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A [[Riemannian metric|Riemannian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522601.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522602.png" /> that does not change under any of the transformations of a given Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522603.png" /> of transformations. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522604.png" /> itself is called a group of motions (isometries) of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522605.png" /> (or of the Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522606.png" />).
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$#C+1 = 149 : ~/encyclopedia/old_files/data/I052/I.0502260 Invariant metric
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A Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522607.png" /> of transformations of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522608.png" /> acting properly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i0522609.png" /> (that is, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226011.png" /> is proper) has an invariant metric. Conversely, the group of all motions of any Riemannian metric (as well as any closed subgroup of it) is a proper Lie group of transformations. In this case the stabilizer (or isotropy group)
+
{{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/i/i052/i052260/i05226012.png" /></td> </tr></table>
+
A [[Riemannian metric|Riemannian metric]]  $  m $
 +
on a manifold  $  M $
 +
that does not change under any of the transformations of a given Lie group  $  G $
 +
of transformations. The group  $  G $
 +
itself is called a group of motions (isometries) of the metric  $  m $(
 +
or of the Riemannian space  $  ( M , m ) $).
  
of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226013.png" /> is a compact subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226015.png" /> itself is compact, then a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226016.png" />-invariant metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226017.png" /> can be constructed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226018.png" /> by averaging any metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226020.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226021.png" />:
+
A Lie group  $  G $
 +
of transformations of a manifold  $  M $
 +
acting properly on  $  M $(
 +
that is, the mapping  $  G \times M \rightarrow M \times M $,
 +
$  ( g , x ) \rightarrow ( gx , x ) $
 +
is proper) has an invariant metric. Conversely, the group of all motions of any Riemannian metric (as well as any closed subgroup of it) is a proper Lie group of transformations. In this case the stabilizer (or isotropy group)
  
<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/i/i052/i052260/i05226022.png" /></td> </tr></table>
+
$$
 +
G _ {x}  = \{ {g \in G } : {gx = x } \}
 +
$$
 +
 
 +
of any point  $  x \in M $
 +
is a compact subgroup of  $  G $.
 +
If  $  G $
 +
itself is compact, then a  $  G $-
 +
invariant metric  $  m _ {0} $
 +
can be constructed on  $  M $
 +
by averaging any metric  $  m $
 +
on  $  M $
 +
over  $  G $:
 +
 
 +
$$
 +
m _ {0= \int\limits _ { G } ( g  ^ {*} m )  dg ,
 +
$$
  
 
where the integral is taken with respect to the Haar measure.
 
where the integral is taken with respect to the Haar measure.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226023.png" /> is transitive, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226024.png" /> can be identified with the space of cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226026.png" /> with respect to the stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226027.png" /> of a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226028.png" />, and in order that there exist a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226029.png" />-invariant metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226030.png" /> it is necessary and sufficient that the linear isotropy group (see [[Isotropy representation|Isotropy representation]]) has compact closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226031.png" /> (in particular, it is sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226032.png" /> be compact). In this case the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226033.png" /> is reductive, that is, the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226035.png" /> admits a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226037.png" /> is the subalgebra corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226039.png" /> is a subspace that is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226040.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226041.png" /> is the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226042.png" /> (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226043.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226044.png" />, then any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226045.png" />-invariant metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226046.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226047.png" /> is obtained from some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226048.png" />-invariant Euclidean metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226050.png" /> in the following way:
+
If $  G $
 +
is transitive, $  M $
 +
can be identified with the space of cosets $  G / H $
 +
of $  G $
 +
with respect to the stabilizer $  H = G _ {x _ {0}  } $
 +
of a fixed point $  x _ {0} \in M $,  
 +
and in order that there exist a $  G $-
 +
invariant metric on $  M $
 +
it is necessary and sufficient that the linear isotropy group (see [[Isotropy representation|Isotropy representation]]) has compact closure in $  \mathop{\rm GL} ( T _ {x _ {0}  } M ) $(
 +
in particular, it is sufficient that $  H $
 +
be compact). In this case the space $  G / H $
 +
is reductive, that is, the Lie algebra $  \mathfrak G $
 +
of $  G $
 +
admits a decomposition $  \mathfrak G = \mathfrak H + \mathfrak M $,  
 +
where $  \mathfrak H $
 +
is the subalgebra corresponding to $  H $
 +
and $  \mathfrak M $
 +
is a subspace that is invariant under $  \mathop{\rm Ad}  H $
 +
where $  \mathop{\rm Ad} $
 +
is the adjoint representation of $  G $(
 +
cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). If $  \mathfrak M $
 +
is identified with $  T _ {x _ {0}  } M $,  
 +
then any $  G $-
 +
invariant metric $  m $
 +
on $  M $
 +
is obtained from some $  \mathop{\rm Ad}  H $-
 +
invariant Euclidean metric $  \langle  , \rangle $
 +
on $  \mathfrak M $
 +
in the following way:
  
