Namespaces
Variants
Actions

Difference between revisions of "Riemannian space, homogeneous"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A [[Riemannian space|Riemannian space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822101.png" /> together with a transitive effective group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822102.png" /> of motions (cf. [[Motion|Motion]]) on it. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822103.png" /> be the isotropy subgroup of a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822104.png" />. Then the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822105.png" /> is identified with the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822106.png" /> by the bijection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822107.png" />, and the Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822108.png" /> is considered as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r0822109.png" />-invariant metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221010.png" />. Usually one assumes in addition that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221011.png" /> is closed in the complete group of motions. In this case the isotropy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221012.png" /> is compact.
+
<!--
 +
r0822101.png
 +
$#A+1 = 70 n = 0
 +
$#C+1 = 70 : ~/encyclopedia/old_files/data/R082/R.0802210 Riemannian space, homogeneous
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221013.png" /> be a compact subgroup of a [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221014.png" /> that does not contain normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221015.png" />. Then the [[Homogeneous space|homogeneous space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221016.png" /> admits an invariant Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221017.png" /> defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221018.png" /> be a reductive structure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221019.png" />, i.e. a decomposition of the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221021.png" /> into the direct sum of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221023.png" /> and a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221024.png" /> that is invariant under the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221027.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221028.png" /> is naturally identified with the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221029.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221030.png" />, and the isotropy representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221032.png" /> with the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221033.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221034.png" />-invariant Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221036.png" /> is obtained from some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221037.png" />-invariant scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221039.png" /> by translations from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221040.png" />:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/r/r082/r082210/r08221041.png" /></td> </tr></table>
+
A [[Riemannian space|Riemannian space]]  $  ( M, \gamma ) $
 +
together with a transitive effective group  $  G $
 +
of motions (cf. [[Motion|Motion]]) on it. Let  $  K $
 +
be the isotropy subgroup of a fixed point  $  o \in M $.  
 +
Then the manifold  $  M $
 +
is identified with the quotient space  $  G/K $
 +
by the bijection  $  G/K \ni gK \iff go \in M $,
 +
and the Riemannian metric  $  \gamma $
 +
is considered as a  $  G $-
 +
invariant metric on  $  G/K $.
 +
Usually one assumes in addition that the group  $  G $
 +
is closed in the complete group of motions. In this case the isotropy group  $  K $
 +
is compact.
  
The existence of such a scalar product follows from the fact that the isotropy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221042.png" /> is compact.
+
Let  $  K $
 +
be a compact subgroup of a [[Lie group|Lie group]]  $  G $
 +
that does not contain normal subgroups of  $  G $.
 +
Then the [[Homogeneous space|homogeneous space]]  $  M = G/K $
 +
admits an invariant Riemannian metric  $  \gamma $
 +
defined as follows. Let  $  \mathfrak G = \mathfrak K + \mathfrak M $
 +
be a reductive structure in  $  M $,
 +
i.e. a decomposition of the [[Lie algebra|Lie algebra]]  $  \mathfrak G $
 +
of  $  G $
 +
into the direct sum of the Lie algebra  $  \mathfrak K $
 +
of  $  K $
 +
and a subspace  $  \mathfrak M $
 +
that is invariant under the adjoint representation  $  \mathop{\rm Ad}  K $
 +
of  $  K $
 +
in  $  \mathfrak G $.
 +
The space  $  \mathfrak M $
 +
is naturally identified with the tangent space  $  T _ {0} M \approx \mathfrak G / \mathfrak K $
 +
at the point  $  o = eK $,
 +
and the isotropy representation of the group $  K $
 +
in  $  T _ {0} M $
 +
with the representation  $  \mathop{\rm Ad}  K \ \mid  _ {\mathfrak M} $.  
 +
Any  $  G $-
 +
invariant Riemannian metric  $  \gamma $
 +
on  $  M $
 +
is obtained from some  $  \mathop{\rm Ad}  K $-
 +
invariant scalar product  $  \gamma _ {0} $
 +
in  $  \mathfrak M $
 +
by translations from  $  G $:
  
