +
is fixed-point free, in particular, $  \pi $
 +
is a [[Faithful representation|faithful representation]]. According to the program developed in [[#References|[4]]], the solution of the Clifford–Klein problem for spherical space forms can be subdivided into several stages. Firstly, one has to find necessary and sufficient conditions on an abstract group $  G $
 +
so that it be the [[Fundamental group|fundamental group]] of a spherical space form, and to classify such groups; one obtains a certain family of groups $  \{ G _  \lambda  \} $.  
 +
Secondly, one has to describe all inequivalent irreducible orthogonal representations of each group in $  \{ G _  \lambda  \} $,  
 +
and distinguish among them those representations that are fixed-point free. Finally, one has to determine all automorphisms of the groups in $  \{ G _  \lambda  \} $
 +
and to clarify which of the representations found are equivalent modulo the automorphisms of the corresponding group. This program has been realized completely in [[#References|[5]]], and has led to an exhaustive classification of spherical space forms. Any finite cyclic group belongs to the family $  \{ G _  \lambda  \} $;  
 +
a non-cyclic group of order $  N $
 +
is the fundamental group of an $  n $-
 +
dimensional spherical space form if (but not only if) $  N $
 +
is relatively prime to $  n + 1 $
 +
and is divisible by the square of an integer.
  
The global theory of Euclidean space forms arose as an application of some results in geometric crystallography (cf. [[Crystallography, mathematical|Crystallography, mathematical]]). In [[#References|[3]]] the list of crystallographic groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619066.png" /> known at the end of the 19th century was used to obtain a topological, and in the compact case an affine, classification of three-dimensional Euclidean space forms. Bieberbach's theorems on crystallographic groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619067.png" /> led to the structure theory of compact Euclidean space forms of arbitrary dimensions. In particular, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619068.png" /> there is only a finite number of different equivalence classes of compact Euclidean space forms of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619069.png" />; moreover, two compact Euclidean space forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619071.png" /> are affinely equivalent if and only if their fundamental groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619073.png" /> are isomorphic. E.g., any two-dimensional compact Euclidean space form is homeomorphic (hence, affinely equivalent) either to a flat torus or to the Klein bottle. An abstract group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619074.png" /> is the fundamental group of a compact Euclidean space form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619075.png" /> if and only if: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619076.png" /> has a normal Abelian subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619077.png" /> of finite index isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619078.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619079.png" /> coincides with the centralizer subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619080.png" />; and c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619081.png" /> does not have elements of finite order. If such a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619082.png" /> is realized as a discrete subgroup in the group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619083.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619084.png" /> coincides with the set of translations belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619085.png" />, and there is a normal covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619086.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619087.png" /> by the flat torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619088.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619089.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619090.png" />. The finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619091.png" /> is isomorphic to the group of covering transformations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619092.png" />, which is, in turn, isomorphic to the [[Holonomy group|holonomy group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619093.png" />. A compact Euclidean space form always has a finite homology group. The converse statement also holds: A compact Riemannian space whose holonomy group is finite is flat. It has been proved that every finite group is isomorphic to the holonomy group of a compact Euclidean space form. The affine classification of compact Euclidean space forms of a given dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619094.png" /> is known (1983) only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619095.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619096.png" /> there are 6 orientable and 4 non-orientable classes of affinely-equivalent compact Euclidean space forms. The compact Euclidean space forms with a cyclic holonomy group of prime order have been classified. The family of non-isometric flat tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619097.png" /> can be parametrized by the elements of
+
The global theory of Euclidean space forms arose as an application of some results in geometric crystallography (cf. [[Crystallography, mathematical|Crystallography, mathematical]]). In [[#References|[3]]] the list of crystallographic groups in $  E  ^ {3} $
 +
known at the end of the 19th century was used to obtain a topological, and in the compact case an affine, classification of three-dimensional Euclidean space forms. Bieberbach's theorems on crystallographic groups in $  E  ^ {3} $
 +
led to the structure theory of compact Euclidean space forms of arbitrary dimensions. In particular, for any $  n \geq  2 $
 +
there is only a finite number of different equivalence classes of compact Euclidean space forms of dimension $  n $;  
 +
moreover, two compact Euclidean space forms $  M  ^ {n} = E  ^ {n} / \Gamma $
 +
and $  M _ {1}  ^ {n} = E  ^ {n} / \Gamma _ {1} $
 +
are affinely equivalent if and only if their fundamental groups $  \Gamma $
 +
and $  \Gamma _ {1} $
 +
are isomorphic. E.g., any two-dimensional compact Euclidean space form is homeomorphic (hence, affinely equivalent) either to a flat torus or to the Klein bottle. An abstract group $  \Gamma $
 +
is the fundamental group of a compact Euclidean space form $  M  ^ {n} $
 +
if and only if: a) $  \Gamma $
 +
has a normal Abelian subgroup $  \Gamma  ^ {*} $
 +
of finite index isomorphic to $  \mathbf Z  ^ {n} $;  
 +
b) $  \Gamma  ^ {*} $
 +
coincides with the centralizer subgroup in $  \Gamma $;  
 +
and c) $  \Gamma $
 +
does not have elements of finite order. If such a group $  \Gamma $
 +
is realized as a discrete subgroup in the group of motions of $  E  ^ {n} $,  
 +
then $  \Gamma  ^ {*} $
 +
coincides with the set of translations belonging to $  \Gamma $,  
 +
and there is a normal covering $  p $
 +
of $  M  ^ {n} = E  ^ {n} / \Gamma $
 +
by the flat torus $  T  ^ {n} = E  ^ {n} / \Gamma  ^ {*} $,  
 +
defined by $  p ( \Gamma  ^ {*} ( x) ) = \Gamma ( x) $
 +
for all $  x \in E  ^ {n} $.  
 +
The finite group $  \Gamma / \Gamma  ^ {*} $
 +
is isomorphic to the group of covering transformations for $  p $,  
 +
which is, in turn, isomorphic to the [[Holonomy group|holonomy group]] of $  M  ^ {n} $.  
 +
A compact Euclidean space form always has a finite homology group. The converse statement also holds: A compact Riemannian space whose holonomy group is finite is flat. It has been proved that every finite group is isomorphic to the holonomy group of a compact Euclidean space form. The affine classification of compact Euclidean space forms of a given dimension $  n $
 +
is known (1983) only for $  n \leq  4 $.  
 +
For $  n = 3 $
 +
there are 6 orientable and 4 non-orientable classes of affinely-equivalent compact Euclidean space forms. The compact Euclidean space forms with a cyclic holonomy group of prime order have been classified. The family of non-isometric flat tori $  T  ^ {n} $
 +
can be parametrized by the elements of
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619098.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm SL} ( n , \mathbf Z ) \setminus
 +
\mathop{\rm GL}  ^ {+} ( n , \mathbf R ) / \mathop{\rm SO} ( n) .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s08619099.png" /> is the connected component of the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190100.png" />. The isometric classification of compact Euclidean space forms of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190101.png" /> immediately follows from their affine classification and the isometric classification of the tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190102.png" />. Non-compact Euclidean space forms have been classified (up to an isometry) only in dimensions 2 and 3. In particular, a two-dimensional non-compact Euclidean space form, different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190103.png" />, is homeomorphic to either a cylinder or the Möbius strip. Any non-compact Euclidean space form admits a real-analytic retraction onto a compact totally-geodesic flat submanifold; the class of fundamental groups of non-compact Euclidean space forms coincides with the class of fundamental groups of compact Euclidean space forms.
+
Here $  \mathop{\rm GL}  ^ {+} ( n , \mathbf R ) $
 +
is the connected component of the identity in $  \mathop{\rm GL} ( n , \mathbf R ) $.  
 +
The isometric classification of compact Euclidean space forms of dimension $  n $
 +
immediately follows from their affine classification and the isometric classification of the tori $  T  ^ {n} $.  
 +
Non-compact Euclidean space forms have been classified (up to an isometry) only in dimensions 2 and 3. In particular, a two-dimensional non-compact Euclidean space form, different from $  E  ^ {2} $,  
 +
is homeomorphic to either a cylinder or the Möbius strip. Any non-compact Euclidean space form admits a real-analytic retraction onto a compact totally-geodesic flat submanifold; the class of fundamental groups of non-compact Euclidean space forms coincides with the class of fundamental groups of compact Euclidean space forms.
  