<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/i/i052/i052260/i05226051.png" /></td> </tr></table>
+
$$
 +
m _ {x} ( X , Y )  = \langle  ( g  ^ {*} )  ^ {-1} X, ( g  ^ {*} )  ^ {-1} Y \rangle ,\ \
 +
X , Y \in T _ {x} M ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226052.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226053.png" />.
+
where $  g \in G $
 +
is such that $  g x _ {0} = x $.
  
The tensor fields associated with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226054.png" />-invariant metric (the curvature tensor, its covariant derivatives, etc.) are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226055.png" />-invariant fields. In the case of a homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226056.png" />, their value at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226057.png" /> can be expressed in terms of the Nomizu operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226058.png" />, which is defined by the formula
+
The tensor fields associated with a $  G $-
 +
invariant metric (the curvature tensor, its covariant derivatives, etc.) are $  G $-
 +
invariant fields. In the case of a homogeneous space $  M = G / H $,  
 +
their value at a point $  x _ {0} $
 +
can be expressed in terms of the Nomizu operator $  L _ {X} \in  \mathop{\rm End} ( \mathfrak M ) $,  
 +
which is defined by the formula
  
<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/i/i052/i052260/i05226059.png" /></td> </tr></table>
+
$$
 +
L _ {X} Y  = - \nabla _ {Y} X  ^ {*}  = \
 +
( {\mathcal L} _ {X  ^ {*}  } - \nabla _ {X  ^ {*}  } ) _ {x _ {0}  } Y ,\ \
 +
Y \in \mathfrak M ,\  X \in \mathfrak G ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226060.png" /> is the velocity field of the one-parameter group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226062.png" /> is the [[Covariant differentiation|covariant differentiation]] operator of the Riemannian connection and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226063.png" /> is the [[Lie derivative|Lie derivative]] operator. In particular, the [[Curvature|curvature]] operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226064.png" /> and the [[Sectional curvature|sectional curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226065.png" /> in the direction given by the orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226066.png" /> satisfy the following formulas:
+
where $  X  ^ {*} $
 +
is the velocity field of the one-parameter group of transformations $  \mathop{\rm exp}  tX $,  
 +
$  \nabla $
 +
is the [[Covariant differentiation|covariant differentiation]] operator of the Riemannian connection and $  {\mathcal L} $
 +
is the [[Lie derivative|Lie derivative]] operator. In particular, the [[Curvature|curvature]] operator $  \mathop{\rm Riem} ( X , Y ) $
 +
and the [[Sectional curvature|sectional curvature]] $  K ( X, Y ) $
 +
in the direction given by the orthonormal basis $  X , Y \in M $
 +
satisfy the following formulas:
  
<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/i/i052/i052260/i05226067.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Riem} ( X , Y )  = [ L _ {X} , L _ {Y} ] - L _ {[ X , Y ] }  ,
 +
$$
  
<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/i/i052/i052260/i05226068.png" /></td> </tr></table>
+
$$
 +
K ( X , Y )  \equiv  - \langle  \mathop{\rm Riem} ( X , Y ) X , Y \rangle =
 +
$$
  
