Reductive space

A homogeneous space $ G/H $ of a connected Lie group $ G $ such that in the 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). 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) 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 [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 [4]). Reductive and $ \nu $- reductive spaces are characterized as maximally homogeneous $ G $- structures (cf. $ G $- structure) of appropriate type (cf. [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 [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 -structures and filtered Lie algebras" Soviet Math. Dokl. , 37 : 2 (1988) pp. 381–384 Dokl. Akad. Nauk SSSR , 299 : 3 (1988) pp. 521–526

References

[a1]	J.A. Wolf, "Spaces of constant curvature" , McGraw-Hill (1967)