The study of two-dimensional hyperbolic space forms essentially began in 1888, when H. Poincaré, [[#References|[1]]], studied discrete groups of fractional-linear transformations of the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190104.png" /> of the complex plane (Fuchsian groups, cf. [[Fuchsian group|Fuchsian group]]) and noted that they can be treated as the groups of motions of the hyperbolic plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190105.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190106.png" /> be the group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190107.png" /> preserving orientation; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190109.png" />, be a convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190110.png" />-gon in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190111.png" /> with pairwise-congruent geodesic sides
+
The study of two-dimensional hyperbolic space forms essentially began in 1888, when H. Poincaré, [[#References|[1]]], studied discrete groups of fractional-linear transformations of the upper half-plane $  \mathop{\rm Im}  z > 0 $
 +
of the complex plane (Fuchsian groups, cf. [[Fuchsian group|Fuchsian group]]) and noted that they can be treated as the groups of motions of the hyperbolic plane $  \Lambda  ^ {2} $.  
 +
Let $  {\mathcal L} $
 +
be the group of motions of $  \Lambda  ^ {2} $
 +
preserving orientation; let $  A _ {1} \dots A _ {4m} $,  
 +
$  m \geq  2 $,  
 +
be a convex $  4 m $-
 +
gon in $  \Lambda  ^ {2} $
 +
with pairwise-congruent geodesic sides
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190112.png" /></td> </tr></table>
+
$$
 +
A _ {4i-} 3 A _ {4i-} 2  = \
 +
A _ {4i-} 1 A _ {4i} ,\ \
 +
A _ {4i-} 2 A _ {4i-} 1  = \
 +
A _ {4i} A _ {4i+} 1 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190114.png" />, and the sum of the angles of which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190115.png" />. The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190117.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190118.png" /> map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190119.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190121.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190122.png" />, respectively (the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190123.png" /> is drawn in the Fig.).
+
where $  i = 1 \dots m $,  
 +
$  A _ {4m+} 1 = A _ {1} $,  
 +
and the sum of the angles of which is $  2 \pi $.  
 +
The elements $  a _ {i} $
 +
and $  b _ {i} $
 +
in $  {\mathcal L} $
 +
map $  A _ {4i-} 3 A _ {4i-} 2 $
 +
to $  A _ {4i} A _ {4i-} 1 $
 +
and $  A _ {4i-} 2 A _ {4i-} 1 $
 +
to $  A _ {4i-} 1 A _ {4i} $,  
 +
respectively (the case $  m = 2 $
 +
is drawn in the Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s086190a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s086190a.gif" />
Line 24: Line 164:
 