<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/i/i052/i052260/i05226069.png" /></td> </tr></table>
+
$$
 +
= \
 +
\langle  L _ {X} Y , L _ {X} Y \rangle - < [ X
 +
, Y ] _ {\mathfrak M }  , [ X , Y ] _ {\mathfrak M }  > +
 +
$$
  
<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/i/i052/i052260/i05226070.png" /></td> </tr></table>
+
$$
 +
- \langle  [ Y , [ Y , X ] ] _ {\mathfrak M }  , X \rangle - \langle  L _ {X} X , L _ {Y} Y \rangle ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226071.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226072.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226073.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226074.png" />.
+
where $  Z _ {\mathfrak M }  $
 +
is the projection of $  Z \in \mathfrak G $
 +
on $  \mathfrak M $
 +
along $  \mathfrak H $.
  
The Nomizu operators can be expressed in terms of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226075.png" /> and the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226076.png" /> by the formula
+
The Nomizu operators can be expressed in terms of the Lie algebra $  \mathfrak G $
 +
and the metric $  \langle  , \rangle $
 +
by the formula
  
<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/i/i052/i052260/i05226077.png" /></td> </tr></table>
+
$$
 +
2 \langle  L _ {X} Y , Z \rangle =
 +
$$
  
<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/i/i052/i052260/i05226078.png" /></td> </tr></table>
+
$$
 +
= \
 +
\langle  [ X , Y ] _ {\mathfrak M }  , Z \rangle - < X , [ Y ,\
 +
Z ] _ {\mathfrak M }  > - \langle  Y , [ X , Z ] _ {\mathfrak M }  \rangle ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226080.png" />. It follows from the definition of the Nomizu operators that their action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226081.png" />-invariant fields differs only in sign from that of the covariant derivative at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226082.png" />. If the Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226083.png" /> does not contain flat factors in the de Rham decomposition, then the linear Lie algebra generated by the Nomizu operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226085.png" />, is the same as the holonomy algebra (cf. [[Holonomy group|Holonomy group]]) of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226086.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226087.png" />.
+
where $  X \in \mathfrak G $,
 +
$  Y , Z \in \mathfrak M $.  
 +
It follows from the definition of the Nomizu operators that their action on $  G $-
 +
invariant fields differs only in sign from that of the covariant derivative at the point $  x _ {0} $.  
 +
If the Riemannian space $  ( G / H , m) $
 +
does not contain flat factors in the de Rham decomposition, then the linear Lie algebra generated by the Nomizu operators $  L _ {X} $,  
 +
$  X \in \mathfrak G $,  
 +
is the same as the holonomy algebra (cf. [[Holonomy group|Holonomy group]]) of the space $  ( G / H , M) $
 +
at $  x _ {0} $.
  
A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. Suppose, to begin with, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226088.png" /> is a Lie group acting on itself by left translations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226089.png" /> be a left-invariant geodesic of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226090.png" /> on the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226091.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226092.png" /> be the curve in the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226093.png" /> corresponding to it (the velocity hodograph). The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226094.png" /> satisfies the hodograph equation
+
A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. Suppose, to begin with, that $  M = G $
 +
is a Lie group acting on itself by left translations. Let $  \gamma _ {t} $
 +
be a left-invariant geodesic of the metric $  m $
 +
on the Lie group $  G $
 +
and let $  X _ {t} = \gamma  ^ {-1} \dot \gamma  _ {t} $
 +
be the curve in the Lie algebra $  \mathfrak G $
 +
corresponding to it (the velocity hodograph). The curve $  X _ {t} $
 +
satisfies the hodograph equation
  
