Reductive space
A homogeneous space of a connected Lie group
such that in the Lie algebra
of
there is an
-invariant subspace complementary to the subalgebra
, where
is the Lie algebra of the group
. The validity of any of the following conditions is sufficient for the homogeneous space
to be reductive: 1) the linear group
is completely reducible; or 2) in
there is an
-invariant bilinear form whose restriction to
is non-degenerate. In particular, any homogeneous Riemannian space is reductive. If
is a reductive space and the group
acts effectively on
, then the linear representation of the isotropy group
in the tangent space
to the manifold
at the point
is faithful (cf. Faithful representation). Two important
-invariant affine connections on
are associated with each
-invariant subspace
complementary to
: the canonical connection and the natural torsion-free connection. The canonical connection on the reductive space
with a fixed
-invariant decomposition
is the unique
-invariant affine connection on
such that for any vector
and any frame
at the point 0, the curve
in the principal fibration of frames over
is horizontal. The canonical connection is complete and the set of its geodesics through 0 coincides with the set of curves of the type
, where
. After the natural identification of the spaces
and
, the curvature tensor
and torsion tensor
of the canonical connection are defined by the formulas
and
, where
and
and
denote the projections of the vector
onto
and
, respectively.
The tensor fields and
are parallel relative to the canonical connection, as is any other
-invariant tensor field on
. The Lie algebra of the linear holonomy group (see Holonomy group) of the canonical connections on
with supporting point 0 is generated by the set
, where
is the linear representation of the isotropy Lie algebra
in the space
. 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
with fixed
-invariant decomposition
there is a unique
-invariant affine connection with zero torsion having the same geodesics as the canonical connection. This connection is called the natural torsion-free connection on
(relative to the decomposition
). A homogeneous Riemannian or pseudo-Riemannian space
is naturally reductive if it admits an
-invariant decomposition
such that
![]() | (*) |
for all , where
is the non-degenerate symmetric bilinear form on
induced by the Riemannian (pseudo-Riemannian) structure on
under the natural identification of the spaces
and
. If
is a naturally reductive Riemannian or pseudo-Riemannian space with a fixed
-invariant decomposition
that satisfies condition (*), then the natural torsion-free connection coincides with the corresponding Riemannian or pseudo-Riemannian connection on
. If
is a simply-connected naturally reductive homogeneous Riemannian space and
is its de Rham decomposition, then
can be represented in the form
; moreover,
,
and
.
An important generalization of reductive spaces are -reductive homogeneous spaces [4]. A homogeneous space
is called
-reductive if its stationary subalgebra
equals
, where
, and if there is a subspace
in
complementary to
such that
,
, where
. The
-reductive homogeneous spaces are in fact reductive spaces; examples of
-reductive homogeneous spaces are projective (and conformal) spaces on which a group of projective (or conformal) transformations acts. If
there is a
-reductive homogeneous space and if
, then the linear representation of the isotropy Lie algebra
is not faithful (since
when
); consequently, there is no
-invariant affine connection on
. However, there is a canonical
-invariant connection on a
-reductive homogeneous space with the homogeneous space of some transitive-differential group of order
as fibre (see [4]). Reductive and
-reductive spaces are characterized as maximally homogeneous
-structures (cf.
-structure) of appropriate type (cf. [6]).
In addition to reductive spaces, partially reductive spaces are also examined, i.e. homogeneous spaces such that there is a decomposition of the Lie algebra
into a direct sum of two non-zero
-invariant subspaces, one of which contains the subalgebra
(see [5]).
References
[1] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969) |
[2] | P.K. Rashevskii, "On the geometry of homogeneous spaces" Trudy Sem. Vektor. i Tenzor. Anal. , 9 (1952) pp. 49–74 |
[3] | K. Nomizu, "Invariant affine connections on homogeneous spaces" Amer. J. Math. , 76 : 1 (1954) pp. 33–65 |
[4] | I.L. Kantor, "Transitive differential groups and invariant connections in homogeneous spaces" Trudy Sem. Vektor. i Tenzor. Anal. , 13 (1966) pp. 310–398 |
[5] | E.B. Vinberg, "Invariant linear connections in a homogeneous space" Trudy Moskov. Mat. Obshch. , 9 (1960) pp. 191–210 (In Russian) |
[6] | D.V. Alekseevskii, "Maximally homogeneous ![]() |
Comments
References
[a1] | J.A. Wolf, "Spaces of constant curvature" , McGraw-Hill (1967) |
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