Figure: s086190a
 
Figure: s086190a
  
The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190124.png" /> generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190125.png" /> then acts fixed-point free on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190126.png" />, and the given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190127.png" />-gon is the fundamental domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190128.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190129.png" /> has the unique defining relation
+
The subgroup $  \Gamma \subset  {\mathcal L} $
 +
generated by the $  a _ {i} , b _ {i} $
 +
then acts fixed-point free on $  \Lambda  ^ {2} $,  
 +
and the given $  4 m $-
 +
gon is the fundamental domain of $  \Gamma $.  
 +
Moreover, $  \Gamma $
 +
has the unique defining relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190130.png" /></td> </tr></table>
+
$$
 +
\prod _ { i= } 1 ^ { m }
 +
[ a _ {i} , b _ {i} ]  = 1 .
 +
$$
  
The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190131.png" /> is an orientable compact hyperbolic space form of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190132.png" />, and every two-dimensional orientable compact hyperbolic space form can be obtained in this way. Suppose now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190133.png" /> is an abstract group isomorphic to the fundamental group of an orientable closed surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190134.png" />. Then there is a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190135.png" /> satisfying the conditions: a) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190136.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190137.png" /> is a monomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190138.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190139.png" />; b) two subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190140.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190141.png" /> are conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190142.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190143.png" />; and c) if a discrete subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190144.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190145.png" />, then it is conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190146.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190147.png" />. Thus, the family of non-isomorphic compact hyperbolic space forms of dimension 2 and genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190148.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190149.png" /> real parameters. A two-dimensional compact hyperbolic space form can be naturally endowed with the structure of a [[Riemann surface|Riemann surface]], and the statement just formulated was originally proved by tools of the theory of uniformization; a geometric proof was given in [[#References|[7]]]. The results given can be generalized to non-compact hyperbolic space forms, which are homeomorphic to a sphere with a finite number of handles and holes, as well as to non-oriented hyperbolic space forms of dimension 2. Contrary to the two-dimensional case, there do not exist continuous families of non-isometric compact hyperbolic space forms of dimension exceeding two. More precisely, compact hyperbolic space forms of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190150.png" /> having isomorphic fundamental groups are isomorphic. Other general results, immediately related to the classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190151.png" />-dimensional hyperbolic space forms, do not exist (1983); examples of hyperbolic space forms of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190152.png" /> have been given in [[#References|[6]]] and [[#References|[8]]].
+
The quotient group $  \Lambda  ^ {2} / \Gamma $
 +
is an orientable compact hyperbolic space form of genus $  m $,  
 +
and every two-dimensional orientable compact hyperbolic space form can be obtained in this way. Suppose now that $  \Gamma $
 +
is an abstract group isomorphic to the fundamental group of an orientable closed surface of genus $  m $.  
 +
Then there is a continuous mapping $  \phi : \Gamma \times \mathbf R  ^ {6m-} 6 \rightarrow {\mathcal L} $
 +
satisfying the conditions: a) for all $  x \in \mathbf R  ^ {6m-} 6 $
 +
the mapping $  \phi _ {x} : g \mapsto \phi ( g , x ) $
 +
is a monomorphism of $  \Gamma $
 +
into $  h $;  
 +
b) two subgroups $  \Gamma _ {x} = \phi _ {x} ( \Gamma ) $
 +
and $  \Gamma _ {y} = \phi _ {y} ( \Gamma ) $
 +
are conjugate in $  {\mathcal L} $
 +
if and only if $  x = y $;  
 +
and c) if a discrete subgroup $  \Gamma _ {1} \subset  {\mathcal L} $
 +
is isomorphic to $  \Gamma $,  
 +
then it is conjugate to $  \Gamma _ {x} $
 +
for some $  x \in \mathbf R  ^ {6m-} 6 $.  
 +
Thus, the family of non-isomorphic compact hyperbolic space forms of dimension 2 and genus $  m $
 +
depends on $  6 m - 6 $
 +
real parameters. A two-dimensional compact hyperbolic space form can be naturally endowed with the structure of a [[Riemann surface|Riemann surface]], and the statement just formulated was originally proved by tools of the theory of uniformization; a geometric proof was given in [[#References|[7]]]. The results given can be generalized to non-compact hyperbolic space forms, which are homeomorphic to a sphere with a finite number of handles and holes, as well as to non-oriented hyperbolic space forms of dimension 2. Contrary to the two-dimensional case, there do not exist continuous families of non-isometric compact hyperbolic space forms of dimension exceeding two. More precisely, compact hyperbolic space forms of dimension $  n \geq  3 $
 +
having isomorphic fundamental groups are isomorphic. Other general results, immediately related to the classification of $  n $-
 +
dimensional hyperbolic space forms, do not exist (1983); examples of hyperbolic space forms of dimension $  \geq  3 $
 +
have been given in [[#References|[6]]] and [[#References|[8]]].
  
 
Besides the Riemannian space forms their generalization have also been studied: pseudo-Riemannian, affine and complex space forms, as well as space forms of symmetric spaces (cf., e.g., [[#References|[9]]]).
 