<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/i/i052/i052260/i05226095.png" /></td> </tr></table>
+
$$
 +
\dot{X} _ {t} - L _ {X _ {t}  } X _ {t}  = \dot{X} _ {t} -
 +
(  \mathop{\rm ad}  ^ {*}  X _ {t} ) X _ {t}  = 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226096.png" /> is the operator dual to the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226097.png" />. The geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226098.png" /> can be recovered in terms of its velocity hodograph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i05226099.png" /> from the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260100.png" /> (which is linear if the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260101.png" /> is linear) or from the functional relations
+
where $  \mathop{\rm ad}  ^ {*}  X $
 +
is the operator dual to the adjoint representation $  \mathop{\rm ad}  X $.  
 +
The geodesic $  \gamma _ {t} $
 +
can be recovered in terms of its velocity hodograph $  X _ {t} $
 +
from the differential equation $  \dot \gamma  _ {t} = \gamma _ {t} X _ {t} $(
 +
which is linear if the group $  G $
 +
is linear) or from the functional relations
  
<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/i/i052/i052260/i052260102.png" /></td> </tr></table>
+
$$
 +
\langle  X _ {t} , (  \mathop{\rm Ad}  \gamma ( t) ) Y \rangle  = \textrm{ const } ,\  Y \in
 +
\mathfrak G ,
 +
$$
  
giving the first integrals of this equation. Thus, the description of the geodesics of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260103.png" /> reduces to the integration of the hodograph equation, which sometimes can be completely integrated. For example, in the case when the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260104.png" /> is also invariant with respect to right translations, the geodesics passing through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260105.png" /> are the one-parameter subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260106.png" />. Such a metric exists on any compact Lie group. In the case of an arbitrary homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260107.png" /> an invariant metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260108.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260109.png" /> can be  "lifted"  to a left-invariant metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260110.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260111.png" /> for which the natural bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260112.png" /> of the Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260113.png" /> over the Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260114.png" /> is a Riemannian bundle, that is, the length of tangent vectors orthogonal to the fibre remains unaltered under projection. For this it is sufficient to extend the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260115.png" /> to the entire algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260116.png" /> by setting
+
giving the first integrals of this equation. Thus, the description of the geodesics of the metric $  m $
 +
reduces to the integration of the hodograph equation, which sometimes can be completely integrated. For example, in the case when the metric $  m $
 +
is also invariant with respect to right translations, the geodesics passing through the point $  e $
 +
are the one-parameter subgroups of $  G $.  
 +
Such a metric exists on any compact Lie group. In the case of an arbitrary homogeneous space $  M = G / H $
 +
an invariant metric $  m $
 +
on $  G / H $
 +
can be  "lifted"  to a left-invariant metric $  \widetilde{m}  $
 +
on $  G $
 +
for which the natural bundle $  G \rightarrow G / H $
 +
of the Riemannian space $  ( G , \widetilde{m}  ) $
 +
over the Riemannian space $  ( G / H , m ) $
 +
is a Riemannian bundle, that is, the length of tangent vectors orthogonal to the fibre remains unaltered under projection. For this it is sufficient to extend the metric $  \langle  , \rangle $
 +
to the entire algebra $  \mathfrak G $
 +
by setting
  
<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/i/i052/i052260/i052260117.png" /></td> </tr></table>
+
$$
 +
\langle  \mathfrak H , \mathfrak M \rangle  = 0 \  \textrm{ and } \  \langle  X , Y \rangle  = \
 +
- \mathop{\rm Tr}  L _ {X} L _ {Y} \  ( X , Y \in \mathfrak H ) ,
 +
$$
  
and carrying it over by left translations to a metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260118.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260119.png" />. The geodesics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260120.png" /> are projections of geodesics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260121.png" /> that are orthogonal to the fibres.
+
and carrying it over by left translations to a metric $  \widetilde{m}  $
 +
on $  G $.  
 +
The geodesics of $  ( G / H , \widetilde{m}  ) $
 +
are projections of geodesics of $  ( G , m ) $
 +
that are orthogonal to the fibres.
  
