UDK 512.816

Weakly symmetric Riemannian spaces with reductive isometry group.

O.S. Yakimova

1. Introduction

Let M be a connected Riemannian manifold and Isom(M) be the full isometry group of M.

Definition 1. The manifold M is said to be symmetric, if for every point x ∈ M there is an isometry s ∈ Isom(M) such that s(x) = x and d_{x}s = −id.

The theory of symmetric manifolds is well developed. Works of Élie Cartan and Helgason, see [12], [13] describe their structure and also deal with harmonic analysis on such manifolds. We will freely use results of these classical works.

The notion of a weakly symmetric manifold is a generalisation of a notion of a symmetric manifold.

Definition 2. The manifold M is said to be weakly symmetric, if the following equivalent conditions hold:

i) ∀x, y ∈ M ∃s ∈ Isom(M) : s(x) = y, s(y) = x.

ii) ∀x ∈ M ∀ξ ∈ T_{x}(M) ∃s ∈ Isom(M) : s(x) = x, ds_{x}(ξ) = −ξ. (The equivalence of the given conditions is proved, for example, in [8].)

The geometric meaning of Definition 2 is that for every point x ∈ M and for every geodesic line containing x there is an isometry stabilising x and reversing the geodesic line.

The notion of a weakly symmetric manifold was introduced by Selberg in [16]. It is well known that there are weakly symmetric Riemannian manifolds which are not symmetric. For example, as was noticed by Selberg, a homogeneous space (SL_{2}(ℝ) × SO_{2})/SO_{2} is weakly symmetric but not symmetric. Recently weakly symmetric Riemannian manifold were intensively studied by several people, see, e.g., [8], [9], [10], [15], [7], [2], [5]. These works show that weakly symmetric spaces possess a fairly interesting geometry. Examples of non-symmetric weakly symmetric manifolds were known before, see [8], [15], but they were constructed as Riemannian manifolds rather then as homogeneous spaces. The proofs of non-symmetry were essentially based on the explicit description of the metric and its curvature tensor.

The main object of the present paper is a consideration of Riemannian manifolds as a homogeneous spaces of Lie groups. This allows us to construct numerous new examples of nonsymmetric weakly symmetric Riemannian manifolds. Among weakly symmetric homogeneous spaces of reductive Lie groups, we distinguish non-symmetric Riemannian manifolds. Thereby, we obtain a classification, up to local isomorphism, of all non-symmetric weakly symmetric Riemannian manifolds having reductive isometry group; in particular, the compact ones. Moreover, for each of these manifolds, dimensions of sets of symmetric and non-symmetric Riemannian metrics (see Definition 4) are given. The classification is presented in Table 1. We also prove that an irreducible non-compact non-symmetric homogeneous space of a semisimple Lie group is not a symmetric manifold regardless of the choice of an invariant metric, see Theorem 2.

It is likely that this approach to weakly symmetric Riemannian manifolds can lead to their complete classification.

The main idea of this paper is the following. A weakly symmetric Riemannian manifold having reductive isometry group is a weakly symmetric homogeneous space, and hence, a real form of some spherical homogeneous space, [7]. Spherical homogeneous spaces and their real forms are classified, [11], [3], [6]. If a homogeneous space G/K is a symmetric Riemannian manifold for some choice of a G-invariant Riemannian metric, then there is a decomposition Isom(G/K) = GQ, where Q is a symmetric subgroup of Isom(G/K). The decompositions of the reductive groups are described in a work by A.L. Onishchick, see [4], that gives us a clue to understanding whether G/K can be a symmetric Riemannian manifold. Via a sequence of reductions, the problem of classifying non-symmetric weakly symmetric Riemannian manifolds is reduced to the consideration of homogeneous spaces from Table 1.

Let us denote Lie groups by capital letters and their Lie algebras by the corresponding lowercase gothic letters. Let G′ stand for the commutator group of G and N_{G}(K) stand for the normaliser of K in G. If it is clear which group G is meant we will write simply N(K).

Acknowledgements. I am grateful to my scientific advisor Professor E.B. Vinberg for attentive guidance during all stages of this work. This paper was completed during my stay at the Max-Planck-Institut für Mathematik (Bonn). I would like to thank the Institut for hospitality and excelent working conditions.

2. Premises and results

In this section G is a real Lie group, K ⊂ G a compact subgroup such that the manifold M = G/K is connected.

Definition 1′. The homogeneous space M is called symmetric, if there is an involution σ of G such that (G^{σ})^{0} ⊂ K ⊂ G^{σ}.

Definition 2′. The homogeneous space M is called weakly symmetric, if there is an automorphism σ ∈ Aut(G), normalising K, such that iii) ∀ x, y ∈ G ∃g ∈ G : σ(x)K = gyK, σ(y)K = gxK.