Besides the Riemannian space forms their generalization have also been studied: pseudo-Riemannian, affine and complex space forms, as well as space forms of symmetric spaces (cf., e.g., [[#References|[9]]]).
Line 34: Line 205:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Oeuvres" , '''3''' , Gauthier-Villars  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Threlfall,  H. Seifert,  "Topologische Untersuchungen der Diskontinuitätsbereiche endlicher Bewegungsgruppen der dreidimensionalen sphärischen Raumes"  ''Math. Ann.'' , '''104'''  (1931)  pp. 1–70</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Nowacki,  "Euklidischen, dreidimensionalen, geschlossenen und offenen Raumformen"  ''Comm. Math. Helvetica'' , '''7'''  (1934)  pp. 81–93</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Vincent,  "Les groupes linéaires finis sans points fixes"  ''Comm. Math. Helvetica'' , '''20'''  (1947)  pp. 117–171</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.A. Wolf,  "Spaces of constant curvature" , Publish or Perish  (1984)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.B. Vinberg,  "Some examples of crystallographic groups in Lobachevskii spaces"  ''Math. USSR Sb.'' , '''7'''  (1969)  pp. 617–622  ''Mat. Sb.'' , '''78''' :  4  (1969)  pp. 633–639</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.M. Natanzon,  "Invariant lines on Fuchsian groups"  ''Russian Math. Surveys'' , '''27''' :  4  (1972)  pp. 161–177  ''Uspekhi Mat. Nauk'' :  4  (1972)  pp. 145–160</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.J. Millson,  "On the first Betti number of a constant negatively cuved manifold"  ''Ann. of Math.'' , '''104'''  (1976)  pp. 235–247</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Borel,  "Compact Clifford–Klein forms of symmetric spaces"  ''Topology'' , '''2'''  (1963)  pp. 111–122</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Oeuvres" , '''3''' , Gauthier-Villars  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Threlfall,  H. Seifert,  "Topologische Untersuchungen der Diskontinuitätsbereiche endlicher Bewegungsgruppen der dreidimensionalen sphärischen Raumes"  ''Math. Ann.'' , '''104'''  (1931)  pp. 1–70</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Nowacki,  "Euklidischen, dreidimensionalen, geschlossenen und offenen Raumformen"  ''Comm. Math. Helvetica'' , '''7'''  (1934)  pp. 81–93</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Vincent,  "Les groupes linéaires finis sans points fixes"  ''Comm. Math. Helvetica'' , '''20'''  (1947)  pp. 117–171</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.A. Wolf,  "Spaces of constant curvature" , Publish or Perish  (1984)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.B. Vinberg,  "Some examples of crystallographic groups in Lobachevskii spaces"  ''Math. USSR Sb.'' , '''7'''  (1969)  pp. 617–622  ''Mat. Sb.'' , '''78''' :  4  (1969)  pp. 633–639</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.M. Natanzon,  "Invariant lines on Fuchsian groups"  ''Russian Math. Surveys'' , '''27''' :  4  (1972)  pp. 161–177  ''Uspekhi Mat. Nauk'' :  4  (1972)  pp. 145–160</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.J. Millson,  "On the first Betti number of a constant negatively cuved manifold"  ''Ann. of Math.'' , '''104'''  (1976)  pp. 235–247</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Borel,  "Compact Clifford–Klein forms of symmetric spaces"  ''Topology'' , '''2'''  (1963)  pp. 111–122</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A group that satisfies the three conditions a), b), c) above for being the fundamental group of a compact Euclidean space form is called a Bieberbach group.
 
A group that satisfies the three conditions a), b), c) above for being the fundamental group of a compact Euclidean space form is called a Bieberbach group.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190153.png" /> be the group of rigid motions of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190154.png" />, i.e. the group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190155.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190156.png" />, the [[Orthogonal group|orthogonal group]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190157.png" />, a translation. There is an exact sequence
+
Let $  {\mathcal R} _ {n} $
 +
be the group of rigid motions of the Euclidean space $  E  ^ {n} $,  
 +
i.e. the group of transformations $  ( m, s) x = mx+ s $
 +
with $  m \in O _ {n} $,  
 +
the [[Orthogonal group|orthogonal group]], and s \in E  ^ {n} $,  
 +
a translation. There is an exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190158.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  T _ {n}  \rightarrow  {\mathcal R} _ {n}  \rightarrow ^ { r }  O _ {n}  \rightarrow  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190159.png" /> is the subgroup of pure translations: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190160.png" />. This is a semi-direct product. An isotropic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190161.png" /> is a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190162.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190163.png" /> spans all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190164.png" />. A uniform subgroup is one such that the orbit space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190165.png" /> is compact; finally, a direct subgroup is one which is discrete as a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190166.png" />. A crystallographic subgroup is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190167.png" /> that is uniform and discrete, and a Bieberbach subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190168.png" /> is a torsion-free crystallographic subgroup. The crystallographic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190169.png" /> are also known as the space groups. Cf. also [[Crystallographic group|Crystallographic group]]. An (abstract) crystallographic group is a group that contains a finitely-generated Abelian torsion-free subgroup of finite index. An (abstract) Bieberbach group is a torsion-free crystallographic subgroup. An Auslander–Kuranishi theorem says that each crystallographic group arises as a crystallographic subgroup of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190170.png" />, and hence that each Bieberbach group arises as a Bieberbach subgroup. A second Auslander–Kuranishi theorem says that for any finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190171.png" /> there is a Bieberbach group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190172.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190173.png" />, and that any finite group arises as a holonomy group of a compact Euclidean space form (cf. above). The three Bieberbach theorems on crystallographic subgroups are as follows: i) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190174.png" /> is a crystallographic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190175.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190176.png" /> is finite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190177.png" /> is isotropic; ii) any isomorphism of crystallographic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190178.png" /> can be realized by an affine change of coordinates: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190179.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190181.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190182.png" />; iii) up to affine coordinate changes there are only finitely many crystallographic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190183.png" />. These last two statements readily lead to corresponding statements concerning Euclidean space forms, as in the main article above.
+
where $  T _ {n} $
 +
is the subgroup of pure translations: $  r( m, s)= m $.  
 +
This is a semi-direct product. An isotropic subgroup of $  {\mathcal R} _ {n} $
 +
is a subgroup $  \pi $
 +
such that $  \pi \cap T _ {n} $
 +
spans all of $  E  ^ {n} $.  
 +
A uniform subgroup is one such that the orbit space $  E  ^ {n} / \pi $
 +
is compact; finally, a direct subgroup is one which is discrete as a subspace of $  {\mathcal R} _ {n} $.  
 +
A crystallographic subgroup is a subgroup of $  {\mathcal R} _ {n} $
 +
that is uniform and discrete, and a Bieberbach subgroup of $  {\mathcal R} _ {n} $
 +
is a torsion-free crystallographic subgroup. The crystallographic subgroups of $  {\mathcal R} _ {3} $
 +
are also known as the space groups. Cf. also [[Crystallographic group|Crystallographic group]]. An (abstract) crystallographic group is a group that contains a finitely-generated Abelian torsion-free subgroup of finite index. An (abstract) Bieberbach group is a torsion-free crystallographic subgroup. An Auslander–Kuranishi theorem says that each crystallographic group arises as a crystallographic subgroup of an $  {\mathcal R} _ {n} $,  
 +
and hence that each Bieberbach group arises as a Bieberbach subgroup. A second Auslander–Kuranishi theorem says that for any finite group $  \pi  ^  \prime  $
 +
there is a Bieberbach group $  \pi $
 +
such that $  r ( \pi ) = \pi  ^  \prime  $,  
 +
and that any finite group arises as a holonomy group of a compact Euclidean space form (cf. above). The three Bieberbach theorems on crystallographic subgroups are as follows: i) if $  \pi $
 +
is a crystallographic subgroup of $  {\mathcal R} _ {n} $,  
 +
then $  r( \pi ) $
 +
is finite and $  \pi $
 +
is isotropic; ii) any isomorphism of crystallographic subgroups of $  {\mathcal R} _ {n} $
 +
can be realized by an affine change of coordinates: $  \beta \mapsto \alpha \beta \alpha  ^ {-} 1 $,
 +
$  \alpha = ( m, s) $,  
 +
$  m \in  \mathop{\rm GL} _ {n} ( \mathbf R ) $,  
 +
s \in E  ^ {n} $;  
 +
iii) up to affine coordinate changes there are only finitely many crystallographic subgroups of $  {\mathcal R} _ {n} $.  
 +
These last two statements readily lead to corresponding statements concerning Euclidean space forms, as in the main article above.
  