Since the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260122.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260123.png" /> is always a first integral of the hodograph equation (the energy integral), the corresponding equation of the vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260124.png" /> is tangent to the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260125.png" />. This implies the completeness of the hodograph equation and therefore also the completeness of any invariant Riemannian metric on a homogeneous space. For a pseudo-Riemannian metric the completeness property does not hold, in general. On the other hand, any invariant pseudo-Riemannian metric on a compact homogeneous space is complete.
+
Since the function $  X \rightarrow \langle  X , X \rangle $
 +
on $  \mathfrak G $
 +
is always a first integral of the hodograph equation (the energy integral), the corresponding equation of the vector field on $  \mathfrak G $
 +
is tangent to the spheres $  \langle  X , X \rangle = \textrm{ const } $.  
 +
This implies the completeness of the hodograph equation and therefore also the completeness of any invariant Riemannian metric on a homogeneous space. For a pseudo-Riemannian metric the completeness property does not hold, in general. On the other hand, any invariant pseudo-Riemannian metric on a compact homogeneous space is complete.
  
 
See also [[Symmetric space|Symmetric space]].
 
See also [[Symmetric space|Symmetric space]].
Line 61: Line 216:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Z. Petrov,  "New methods in general relativity theory" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kobayashi,  "Transformation groups in differential geometry" , Springer  (1972)</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">  A. Lichnerowicz,  "Geometry of groups of transformations" , Noordhoff  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.L. Besse,  "Einstein manifolds" , Springer  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Z. Petrov,  "New methods in general relativity theory" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kobayashi,  "Transformation groups in differential geometry" , Springer  (1972)</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">  A. Lichnerowicz,  "Geometry of groups of transformations" , Noordhoff  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.L. Besse,  "Einstein manifolds" , Springer  (1987)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A de Rham decomposition (of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260126.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260127.png" />) is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260128.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260129.png" /> be the tangent space at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260130.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260131.png" /> be the [[Holonomy group|holonomy group]] of the Riemannian connection at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260132.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260133.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260134.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260135.png" /> be the subspace of tangent vectors that are left invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260136.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260137.png" /> be the orthogonal complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260138.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260139.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260140.png" /> be a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260141.png" /> into mutually-orthogonal invariant irreducible subspaces. The decomposition
+
A de Rham decomposition (of the tangent space $  T _ {x} M $
 +
at a point $  x $)  
 +
is defined as follows. Let $  x \in M $,  
 +
let $  T _ {x} M $
 +
be the tangent space at $  x $
 +
and let $  \Phi _ {x} $
 +
be the [[Holonomy group|holonomy group]] of the Riemannian connection at $  x $.  
 +
The group $  \Phi _ {x} $
 +
acts on $  T _ {x} M $.  
 +
Let $  T _ {x}  ^ {(} 0) M $
 +
be the subspace of tangent vectors that are left invariant under $  \Phi _ {x} $.  
 +
Let $  T _ {x}  ^  \prime  M $
 +
be the orthogonal complement of $  T _ {x}  ^ {(} 0) M $
 +
in $  T _ {x} M $
 +
and let $  T _ {x}  ^  \prime  M = \sum_{i=1} ^ {r} T _ {x}  ^ {(} i) M $
 +
be a decomposition of $  T _ {x}  ^  \prime  M $
 +
into mutually-orthogonal invariant irreducible subspaces. The decomposition
  
<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/i/i052/i052260/i052260142.png" /></td> </tr></table>
+
$$
 +
T _ {x} M  = \sum_{i=0}^ { r }  T _ {x}  ^ {(i)} M
 +
$$
  
 
is called a de Rham decomposition or a canonical decomposition.
 
is called a de Rham decomposition or a canonical decomposition.
  
An irreducible Riemannian manifold is one for which the holonomy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260143.png" /> acts irreducibly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260144.png" /> (so that there is only one factor in the Rham decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260145.png" />).
+
An irreducible Riemannian manifold is one for which the holonomy group $  \Phi _ {x} $
 +
acts irreducibly on $  T _ {x} M $(
 +
so that there is only one factor in the Rham decomposition of $  T _ {x} M $).
  
