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A [[Homogeneous space|homogeneous space]]  $  G/H $
of a connected [[Lie group|Lie group]]  $  G $
such that in the [[Lie algebra|Lie algebra]]  $  \mathfrak g $
of  $  G $
there is an  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $-
invariant subspace complementary to the subalgebra  $  \mathfrak h \subset  \mathfrak g $, 
where  $  \mathfrak h $
is the Lie algebra of the group  $  H $. 
The validity of any of the following conditions is sufficient for the homogeneous space  $  G/H $
to be reductive: 1) the linear group  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $
is completely reducible; or 2) in  $  \mathfrak g $
there is an  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $-
invariant bilinear form whose restriction to  $  \mathfrak h $
is non-degenerate. In particular, any homogeneous Riemannian space is reductive. If  $  M = G/H $
is a reductive space and the group  $  G $
acts effectively on  $  M $, 
then the linear representation of the isotropy group  $  H $
in the tangent space  $  M _ {0} $
to the manifold  $  M $
at the point  $  0 = eH \in M $
is faithful (cf. [[Faithful representation|Faithful representation]]). Two important  $  G $-
invariant affine connections on  $  M $
are associated with each  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $-
invariant subspace  $  \mathfrak m \subset  \mathfrak g $
complementary to  $  \mathfrak h $: 
the canonical connection and the natural torsion-free connection. The canonical connection on the reductive space  $  M = G/H $
with a fixed  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $-
invariant decomposition  $  \mathfrak g = \mathfrak h \dot{+} \mathfrak m $
is the unique  $  G $-
invariant affine connection on  $  M $
such that for any vector  $  X \in \mathfrak m $
and any frame  $  u $
at the point 0, the curve  $  (  \mathop{\rm exp}  tX) u $
in the principal fibration of frames over  $  M $
is horizontal. The canonical connection is complete and the set of its geodesics through 0 coincides with the set of curves of the type  $  (  \mathop{\rm exp}  tX) 0 $, 
where  $  X \in \mathfrak m $. 
After the natural identification of the spaces  $  \mathfrak m $
and  $  M _ {0} $, 
the curvature tensor  $  R $
and torsion tensor  $  T $
of the canonical connection are defined by the formulas  $  ( R( X, Y) Z) _ {0} = - [[ X, Y] _ {\mathfrak h} , Z] $
and  $  T( X, Y) _ {0} = -[ X, Y] _ {\mathfrak m} $, 
where  $  X, Y, Z \in \mathfrak m $
and  $  W _ {\mathfrak h} $
and  $  W _ {\mathfrak m} $
denote the projections of the vector  $  W \in \mathfrak g $
onto  $  \mathfrak h $
and  $  \mathfrak m $, 
respectively.

The tensor fields  $  R $
and  $  T $
are parallel relative to the canonical connection, as is any other  $  G $-
invariant tensor field on  $  M $. 
The Lie algebra of the linear holonomy group (see [[Holonomy group|Holonomy group]]) of the canonical connections on  $  M $
with supporting point 0 is generated by the set  $  \{ {\lambda ([ X, Y] _ {\mathfrak h} ) } : {X, Y \in \mathfrak m } \} $, 
where  $  \lambda $
is the linear representation of the isotropy Lie algebra  $  \mathfrak h $
in the space  $  M _ {0} $. 
Any connected simply-connected manifold with a complete affine connection with parallel curvature and torsion fields can be represented as a reductive space whose canonical connection coincides with the given affine connection. In the reductive space  $  M = G/H $
with fixed  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $-
invariant decomposition  $  \mathfrak g = \mathfrak h \dot{+} \mathfrak m $
there is a unique  $  G $-
invariant affine connection with zero torsion having the same geodesics as the canonical connection. This connection is called the natural torsion-free connection on  $  M $(
relative to the decomposition  $  \mathfrak g = \mathfrak h \dot{+} \mathfrak m $). 
A homogeneous Riemannian or pseudo-Riemannian space  $  M = G/H $
is naturally reductive if it admits an  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $-
invariant decomposition  $  \mathfrak g = \mathfrak h \dot{+} \mathfrak m $
such that

$$ \tag{* }
B( X, [ Z, Y] _ {\mathfrak m} ) + B ([ Z, X] _ {\mathfrak m} , Y)  =  0
$$