The numbers of crystallographic and Bieberbach subgroups (up to isomorphism) in the first few dimensions are as follows.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">dimension</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1">3</td> <td colname="5" style="background-color:white;" colspan="1">4</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"># crystallographic subgroups</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">17</td> <td colname="4" style="background-color:white;" colspan="1">219</td> <td colname="5" style="background-color:white;" colspan="1">4783</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"># Bieberbach subgroups</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1">10</td> <td colname="5" style="background-color:white;" colspan="1">74</td> </tr> </tbody> </table>
+
The numbers of crystallographic and Bieberbach subgroups (up to isomorphism) in the first few dimensions are as follows.<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">dimension</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1">3</td> <td colname="5" style="background-color:white;" colspan="1">4</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"># crystallographic subgroups</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">17</td> <td colname="4" style="background-color:white;" colspan="1">219</td> <td colname="5" style="background-color:white;" colspan="1">4783</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"># Bieberbach subgroups</td> <td colname="2" style="background-color:white;" colspan="1">1</td> <td colname="3" style="background-color:white;" colspan="1">2</td> <td colname="4" style="background-color:white;" colspan="1">10</td> <td colname="5" style="background-color:white;" colspan="1">74</td> </tr> </tbody> </table>
  
 
</td></tr> </table>
 
</td></tr> </table>
  
If one considers the crystallographic groups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190184.png" /> up to orientation preserving affine conjugacy, the more familiar number of 230 equivalence classes arises. (The 230 space groups.)
+
If one considers the crystallographic groups in $  {\mathcal R} _ {3} $
 +
up to orientation preserving affine conjugacy, the more familiar number of 230 equivalence classes arises. (The 230 space groups.)
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.S. Charlap,  "Bieberbach groups and flat manifolds" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Auslander,  M. Kuranishi,  "On the holonomy groups of locally Euclidean spaces"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 411</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.L.E. Schwarzenberger,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190185.png" />-dimensional crystallography" , Pitman  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.S. Charlap,  "Bieberbach groups and flat manifolds" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Auslander,  M. Kuranishi,  "On the holonomy groups of locally Euclidean spaces"  ''Ann. of Math.'' , '''65'''  (1957)  pp. 411</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.L.E. Schwarzenberger,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086190/s086190185.png" />-dimensional crystallography" , Pitman  (1980)</TD></TR></table>

Latest revision as of 08:22, 6 June 2020


Connected complete Riemannian spaces of constant curvature (cf. Complete Riemannian space). The problem of classifying the $ n $- dimensional Riemannian spaces of arbitrary constant curvature was formulated by W. Killing (1891), who called it the Clifford–Klein problem of space forms. The contemporary formulation of this problem is due to H. Hopf (1925).

Examples of space forms.

The Euclidean space $ E ^ {n} $ of dimension $ n $ is a space form of zero curvature (a so-called flat space); the sphere $ S ^ {n} $ in $ E ^ {n+} 1 $ of radius $ r > 0 $ is a space form of positive curvature $ 1 / r ^ {2} $; the Lobachevskii space $ \Lambda ^ {n} $( a hyperbolic space) is a space form of negative curvature; the flat torus $ T ^ {n} = E ^ {n} / \Gamma $, where $ \Gamma $ is an $ n $- dimensional lattice in $ E ^ {n} $, is a space form of zero curvature (a flat space).