The de Rham decomposition theorem says that a connected simply-connected complete Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260146.png" /> is isometric to a direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260147.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260148.png" /> is a Euclidean space (possibly of dimension zero) and where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052260/i052260149.png" /> are all simply-connected complete irreducible Riemannian manifolds. Such a decomposition is unique up to the order of the factors, [[#References|[a1]]], Sect. IV. 6.
+
The de Rham decomposition theorem says that a connected simply-connected complete Riemannian manifold $  M $
 +
is isometric to a direct product $  M _ {0} \times M _ {1} \times \dots \times M _ {r} $
 +
where $  M _ {0} $
 +
is a Euclidean space (possibly of dimension zero) and where the $  M _ {1} \dots M _ {r} $
 +
are all simply-connected complete irreducible Riemannian manifolds. Such a decomposition is unique up to the order of the factors, [[#References|[a1]]], Sect. IV. 6.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR></table>

Latest revision as of 19:04, 9 January 2024


A Riemannian metric $ m $ on a manifold $ M $ that does not change under any of the transformations of a given Lie group $ G $ of transformations. The group $ G $ itself is called a group of motions (isometries) of the metric $ m $( or of the Riemannian space $ ( M , m ) $).

A Lie group $ G $ of transformations of a manifold $ M $ acting properly on $ M $( that is, the mapping $ G \times M \rightarrow M \times M $, $ ( g , x ) \rightarrow ( gx , x ) $ is proper) has an invariant metric. Conversely, the group of all motions of any Riemannian metric (as well as any closed subgroup of it) is a proper Lie group of transformations. In this case the stabilizer (or isotropy group)

$$ G _ {x} = \{ {g \in G } : {gx = x } \} $$

of any point $ x \in M $ is a compact subgroup of $ G $. If $ G $ itself is compact, then a $ G $- invariant metric $ m _ {0} $ can be constructed on $ M $ by averaging any metric $ m $ on $ M $ over $ G $:

$$ m _ {0} = \int\limits _ { G } ( g ^ {*} m ) dg , $$

where the integral is taken with respect to the Haar measure.

If $ G $ is transitive, $ M $ can be identified with the space of cosets $ G / H $ of $ G $ with respect to the stabilizer $ H = G _ {x _ {0} } $ of a fixed point $ x _ {0} \in M $, and in order that there exist a $ G $- invariant metric on $ M $ it is necessary and sufficient that the linear isotropy group (see Isotropy representation) has compact closure in $ \mathop{\rm GL} ( T _ {x _ {0} } M ) $( in particular, it is sufficient that $ H $ be compact). In this case the space $ G / H $ is reductive, that is, the Lie algebra $ \mathfrak G $ of $ G $ admits a decomposition $ \mathfrak G = \mathfrak H + \mathfrak M $, where $ \mathfrak H $ is the subalgebra corresponding to $ H $ and $ \mathfrak M $ is a subspace that is invariant under $ \mathop{\rm Ad} H $ where $ \mathop{\rm Ad} $ is the adjoint representation of $ G $( cf. Adjoint representation of a Lie group). If $ \mathfrak M $ is identified with $ T _ {x _ {0} } M $, then any $ G $- invariant metric $ m $ on $ M $ is obtained from some $ \mathop{\rm Ad} H $- invariant Euclidean metric $ \langle , \rangle $ on $ \mathfrak M $ in the following way:

$$ m _ {x} ( X , Y ) = \langle ( g ^ {*} ) ^ {-1} X, ( g ^ {*} ) ^ {-1} Y \rangle ,\ \ X , Y \in T _ {x} M , $$

where $ g \in G $ is such that $ g x _ {0} = x $.