Any homogeneous Riemannian space locally isometric to a simply-connected homogeneous Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221043.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221044.png" /> by factorization with respect to an arbitrary Clifford–Wolf discrete group of isometries (i.e. motions of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221045.png" /> that displace all points by equal distances [[#References|[2]]]).
+
$$
 +
\gamma _ {go} ( X, Y)  = \gamma _ {0} ( g  ^ {-} 1 X, g  ^ {-} 1 Y),\ \
 +
X, Y \in T _ {go} M,\ \
 +
g \in G.
 +
$$
  
The best studied classes of homogeneous Riemannian spaces are the Riemannian symmetric spaces (cf. also [[Symmetric space|Symmetric space]]); homogeneous Kähler spaces (cf. [[Kähler manifold|Kähler manifold]]) and homogeneous quaternionic spaces; isotropically-irreducible homogeneous Riemannian spaces (classified in [[#References|[9]]], [[#References|[10]]]); normal homogeneous Riemannian spaces, in which the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221047.png" /> is defined by a non-degenerate symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221048.png" />-invariant bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221049.png" />; and naturally-reductive homogeneous Riemannian spaces, characterized by the fact that any geodesic in them is the trajectory of a one-parameter group of motions.
+
The existence of such a scalar product follows from the fact that the isotropy group $  \mathop{\rm Ad}  K \ \mid  _ {\mathfrak M} $
 +
is compact.
  
The structure of homogeneous Riemannian spaces with different conditions on the curvature tensor is well studied. For instance, the classification of homogeneous Riemannian spaces of positive sectional curvature is known [[#References|[5]]]. The structure of simply-transitive groups of motions of a homogeneous Riemannian space of non-positive curvature [[#References|[8]]], of non-negative curvature and of non-negative Ricci curvature [[#References|[4]]] has been described. A homogeneous Riemannian space with a solvable group of motions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221050.png" /> always has a non-positive scalar curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221051.png" />, and the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221052.png" /> is possible only for locally Euclidean spaces. Any invariant Riemannian metric on a simply-connected homogeneous Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221053.png" /> has non-positive scalar curvature if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221054.png" /> is a maximal compact subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221055.png" /> (see [[#References|[4]]]).
+
Any homogeneous Riemannian space locally isometric to a simply-connected homogeneous Riemannian space $  \widetilde{M}  $
 +
is obtained from  $  \widetilde{M}  $
 +
by factorization with respect to an arbitrary Clifford–Wolf discrete group of isometries (i.e. motions of the manifold  $  \widetilde{M}  $
 +
that displace all points by equal distances [[#References|[2]]]).
  
A homogeneous Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221056.png" /> is called Einstein if its [[Ricci tensor|Ricci tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221057.png" /> is proportional to the metric: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221059.png" />. The problem of the description of Einstein homogeneous Riemannian spaces has not yet been solved (1991). One knows a number of particular results. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221060.png" /> be an Einstein homogeneous Riemannian space of scalar curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221061.png" />. 1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221062.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221063.png" /> is a compact manifold. All such spaces have been described: a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221064.png" /> is a quaternionic space; b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221065.png" /> is diffeomorphic to a symmetric space of rank one; and c) for a certain class of naturally-reductive homogeneous Riemannian spaces (see [[#References|[7]]]) and for isotropically-irreducible homogeneous Riemannian spaces (see [[#References|[10]]]). 2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221066.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221067.png" /> is a locally Euclidean space. 3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221069.png" /> is unimodular (i.e. the determinant of its adjoint representation operator is equal to 1), then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082210/r08221070.png" /> is semi-simple.
+
The best studied classes of homogeneous Riemannian spaces are the Riemannian symmetric spaces (cf. also [[Symmetric space|Symmetric space]]); homogeneous Kähler spaces (cf. [[Kähler manifold|Kähler manifold]]) and homogeneous quaternionic spaces; isotropically-irreducible homogeneous Riemannian spaces (classified in [[#References|[9]]], [[#References|[10]]]); normal homogeneous Riemannian spaces, in which the scalar product  $  \gamma _ {0} $
 +
in  $  \mathfrak M $
 +
is defined by a non-degenerate symmetric  $  \mathop{\rm Ad}  G $-
 +
invariant bilinear form on  $  \mathfrak G $;
 +
and naturally-reductive homogeneous Riemannian spaces, characterized by the fact that any geodesic in them is the trajectory of a one-parameter group of motions.
 +
 