Any space form $ M ^ {n} $ of curvature $ \sigma $ can be obtained from a simply-connected space form $ \widetilde{M} {} ^ {n} $ of the same curvature by factorization with respect to a discrete group $ \Gamma $ of freely-acting motions of $ \widetilde{M} {} ^ {n} $( i.e. acting fixed-point free). Two spaces $ M ^ {n} = \widetilde{M} {} ^ {n} / \Gamma $ and $ M _ {1} ^ {n} = \widetilde{M} {} ^ {n} / \Gamma _ {1} $ are, moreover, isometric if and only if $ \Gamma $ and $ \Gamma _ {1} $ are conjugate in the group of all motions of $ \widetilde{M} {} ^ {n} $. Thus, the problem of classifying space forms reduces to the problem of describing all non-conjugate groups of motions of $ S ^ {n} $, $ E ^ {n} $ or $ \Lambda ^ {n} $, acting freely. A space $ M ^ {n} $ is called a spherical space form if $ M ^ {n} = S ^ {n} / \Gamma $, a Euclidean space form if $ M ^ {n} = E ^ {n} / \Gamma $ and a hyperbolic space form if $ M ^ {n} = \Lambda ^ {n} / \Gamma $; the fundamental group of $ M ^ {n} $ is isomorphic to $ \Gamma $. In the study of the classification problem of space forms of non-zero curvature $ \sigma $, only the sign of $ \sigma $ plays a significant role, so one usually puts $ | \sigma | = 1 $.

If $ n $ is even, then the only motions of the sphere $ S ^ {n} $ without fixed points are the central symmetries, mapping each point of $ S ^ {n} $ into the diametrically-opposite point. The quotient space $ S ^ {n} / \Gamma $ by the group $ \Gamma $ generated by these motions is an elliptic space. Any spherical space form of even dimension $ n $ is isometric to either $ S ^ {n} $ or $ P ^ {n} $. The three-dimensional spherical space forms have been classified (cf. [2]). The next step in the direction of classifying spherical space forms is a general program for solving this problem, as well as its applications to the classification of spherical space forms of dimension $ 4 k + 1 $( cf. [4]). Since $ S ^ {n} $ is compact, and the discrete group $ \Gamma $ of motions of $ S ^ {n} $ is finite, in order to classify $ n $- dimensional spherical space forms it is sufficient to describe all non-conjugate finite subgroups of the orthogonal group $ O ( n + 1 ) $ acting freely on $ S ^ {n} $. One says that an orthogonal representation $ \pi $ of a finite group $ G $ in $ E ^ {n+} 1 $ is fixed-point free if for all $ g \in G \setminus \{ 1 \} $ the transformation $ \pi ( g) $ of the sphere $ S ^ {n} $ is fixed-point free, in particular, $ \pi $ is a faithful representation. According to the program developed in [4], the solution of the Clifford–Klein problem for spherical space forms can be subdivided into several stages. Firstly, one has to find necessary and sufficient conditions on an abstract group $ G $ so that it be the fundamental group of a spherical space form, and to classify such groups; one obtains a certain family of groups $ \{ G _ \lambda \} $. Secondly, one has to describe all inequivalent irreducible orthogonal representations of each group in $ \{ G _ \lambda \} $, and distinguish among them those representations that are fixed-point free. Finally, one has to determine all automorphisms of the groups in $ \{ G _ \lambda \} $ and to clarify which of the representations found are equivalent modulo the automorphisms of the corresponding group. This program has been realized completely in [5], and has led to an exhaustive classification of spherical space forms. Any finite cyclic group belongs to the family $ \{ G _ \lambda \} $; a non-cyclic group of order $ N $ is the fundamental group of an $ n $- dimensional spherical space form if (but not only if) $ N $ is relatively prime to $ n + 1 $ and is divisible by the square of an integer.

The global theory of Euclidean space forms arose as an application of some results in geometric crystallography (cf. Crystallography, mathematical). In [3] the list of crystallographic groups in $ E ^ {3} $ known at the end of the 19th century was used to obtain a topological, and in the compact case an affine, classification of three-dimensional Euclidean space forms. Bieberbach's theorems on crystallographic groups in $ E ^ {3} $ led to the structure theory of compact Euclidean space forms of arbitrary dimensions. In particular, for any $ n \geq 2 $ there is only a finite number of different equivalence classes of compact Euclidean space forms of dimension $ n $; moreover, two compact Euclidean space forms $ M ^ {n} = E ^ {n} / \Gamma $ and $ M _ {1} ^ {n} = E ^ {n} / \Gamma _ {1} $ are affinely equivalent if and only if their fundamental groups $ \Gamma $ and $ \Gamma _ {1} $ are isomorphic. E.g., any two-dimensional compact Euclidean space form is homeomorphic (hence, affinely equivalent) either to a flat torus or to the Klein bottle. An abstract group $ \Gamma $ is the fundamental group of a compact Euclidean space form $ M ^ {n} $ if and only if: a) $ \Gamma $ has a normal Abelian subgroup $ \Gamma ^ {*} $ of finite index isomorphic to $ \mathbf Z ^ {n} $; b) $ \Gamma ^ {*} $ coincides with the centralizer subgroup in $ \Gamma $; and c) $ \Gamma $ does not have elements of finite order. If such a group $ \Gamma $ is realized as a discrete subgroup in the group of motions of $ E ^ {n} $, then $ \Gamma ^ {*} $ coincides with the set of translations belonging to $ \Gamma $, and there is a normal covering $ p $ of $ M ^ {n} = E ^ {n} / \Gamma $ by the flat torus $ T ^ {n} = E ^ {n} / \Gamma ^ {*} $, defined by $ p ( \Gamma ^ {*} ( x) ) = \Gamma ( x) $ for all $ x \in E ^ {n} $. The finite group $ \Gamma / \Gamma ^ {*} $ is isomorphic to the group of covering transformations for $ p $, which is, in turn, isomorphic to the holonomy group of $ M ^ {n} $. A compact Euclidean space form always has a finite homology group. The converse statement also holds: A compact Riemannian space whose holonomy group is finite is flat. It has been proved that every finite group is isomorphic to the holonomy group of a compact Euclidean space form. The affine classification of compact Euclidean space forms of a given dimension $ n $ is known (1983) only for $ n \leq 4 $. For $ n = 3 $ there are 6 orientable and 4 non-orientable classes of affinely-equivalent compact Euclidean space forms. The compact Euclidean space forms with a cyclic holonomy group of prime order have been classified. The family of non-isometric flat tori $ T ^ {n} $ can be parametrized by the elements of