The tensor fields associated with a $ G $- invariant metric (the curvature tensor, its covariant derivatives, etc.) are $ G $- invariant fields. In the case of a homogeneous space $ M = G / H $, their value at a point $ x _ {0} $ can be expressed in terms of the Nomizu operator $ L _ {X} \in \mathop{\rm End} ( \mathfrak M ) $, which is defined by the formula

$$ L _ {X} Y = - \nabla _ {Y} X ^ {*} = \ ( {\mathcal L} _ {X ^ {*} } - \nabla _ {X ^ {*} } ) _ {x _ {0} } Y ,\ \ Y \in \mathfrak M ,\ X \in \mathfrak G , $$

where $ X ^ {*} $ is the velocity field of the one-parameter group of transformations $ \mathop{\rm exp} tX $, $ \nabla $ is the covariant differentiation operator of the Riemannian connection and $ {\mathcal L} $ is the Lie derivative operator. In particular, the curvature operator $ \mathop{\rm Riem} ( X , Y ) $ and the sectional curvature $ K ( X, Y ) $ in the direction given by the orthonormal basis $ X , Y \in M $ satisfy the following formulas:

$$ \mathop{\rm Riem} ( X , Y ) = [ L _ {X} , L _ {Y} ] - L _ {[ X , Y ] } , $$

$$ K ( X , Y ) \equiv - \langle \mathop{\rm Riem} ( X , Y ) X , Y \rangle = $$

$$ = \ \langle L _ {X} Y , L _ {X} Y \rangle - < [ X , Y ] _ {\mathfrak M } , [ X , Y ] _ {\mathfrak M } > + $$

$$ - \langle [ Y , [ Y , X ] ] _ {\mathfrak M } , X \rangle - \langle L _ {X} X , L _ {Y} Y \rangle , $$

where $ Z _ {\mathfrak M } $ is the projection of $ Z \in \mathfrak G $ on $ \mathfrak M $ along $ \mathfrak H $.

The Nomizu operators can be expressed in terms of the Lie algebra $ \mathfrak G $ and the metric $ \langle , \rangle $ by the formula

$$ 2 \langle L _ {X} Y , Z \rangle = $$

$$ = \ \langle [ X , Y ] _ {\mathfrak M } , Z \rangle - < X , [ Y ,\ Z ] _ {\mathfrak M } > - \langle Y , [ X , Z ] _ {\mathfrak M } \rangle , $$

where $ X \in \mathfrak G $, $ Y , Z \in \mathfrak M $. It follows from the definition of the Nomizu operators that their action on $ G $- invariant fields differs only in sign from that of the covariant derivative at the point $ x _ {0} $. If the Riemannian space $ ( G / H , m) $ does not contain flat factors in the de Rham decomposition, then the linear Lie algebra generated by the Nomizu operators $ L _ {X} $, $ X \in \mathfrak G $, is the same as the holonomy algebra (cf. Holonomy group) of the space $ ( G / H , M) $ at $ x _ {0} $.

A description of the geodesics of an invariant metric on a homogeneous space can be given in the following way. Suppose, to begin with, that $ M = G $ is a Lie group acting on itself by left translations. Let $ \gamma _ {t} $ be a left-invariant geodesic of the metric $ m $ on the Lie group $ G $ and let $ X _ {t} = \gamma ^ {-1} \dot \gamma _ {t} $ be the curve in the Lie algebra $ \mathfrak G $ corresponding to it (the velocity hodograph). The curve $ X _ {t} $ satisfies the hodograph equation

$$ \dot{X} _ {t} - L _ {X _ {t} } X _ {t} = \dot{X} _ {t} - ( \mathop{\rm ad} ^ {*} X _ {t} ) X _ {t} = 0 , $$

where $ \mathop{\rm ad} ^ {*} X $ is the operator dual to the adjoint representation $ \mathop{\rm ad} X $. The geodesic $ \gamma _ {t} $ can be recovered in terms of its velocity hodograph $ X _ {t} $ from the differential equation $ \dot \gamma _ {t} = \gamma _ {t} X _ {t} $( which is linear if the group $ G $ is linear) or from the functional relations

$$ \langle X _ {t} , ( \mathop{\rm Ad} \gamma ( t) ) Y \rangle = \textrm{ const } ,\ Y \in \mathfrak G , $$

