Riemannian space, homogeneous

A Riemannian space $ ( M, \gamma ) $ together with a transitive effective group $ G $ of motions (cf. Motion) on it. Let $ K $ be the isotropy subgroup of a fixed point $ o \in M $. Then the manifold $ M $ is identified with the quotient space $ G/K $ by the bijection $ G/K \ni gK \iff go \in M $, and the Riemannian metric $ \gamma $ is considered as a $ G $- invariant metric on $ G/K $. Usually one assumes in addition that the group $ G $ is closed in the complete group of motions. In this case the isotropy group $ K $ is compact.

Let $ K $ be a compact subgroup of a Lie group $ G $ that does not contain normal subgroups of $ G $. Then the homogeneous space $ M = G/K $ admits an invariant Riemannian metric $ \gamma $ defined as follows. Let $ \mathfrak G = \mathfrak K + \mathfrak M $ be a reductive structure in $ M $, i.e. a decomposition of the Lie algebra $ \mathfrak G $ of $ G $ into the direct sum of the Lie algebra $ \mathfrak K $ of $ K $ and a subspace $ \mathfrak M $ that is invariant under the adjoint representation $ \mathop{\rm Ad} K $ of $ K $ in $ \mathfrak G $. The space $ \mathfrak M $ is naturally identified with the tangent space $ T _ {0} M \approx \mathfrak G / \mathfrak K $ at the point $ o = eK $, and the isotropy representation of the group $ K $ in $ T _ {0} M $ with the representation $ \mathop{\rm Ad} K \ \mid _ {\mathfrak M} $. Any $ G $- invariant Riemannian metric $ \gamma $ on $ M $ is obtained from some $ \mathop{\rm Ad} K $- invariant scalar product $ \gamma _ {0} $ in $ \mathfrak M $ by translations from $ G $:

$$ \gamma _ {go} ( X, Y) = \gamma _ {0} ( g ^ {-} 1 X, g ^ {-} 1 Y),\ \ X, Y \in T _ {go} M,\ \ g \in G. $$

The existence of such a scalar product follows from the fact that the isotropy group $ \mathop{\rm Ad} K \ \mid _ {\mathfrak M} $ is compact.

Any homogeneous Riemannian space locally isometric to a simply-connected homogeneous Riemannian space $ \widetilde{M} $ is obtained from $ \widetilde{M} $ by factorization with respect to an arbitrary Clifford–Wolf discrete group of isometries (i.e. motions of the manifold $ \widetilde{M} $ that displace all points by equal distances [2]).

The best studied classes of homogeneous Riemannian spaces are the Riemannian symmetric spaces (cf. also Symmetric space); homogeneous Kähler spaces (cf. Kähler manifold) and homogeneous quaternionic spaces; isotropically-irreducible homogeneous Riemannian spaces (classified in [9], [10]); normal homogeneous Riemannian spaces, in which the scalar product $ \gamma _ {0} $ in $ \mathfrak M $ is defined by a non-degenerate symmetric $ \mathop{\rm Ad} G $- invariant bilinear form on $ \mathfrak G $; and naturally-reductive homogeneous Riemannian spaces, characterized by the fact that any geodesic in them is the trajectory of a one-parameter group of motions.

The structure of homogeneous Riemannian spaces with different conditions on the curvature tensor is well studied. For instance, the classification of homogeneous Riemannian spaces of positive sectional curvature is known [5]. The structure of simply-transitive groups of motions of a homogeneous Riemannian space of non-positive curvature [8], of non-negative curvature and of non-negative Ricci curvature [4] has been described. A homogeneous Riemannian space with a solvable group of motions $ G $ always has a non-positive scalar curvature $ \mathop{\rm sc} $, and the case $ \mathop{\rm sc} = 0 $ is possible only for locally Euclidean spaces. Any invariant Riemannian metric on a simply-connected homogeneous Riemannian space $ G/K $ has non-positive scalar curvature if and only if $ K $ is a maximal compact subgroup of $ G $( see [4]).

A homogeneous Riemannian space $ ( M, \gamma ) $ is called Einstein if its Ricci tensor $ \rho $ is proportional to the metric: $ \rho = \lambda \gamma $, $ \lambda = \textrm{ const } $. The problem of the description of Einstein homogeneous Riemannian spaces has not yet been solved (1991). One knows a number of particular results. Let $ ( M = G/K, \gamma ) $ be an Einstein homogeneous Riemannian space of scalar curvature $ \mathop{\rm sc} $. 1) If $ \mathop{\rm sc} > 0 $, then $ M $ is a compact manifold. All such spaces have been described: a) if $ ( M, \gamma ) $ is a quaternionic space; b) if $ M $ is diffeomorphic to a symmetric space of rank one; and c) for a certain class of naturally-reductive homogeneous Riemannian spaces (see [7]) and for isotropically-irreducible homogeneous Riemannian spaces (see [10]). 2) If $ \mathop{\rm sc} = 0 $, then $ M $ is a locally Euclidean space. 3) If $ \mathop{\rm sc} < 0 $ and $ G $ is unimodular (i.e. the determinant of its adjoint representation operator is equal to 1), then the group $ G $ is semi-simple.

References

[1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1968)
[2]	J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1977)
[3]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[4]	L. Berard Bergery, "Sur le courbure des métriques riemanniennes invariantes des groupes de Lie et des espaces homogènes" Ann. Sci. Ecole Norm. Sup. , 11 : 4 (1978) pp. 545–576
[5]	L. Berard Bergery, "Les variétés riemanniennes simplement connexes de dimension impairé à courbure strictement positive" J. Math. Pures Appl. , 55 (1976) pp. 47–67
[6]	G.R. Jensen, "Einstein metrics on principle fiber bundles" J. Dif. Geom. , 8 (1973) pp. 599–614
[7]	J.E. d'Atri, W. Ziller, "Naturally reductive metrics and Einstein metrics on compact Lie groups" Mem. Amer. Math. Soc. , 18 (1979) pp. 1–72
[8]	R. Azencott, E.N. Wilson, "Homogeneous manifolds with negative curvature II" Mem. Amer. Math. Soc. , 8 (1976) pp. 1–102
[9]	O.V. Manturov, "Homogeneous Riemannian spaces with an irreducible rotation group" Trudy Sem. Vektor. i Tenzor. Anal. , 13 (1966) pp. 68–145 (In Russian)
[10]	J. Wolf, "The geometry and structure of isotropy irreducible homogeneous spaces" Acta Math. , 120 (1968) pp. 59–148

Comments

For a quite exhaustive treatment of Einstein manifolds see [a1], esp. Chapts. 7, 8.

Usually, an isometry of a Riemannian space is used as a synonym for motion, while the isometries in a Clifford–Wolf discrete group are known as Clifford translations, [2].

References

[a1]	A.L. Besse, "Einstein manifolds" , Springer (1987)