$$ \mathop{\rm SL} ( n , \mathbf Z ) \setminus \mathop{\rm GL} ^ {+} ( n , \mathbf R ) / \mathop{\rm SO} ( n) . $$

Here $ \mathop{\rm GL} ^ {+} ( n , \mathbf R ) $ is the connected component of the identity in $ \mathop{\rm GL} ( n , \mathbf R ) $. The isometric classification of compact Euclidean space forms of dimension $ n $ immediately follows from their affine classification and the isometric classification of the tori $ T ^ {n} $. Non-compact Euclidean space forms have been classified (up to an isometry) only in dimensions 2 and 3. In particular, a two-dimensional non-compact Euclidean space form, different from $ E ^ {2} $, is homeomorphic to either a cylinder or the Möbius strip. Any non-compact Euclidean space form admits a real-analytic retraction onto a compact totally-geodesic flat submanifold; the class of fundamental groups of non-compact Euclidean space forms coincides with the class of fundamental groups of compact Euclidean space forms.

The study of two-dimensional hyperbolic space forms essentially began in 1888, when H. Poincaré, [1], studied discrete groups of fractional-linear transformations of the upper half-plane $ \mathop{\rm Im} z > 0 $ of the complex plane (Fuchsian groups, cf. Fuchsian group) and noted that they can be treated as the groups of motions of the hyperbolic plane $ \Lambda ^ {2} $. Let $ {\mathcal L} $ be the group of motions of $ \Lambda ^ {2} $ preserving orientation; let $ A _ {1} \dots A _ {4m} $, $ m \geq 2 $, be a convex $ 4 m $- gon in $ \Lambda ^ {2} $ with pairwise-congruent geodesic sides

$$ A _ {4i-} 3 A _ {4i-} 2 = \ A _ {4i-} 1 A _ {4i} ,\ \ A _ {4i-} 2 A _ {4i-} 1 = \ A _ {4i} A _ {4i+} 1 , $$

where $ i = 1 \dots m $, $ A _ {4m+} 1 = A _ {1} $, and the sum of the angles of which is $ 2 \pi $. The elements $ a _ {i} $ and $ b _ {i} $ in $ {\mathcal L} $ map $ A _ {4i-} 3 A _ {4i-} 2 $ to $ A _ {4i} A _ {4i-} 1 $ and $ A _ {4i-} 2 A _ {4i-} 1 $ to $ A _ {4i-} 1 A _ {4i} $, respectively (the case $ m = 2 $ is drawn in the Fig.).

Figure: s086190a

The subgroup $ \Gamma \subset {\mathcal L} $ generated by the $ a _ {i} , b _ {i} $ then acts fixed-point free on $ \Lambda ^ {2} $, and the given $ 4 m $- gon is the fundamental domain of $ \Gamma $. Moreover, $ \Gamma $ has the unique defining relation

$$ \prod _ { i= } 1 ^ { m } [ a _ {i} , b _ {i} ] = 1 . $$

The quotient group $ \Lambda ^ {2} / \Gamma $ is an orientable compact hyperbolic space form of genus $ m $, and every two-dimensional orientable compact hyperbolic space form can be obtained in this way. Suppose now that $ \Gamma $ is an abstract group isomorphic to the fundamental group of an orientable closed surface of genus $ m $. Then there is a continuous mapping $ \phi : \Gamma \times \mathbf R ^ {6m-} 6 \rightarrow {\mathcal L} $ satisfying the conditions: a) for all $ x \in \mathbf R ^ {6m-} 6 $ the mapping $ \phi _ {x} : g \mapsto \phi ( g , x ) $ is a monomorphism of $ \Gamma $ into $ h $; b) two subgroups $ \Gamma _ {x} = \phi _ {x} ( \Gamma ) $ and $ \Gamma _ {y} = \phi _ {y} ( \Gamma ) $ are conjugate in $ {\mathcal L} $ if and only if $ x = y $; and c) if a discrete subgroup $ \Gamma _ {1} \subset {\mathcal L} $ is isomorphic to $ \Gamma $, then it is conjugate to $ \Gamma _ {x} $ for some $ x \in \mathbf R ^ {6m-} 6 $. Thus, the family of non-isomorphic compact hyperbolic space forms of dimension 2 and genus $ m $ depends on $ 6 m - 6 $ real parameters. A two-dimensional compact hyperbolic space form can be naturally endowed with the structure of a Riemann surface, and the statement just formulated was originally proved by tools of the theory of uniformization; a geometric proof was given in [7]. The results given can be generalized to non-compact hyperbolic space forms, which are homeomorphic to a sphere with a finite number of handles and holes, as well as to non-oriented hyperbolic space forms of dimension 2. Contrary to the two-dimensional case, there do not exist continuous families of non-isometric compact hyperbolic space forms of dimension exceeding two. More precisely, compact hyperbolic space forms of dimension $ n \geq 3 $ having isomorphic fundamental groups are isomorphic. Other general results, immediately related to the classification of $ n $- dimensional hyperbolic space forms, do not exist (1983); examples of hyperbolic space forms of dimension $ \geq 3 $ have been given in [6] and [8].