giving the first integrals of this equation. Thus, the description of the geodesics of the metric $ m $ reduces to the integration of the hodograph equation, which sometimes can be completely integrated. For example, in the case when the metric $ m $ is also invariant with respect to right translations, the geodesics passing through the point $ e $ are the one-parameter subgroups of $ G $. Such a metric exists on any compact Lie group. In the case of an arbitrary homogeneous space $ M = G / H $ an invariant metric $ m $ on $ G / H $ can be "lifted" to a left-invariant metric $ \widetilde{m} $ on $ G $ for which the natural bundle $ G \rightarrow G / H $ of the Riemannian space $ ( G , \widetilde{m} ) $ over the Riemannian space $ ( G / H , m ) $ is a Riemannian bundle, that is, the length of tangent vectors orthogonal to the fibre remains unaltered under projection. For this it is sufficient to extend the metric $ \langle , \rangle $ to the entire algebra $ \mathfrak G $ by setting

$$ \langle \mathfrak H , \mathfrak M \rangle = 0 \ \textrm{ and } \ \langle X , Y \rangle = \ - \mathop{\rm Tr} L _ {X} L _ {Y} \ ( X , Y \in \mathfrak H ) , $$

and carrying it over by left translations to a metric $ \widetilde{m} $ on $ G $. The geodesics of $ ( G / H , \widetilde{m} ) $ are projections of geodesics of $ ( G , m ) $ that are orthogonal to the fibres.

Since the function $ X \rightarrow \langle X , X \rangle $ on $ \mathfrak G $ is always a first integral of the hodograph equation (the energy integral), the corresponding equation of the vector field on $ \mathfrak G $ is tangent to the spheres $ \langle X , X \rangle = \textrm{ const } $. This implies the completeness of the hodograph equation and therefore also the completeness of any invariant Riemannian metric on a homogeneous space. For a pseudo-Riemannian metric the completeness property does not hold, in general. On the other hand, any invariant pseudo-Riemannian metric on a compact homogeneous space is complete.

See also Symmetric space.

References

[1] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[2] A.Z. Petrov, "New methods in general relativity theory" , Moscow (1966) (In Russian)
[3] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)
[4] S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
[5] J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1984)
[6] A. Lichnerowicz, "Geometry of groups of transformations" , Noordhoff (1977) (Translated from French)
[7] A.L. Besse, "Einstein manifolds" , Springer (1987)

Comments

A de Rham decomposition (of the tangent space $ T _ {x} M $ at a point $ x $) is defined as follows. Let $ x \in M $, let $ T _ {x} M $ be the tangent space at $ x $ and let $ \Phi _ {x} $ be the holonomy group of the Riemannian connection at $ x $. The group $ \Phi _ {x} $ acts on $ T _ {x} M $. Let $ T _ {x} ^ {(} 0) M $ be the subspace of tangent vectors that are left invariant under $ \Phi _ {x} $. Let $ T _ {x} ^ \prime M $ be the orthogonal complement of $ T _ {x} ^ {(} 0) M $ in $ T _ {x} M $ and let $ T _ {x} ^ \prime M = \sum_{i=1} ^ {r} T _ {x} ^ {(} i) M $ be a decomposition of $ T _ {x} ^ \prime M $ into mutually-orthogonal invariant irreducible subspaces. The decomposition

$$ T _ {x} M = \sum_{i=0}^ { r } T _ {x} ^ {(i)} M $$

is called a de Rham decomposition or a canonical decomposition.

An irreducible Riemannian manifold is one for which the holonomy group $ \Phi _ {x} $ acts irreducibly on $ T _ {x} M $( so that there is only one factor in the Rham decomposition of $ T _ {x} M $).

The de Rham decomposition theorem says that a connected simply-connected complete Riemannian manifold $ M $ is isometric to a direct product $ M _ {0} \times M _ {1} \times \dots \times M _ {r} $ where $ M _ {0} $ is a Euclidean space (possibly of dimension zero) and where the $ M _ {1} \dots M _ {r} $ are all simply-connected complete irreducible Riemannian manifolds. Such a decomposition is unique up to the order of the factors, [a1], Sect. IV. 6.

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
How to Cite This Entry:
Invariant metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_metric&oldid=18868
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article