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 -structures and filtered Lie algebras" Soviet Math. Dokl. , 37 : 2 (1988) pp. 381–384 Dokl. Akad. Nauk SSSR , 299 : 3 (1988) pp. 521–526