 +
The structure of homogeneous Riemannian spaces with different conditions on the curvature tensor is well studied. For instance, the classification of homogeneous Riemannian spaces of positive sectional curvature is known [[#References|[5]]]. The structure of simply-transitive groups of motions of a homogeneous Riemannian space of non-positive curvature [[#References|[8]]], of non-negative curvature and of non-negative Ricci curvature [[#References|[4]]] has been described. A homogeneous Riemannian space with a solvable group of motions  $  G $
 +
always has a non-positive scalar curvature  $  \mathop{\rm sc} $,
 +
and the case  $  \mathop{\rm sc} = 0 $
 +
is possible only for locally Euclidean spaces. Any invariant Riemannian metric on a simply-connected homogeneous Riemannian space  $  G/K $
 +
has non-positive scalar curvature if and only if  $  K $
 +
is a maximal compact subgroup of  $  G $(
 +
see [[#References|[4]]]).
 +
 
 +
A homogeneous Riemannian space  $  ( M, \gamma ) $
 +
is called Einstein if its [[Ricci tensor|Ricci tensor]] $  \rho $
 +
is proportional to the metric: $  \rho = \lambda \gamma $,  
 +
$  \lambda = \textrm{ const } $.  
 +
The problem of the description of Einstein homogeneous Riemannian spaces has not yet been solved (1991). One knows a number of particular results. Let $  ( M = G/K, \gamma ) $
 +
be an Einstein homogeneous Riemannian space of scalar curvature $  \mathop{\rm sc} $.  
 +
1) If $  \mathop{\rm sc} > 0 $,  
 +
then $  M $
 +
is a compact manifold. All such spaces have been described: a) if $  ( M, \gamma ) $
 +
is a quaternionic space; b) if $  M $
 +
is diffeomorphic to a symmetric space of rank one; and c) for a certain class of naturally-reductive homogeneous Riemannian spaces (see [[#References|[7]]]) and for isotropically-irreducible homogeneous Riemannian spaces (see [[#References|[10]]]). 2) If $  \mathop{\rm sc} = 0 $,  
 +
then $  M $
 +
is a locally Euclidean space. 3) If $  \mathop{\rm sc} < 0 $
 +
and $  G $
 +
is unimodular (i.e. the determinant of its adjoint representation operator is equal to 1), then the group $  G $
 +
is semi-simple.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963–1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Wolf,  "Spaces of constant curvature" , Publish or Perish  (1977)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Berard Bergery,  "Sur le courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes"  ''Ann. Sci. Ecole Norm. Sup.'' , '''11''' :  4  (1978)  pp. 545–576</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Berard Bergery,  "Les variétés riemanniennes simplement connexes de dimension impairé à courbure strictement positive"  ''J. Math. Pures Appl.'' , '''55'''  (1976)  pp. 47–67</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.R. Jensen,  "Einstein metrics on principle fiber bundles"  ''J. Dif. Geom.'' , '''8'''  (1973)  pp. 599–614</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.E. d'Atri,  W. Ziller,  "Naturally reductive metrics and Einstein metrics on compact Lie groups"  ''Mem. Amer. Math. Soc.'' , '''18'''  (1979)  pp. 1–72</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R. Azencott,  E.N. Wilson,  "Homogeneous manifolds with negative curvature II"  ''Mem. Amer. Math. Soc.'' , '''8'''  (1976)  pp. 1–102</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  O.V. Manturov,  "Homogeneous Riemannian spaces with an irreducible rotation group"  ''Trudy Sem. Vektor. i Tenzor. Anal.'' , '''13'''  (1966)  pp. 68–145  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J. Wolf,  "The geometry and structure of isotropy irreducible homogeneous spaces"  ''Acta Math.'' , '''120'''  (1968)  pp. 59–148</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963–1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Wolf,  "Spaces of constant curvature" , Publish or Perish  (1977)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Berard Bergery,  "Sur le courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes"  ''Ann. Sci. Ecole Norm. Sup.'' , '''11''' :  4  (1978)  pp. 545–576</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Berard Bergery,  "Les variétés riemanniennes simplement connexes de dimension impairé à courbure strictement positive"  ''J. Math. Pures Appl.'' , '''55'''  (1976)  pp. 47–67</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  G.R. Jensen,  "Einstein metrics on principle fiber bundles"  ''J. Dif. Geom.'' , '''8'''  (1973)  pp. 599–614</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.E. d'Atri,  W. Ziller,  "Naturally reductive metrics and Einstein metrics on compact Lie groups"  ''Mem. Amer. Math. Soc.'' , '''18'''  (1979)  pp. 1–72</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R. Azencott,  E.N. Wilson,  "Homogeneous manifolds with negative curvature II"  ''Mem. Amer. Math. Soc.'' , '''8'''  (1976)  pp. 1–102</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  O.V. Manturov,  "Homogeneous Riemannian spaces with an irreducible rotation group"  ''Trudy Sem. Vektor. i Tenzor. Anal.'' , '''13'''  (1966)  pp. 68–145  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J. Wolf,  "The geometry and structure of isotropy irreducible homogeneous spaces"  ''Acta Math.'' , '''120'''  (1968)  pp. 59–148</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:11, 6 June 2020