Besides the Riemannian space forms their generalization have also been studied: pseudo-Riemannian, affine and complex space forms, as well as space forms of symmetric spaces (cf., e.g., [9]).

References

[1] H. Poincaré, "Oeuvres" , 3 , Gauthier-Villars (1934)
[2] W. Threlfall, H. Seifert, "Topologische Untersuchungen der Diskontinuitätsbereiche endlicher Bewegungsgruppen der dreidimensionalen sphärischen Raumes" Math. Ann. , 104 (1931) pp. 1–70
[3] W. Nowacki, "Euklidischen, dreidimensionalen, geschlossenen und offenen Raumformen" Comm. Math. Helvetica , 7 (1934) pp. 81–93
[4] G. Vincent, "Les groupes linéaires finis sans points fixes" Comm. Math. Helvetica , 20 (1947) pp. 117–171
[5] J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1984)
[6] E.B. Vinberg, "Some examples of crystallographic groups in Lobachevskii spaces" Math. USSR Sb. , 7 (1969) pp. 617–622 Mat. Sb. , 78 : 4 (1969) pp. 633–639
[7] S.M. Natanzon, "Invariant lines on Fuchsian groups" Russian Math. Surveys , 27 : 4 (1972) pp. 161–177 Uspekhi Mat. Nauk : 4 (1972) pp. 145–160
[8] J.J. Millson, "On the first Betti number of a constant negatively cuved manifold" Ann. of Math. , 104 (1976) pp. 235–247
[9] A. Borel, "Compact Clifford–Klein forms of symmetric spaces" Topology , 2 (1963) pp. 111–122

Comments

A group that satisfies the three conditions a), b), c) above for being the fundamental group of a compact Euclidean space form is called a Bieberbach group.

Let $ {\mathcal R} _ {n} $ be the group of rigid motions of the Euclidean space $ E ^ {n} $, i.e. the group of transformations $ ( m, s) x = mx+ s $ with $ m \in O _ {n} $, the orthogonal group, and $ s \in E ^ {n} $, a translation. There is an exact sequence

$$ 0 \rightarrow T _ {n} \rightarrow {\mathcal R} _ {n} \rightarrow ^ { r } O _ {n} \rightarrow 0 , $$

where $ T _ {n} $ is the subgroup of pure translations: $ r( m, s)= m $. This is a semi-direct product. An isotropic subgroup of $ {\mathcal R} _ {n} $ is a subgroup $ \pi $ such that $ \pi \cap T _ {n} $ spans all of $ E ^ {n} $. A uniform subgroup is one such that the orbit space $ E ^ {n} / \pi $ is compact; finally, a direct subgroup is one which is discrete as a subspace of $ {\mathcal R} _ {n} $. A crystallographic subgroup is a subgroup of $ {\mathcal R} _ {n} $ that is uniform and discrete, and a Bieberbach subgroup of $ {\mathcal R} _ {n} $ is a torsion-free crystallographic subgroup. The crystallographic subgroups of $ {\mathcal R} _ {3} $ are also known as the space groups. Cf. also Crystallographic group. An (abstract) crystallographic group is a group that contains a finitely-generated Abelian torsion-free subgroup of finite index. An (abstract) Bieberbach group is a torsion-free crystallographic subgroup. An Auslander–Kuranishi theorem says that each crystallographic group arises as a crystallographic subgroup of an $ {\mathcal R} _ {n} $, and hence that each Bieberbach group arises as a Bieberbach subgroup. A second Auslander–Kuranishi theorem says that for any finite group $ \pi ^ \prime $ there is a Bieberbach group $ \pi $ such that $ r ( \pi ) = \pi ^ \prime $, and that any finite group arises as a holonomy group of a compact Euclidean space form (cf. above). The three Bieberbach theorems on crystallographic subgroups are as follows: i) if $ \pi $ is a crystallographic subgroup of $ {\mathcal R} _ {n} $, then $ r( \pi ) $ is finite and $ \pi $ is isotropic; ii) any isomorphism of crystallographic subgroups of $ {\mathcal R} _ {n} $ can be realized by an affine change of coordinates: $ \beta \mapsto \alpha \beta \alpha ^ {-} 1 $, $ \alpha = ( m, s) $, $ m \in \mathop{\rm GL} _ {n} ( \mathbf R ) $, $ s \in E ^ {n} $; iii) up to affine coordinate changes there are only finitely many crystallographic subgroups of $ {\mathcal R} _ {n} $. These last two statements readily lead to corresponding statements concerning Euclidean space forms, as in the main article above.

The numbers of crystallographic and Bieberbach subgroups (up to isomorphism) in the first few dimensions are as follows.

<tbody> </tbody>
dimension 1 2 3 4
# crystallographic subgroups 1 17 219 4783
# Bieberbach subgroups 1 2 10 74

If one considers the crystallographic groups in $ {\mathcal R} _ {3} $ up to orientation preserving affine conjugacy, the more familiar number of 230 equivalence classes arises. (The 230 space groups.)

References

[a1] L.S. Charlap, "Bieberbach groups and flat manifolds" , Springer (1986)
[a2] L. Auslander, M. Kuranishi, "On the holonomy groups of locally Euclidean spaces" Ann. of Math. , 65 (1957) pp. 411
[a3] R.L.E. Schwarzenberger, "-dimensional crystallography" , Pitman (1980)
How to Cite This Entry:
Space forms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Space_forms&oldid=11705
This article was adapted from an original article by N.R. Kamyshanskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article