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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 ) $).

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|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:

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

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|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:

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

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

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

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

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|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|Symmetric space]].

====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>

====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|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, [[#References|[a1]]], Sect. IV. 6.

====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>