A Riemannian space $ ( M, \gamma ) $ together with a transitive effective group $ G $ of motions (cf. Motion) on it. Let $ K $ be the isotropy subgroup of a fixed point $ o \in M $. Then the manifold $ M $ is identified with the quotient space $ G/K $ by the bijection $ G/K \ni gK \iff go \in M $, and the Riemannian metric $ \gamma $ is considered as a $ G $- invariant metric on $ G/K $. Usually one assumes in addition that the group $ G $ is closed in the complete group of motions. In this case the isotropy group $ K $ is compact.

Let $ K $ be a compact subgroup of a Lie group $ G $ that does not contain normal subgroups of $ G $. Then the homogeneous space $ M = G/K $ admits an invariant Riemannian metric $ \gamma $ defined as follows. Let $ \mathfrak G = \mathfrak K + \mathfrak M $ be a reductive structure in $ M $, i.e. a decomposition of the Lie algebra $ \mathfrak G $ of $ G $ into the direct sum of the Lie algebra $ \mathfrak K $ of $ K $ and a subspace $ \mathfrak M $ that is invariant under the adjoint representation $ \mathop{\rm Ad} K $ of $ K $ in $ \mathfrak G $. The space $ \mathfrak M $ is naturally identified with the tangent space $ T _ {0} M \approx \mathfrak G / \mathfrak K $ at the point $ o = eK $, and the isotropy representation of the group $ K $ in $ T _ {0} M $ with the representation $ \mathop{\rm Ad} K \ \mid _ {\mathfrak M} $. Any $ G $- invariant Riemannian metric $ \gamma $ on $ M $ is obtained from some $ \mathop{\rm Ad} K $- invariant scalar product $ \gamma _ {0} $ in $ \mathfrak M $ by translations from $ G $:

$$ \gamma _ {go} ( X, Y) = \gamma _ {0} ( g ^ {-} 1 X, g ^ {-} 1 Y),\ \ X, Y \in T _ {go} M,\ \ g \in G. $$

The existence of such a scalar product follows from the fact that the isotropy group $ \mathop{\rm Ad} K \ \mid _ {\mathfrak M} $ is compact.

Any homogeneous Riemannian space locally isometric to a simply-connected homogeneous Riemannian space $ \widetilde{M} $ is obtained from $ \widetilde{M} $ by factorization with respect to an arbitrary Clifford–Wolf discrete group of isometries (i.e. motions of the manifold $ \widetilde{M} $ that displace all points by equal distances [2]).