for all  $  X, Y, Z \in \mathfrak m $, 
where  $  B $
is the non-degenerate symmetric bilinear form on  $  \mathfrak m $
induced by the Riemannian (pseudo-Riemannian) structure on  $  M $
under the natural identification of the spaces  $  \mathfrak m $
and  $  M _ {0} $. 
If  $  M = G/H $
is a naturally reductive Riemannian or pseudo-Riemannian space with a fixed  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $-
invariant decomposition  $  \mathfrak g = \mathfrak h \dot{+} \mathfrak m $
that satisfies condition (*), then the natural torsion-free connection coincides with the corresponding Riemannian or pseudo-Riemannian connection on  $  M $. 
If  $  M $
is a simply-connected naturally reductive homogeneous Riemannian space and  $  M = M _ {0} \times \dots \times M _ {r} $
is its de Rham decomposition, then  $  M $
can be represented in the form  $  M = G/H $; 
moreover,  $  G = G _ {0} \times \dots \times G _ {r} $, 
$  H = H _ {0} \times \dots \times H _ {r} $
and  $  M _ {i} = G _ {i} /H _ {i} $
$  ( i = 0 \dots r) $.

An important generalization of reductive spaces are  $  \nu $-
reductive homogeneous spaces [[#References|[4]]]. A homogeneous space  $  G/H $
is called  $  \nu $-
reductive if its stationary subalgebra  $  \mathfrak h $
equals  $  \mathfrak h _ {1} \dot{+} \dots \dot{+} \mathfrak h _  \nu  $, 
where  $  \mathfrak h _  \nu  \neq \{ 0 \} $, 
and if there is a subspace  $  \mathfrak m $
in  $  \mathfrak h $
complementary to  $  \mathfrak h $
such that  $  [ \mathfrak h _ {i} , \mathfrak m ] \subset  \mathfrak h _ {i-} 1 $, 
$  i = 1 \dots \nu $, 
where  $  \mathfrak h _ {0} = \mathfrak m $. 
The  $  1 $-
reductive homogeneous spaces are in fact reductive spaces; examples of  $  2 $-
reductive homogeneous spaces are projective (and conformal) spaces on which a group of projective (or conformal) transformations acts. If  $  M = G/H $
there is a  $  \nu $-
reductive homogeneous space and if  $  \nu > 1 $, 
then the linear representation of the isotropy Lie algebra  $  \mathfrak h $
is not faithful (since  $  [ \mathfrak h _ {i} , \mathfrak m ] \subset  \mathfrak h $
when  $  i > 1 $); 
consequently, there is no  $  G $-
invariant affine connection on  $  M $. 
However, there is a canonical  $  G $-
invariant connection on a  $  \nu $-
reductive homogeneous space with the homogeneous space of some transitive-differential group of order  $  \nu $
as fibre (see [[#References|[4]]]). Reductive and  $  \nu $-
reductive spaces are characterized as maximally homogeneous  $  G $-
structures (cf. [[G-structure| $  G $-
structure]]) of appropriate type (cf. [[#References|[6]]]).

In addition to reductive spaces, partially reductive spaces are also examined, i.e. homogeneous spaces  $  G/H $
such that there is a decomposition of the Lie algebra  $  \mathfrak g $
into a direct sum of two non-zero  $   \mathop{\rm Ad} _ {\mathfrak g} ( H) $-
invariant subspaces, one of which contains the subalgebra  $  \mathfrak h $(
see [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Rashevskii,   "On the geometry of homogeneous spaces"  ''Trudy Sem. Vektor. i Tenzor. Anal.'' , '''9'''  (1952)  pp. 49–74</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Nomizu,   "Invariant affine connections on homogeneous spaces"  ''Amer. J. Math.'' , '''76''' :  1  (1954)  pp. 33–65</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.L. Kantor,   "Transitive differential groups and invariant connections in homogeneous spaces"  ''Trudy Sem. Vektor. i Tenzor. Anal.'' , '''13'''  (1966)  pp. 310–398</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.B. Vinberg,   "Invariant linear connections in a homogeneous space"  ''Trudy Moskov. Mat. Obshch.'' , '''9'''  (1960)  pp. 191–210  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.V. Alekseevskii,   "Maximally homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080450/r080450119.png" />-structures and filtered Lie algebras"  ''Soviet Math. Dokl.'' , '''37''' :  2  (1988)  pp. 381–384  ''Dokl. Akad. Nauk SSSR'' , '''299''' :  3  (1988)  pp. 521–526</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.A. Wolf,   "Spaces of constant curvature" , McGraw-Hill  (1967)</TD></TR></table>