The best studied classes of homogeneous Riemannian spaces are the Riemannian symmetric spaces (cf. also Symmetric space); homogeneous Kähler spaces (cf. Kähler manifold) and homogeneous quaternionic spaces; isotropically-irreducible homogeneous Riemannian spaces (classified in [9], [10]); normal homogeneous Riemannian spaces, in which the scalar product $ \gamma _ {0} $ in $ \mathfrak M $ is defined by a non-degenerate symmetric $ \mathop{\rm Ad} G $- invariant bilinear form on $ \mathfrak G $; and naturally-reductive homogeneous Riemannian spaces, characterized by the fact that any geodesic in them is the trajectory of a one-parameter group of motions.

The structure of homogeneous Riemannian spaces with different conditions on the curvature tensor is well studied. For instance, the classification of homogeneous Riemannian spaces of positive sectional curvature is known [5]. The structure of simply-transitive groups of motions of a homogeneous Riemannian space of non-positive curvature [8], of non-negative curvature and of non-negative Ricci curvature [4] has been described. A homogeneous Riemannian space with a solvable group of motions $ G $ always has a non-positive scalar curvature $ \mathop{\rm sc} $, and the case $ \mathop{\rm sc} = 0 $ is possible only for locally Euclidean spaces. Any invariant Riemannian metric on a simply-connected homogeneous Riemannian space $ G/K $ has non-positive scalar curvature if and only if $ K $ is a maximal compact subgroup of $ G $( see [4]).

A homogeneous Riemannian space $ ( M, \gamma ) $ is called Einstein if its Ricci tensor $ \rho $ is proportional to the metric: $ \rho = \lambda \gamma $, $ \lambda = \textrm{ const } $. The problem of the description of Einstein homogeneous Riemannian spaces has not yet been solved (1991). One knows a number of particular results. Let $ ( M = G/K, \gamma ) $ be an Einstein homogeneous Riemannian space of scalar curvature $ \mathop{\rm sc} $. 1) If $ \mathop{\rm sc} > 0 $, then $ M $ is a compact manifold. All such spaces have been described: a) if $ ( M, \gamma ) $ is a quaternionic space; b) if $ M $ is diffeomorphic to a symmetric space of rank one; and c) for a certain class of naturally-reductive homogeneous Riemannian spaces (see [7]) and for isotropically-irreducible homogeneous Riemannian spaces (see [10]). 2) If $ \mathop{\rm sc} = 0 $, then $ M $ is a locally Euclidean space. 3) If $ \mathop{\rm sc} < 0 $ and $ G $ is unimodular (i.e. the determinant of its adjoint representation operator is equal to 1), then the group $ G $ is semi-simple.

References

[1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1968)
[2] J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1977)
[3] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[4] L. Berard Bergery, "Sur le courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes" Ann. Sci. Ecole Norm. Sup. , 11 : 4 (1978) pp. 545–576
[5] L. Berard Bergery, "Les variétés riemanniennes simplement connexes de dimension impairé à courbure strictement positive" J. Math. Pures Appl. , 55 (1976) pp. 47–67
[6] G.R. Jensen, "Einstein metrics on principle fiber bundles" J. Dif. Geom. , 8 (1973) pp. 599–614
[7] J.E. d'Atri, W. Ziller, "Naturally reductive metrics and Einstein metrics on compact Lie groups" Mem. Amer. Math. Soc. , 18 (1979) pp. 1–72
[8] R. Azencott, E.N. Wilson, "Homogeneous manifolds with negative curvature II" Mem. Amer. Math. Soc. , 8 (1976) pp. 1–102
[9] O.V. Manturov, "Homogeneous Riemannian spaces with an irreducible rotation group" Trudy Sem. Vektor. i Tenzor. Anal. , 13 (1966) pp. 68–145 (In Russian)
[10] J. Wolf, "The geometry and structure of isotropy irreducible homogeneous spaces" Acta Math. , 120 (1968) pp. 59–148

Comments

For a quite exhaustive treatment of Einstein manifolds see [a1], esp. Chapts. 7, 8.

Usually, an isometry of a Riemannian space is used as a synonym for motion, while the isometries in a Clifford–Wolf discrete group are known as Clifford translations, [2].

References

[a1] A.L. Besse, "Einstein manifolds" , Springer (1987)
How to Cite This Entry:
Riemannian space, homogeneous. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_space,_homogeneous&oldid=13094
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article