If M = G/K is (weakly) symmetric then the pair (G, K) is called (weakly) symmetric.

Let M be a weakly symmetric Riemannian manifold. Then, due to the condition i), Isom(M) acts on M transitively, in particular, M is a complete Riemannian manifold. Denote by St(x) ⊂ Isom(M) the stabiliser of a point x ∈ M. We have

M = Isom(M)/St(x) = Isom(M)^{0}/(St(x) ∩ Isom(M)^{0}),

Isom(M) is a real Lie group, St(x) is a compact subgroup of Isom(M). It is easy to verify, that the homogeneous space M of the group Isom(M) is weakly symmetric with respect to the identity map. The analogues fact concerning symmetric manifolds is well known. A symmetric Riemannian manifold M is a symmetric homogeneous space of the group Isom(M) and also of the group Isom(M)^{0}.

On the other hand, suppose a homogeneous space M = G/K is (weakly) symmetric with respect to some automorphism σ. Let us introduce a G-invariant Riemannian metric on M. (This metric will be also σ-invariant due to condition iii).) Determine an isometry s ∈ Isom(M) by the formula s(gK) = σ(g)K. Then G/K becomes a (weakly) symmetric Riemannian manifold.

Thus, the notions of (weakly) symmetric Riemannian manifolds and (weakly) symmetric homogeneous space are quite close. The difference between them lays in the fact that a weakly symmetric Riemannian manifold M can be a weakly symmetric homogeneous space of several groups.

Let M = G/K be a weakly symmetric homogeneous space. Then M is also a weakly symmetric homogeneous space of G^{0}, see [2, Lemma 1]. From now on assume that G is connected. This restriction is not important for our goal, i.e., classification of non-symmetric weakly symmetric Riemannian manifolds.

A non-symmetric homogeneous space can be a symmetric Riemannian manifold. For instance, a sphere S^{2n−1} = SU_{n}/SU_{n−1} = SO_{2n}/SO_{2n−1} is a symmetric Riemannian manifold and simultaneously a non-symmetric weakly symmetric homogeneous space of the group SU_{n}.

Definition 3. Let M = G/K be a homogeneous space, μ be a G-invariant Riemannian metric on M. We say that μ is (weakly) symmetric, if the pair (M, μ) is a (weakly) symmetric Riemannian manifold.

To understand whether a given Riemannian metric is symmetric it is sufficient to know the isometry group of the pair (M, μ) or its connected component Isom(M)^{0}.

Denote by 𝔪 the tangent space T_{eK}(G/K) ≅ 𝔤/𝔨. The space 𝔪 can be identified with a K-invariant complement of 𝔨 in 𝔤. Let 𝔹(𝔪) be a set of all positive-definite K-invariant scalar products on the vector space 𝔪. In order to determine a G-invariant Riemannian metric on G/K, it is necessary and sufficient to choose an element of 𝔹(𝔪).

As was proved in [2], a connected homogeneous space that is locally isomorphic to a weakly symmetric one is also weakly symmetric. Unfortunately, the analogous statement is not true for symmetric spaces.

Let M̃ = G̃/K̂ be a simply connected covering of M. Here G̃ is a simply connected covering of G and K̂ is a connected subgroup, whose Lie algebra is 𝔨. Any G-invariant Riemannian metric μ on M can be lifted to a G̃-invariant Riemannian metric μ̃ on M̃. Now we will show how to decide whether M is symmetric, assuming that Isom(M̃) is known.

Suppose (M, μ) is a symmetric Riemannian manifold. Then (M̃, μ̃) is also symmetric. Moreover, M is a symmetric homogeneous space of Isom(M̃)^{0}.

On the other hand, suppose we know that (M̃, μ̃) is symmetric. One can easily check if (M, μ) is symmetric or not. Denote by F the group Isom(M̃); by σ an involution of F, determining a symmetric space structure on M̃. Denote by N the ineffective kernel of an action of F on M. We have Isom(M)^{0} = F/N. Evidently, M is symmetric if and only if σ(N) = N.

Thus, the problem of finding all non-symmetric Riemannian manifold among the weakly symmetric homogeneous spaces is reduced to the same problem for simply connected homogeneous spaces. Definition 4. A linear Lie group G ⊂ GL(V), having finitely many connected components, is called reductive, if it is completely reducible. A Lie algebra of a reductive group is called a reductive algebra.

By the definition, any compact group is reductive.

We will deal with homogeneous spaces of reductive groups. But before we restrict ourself to this case, note that it is quite possible that M̃ is not a homogeneous space of any reductive group even if M is. For example, there is no reductive group acting transitively on a real line ℝ = Ũ_{1}.

In particular, having restricted the area of our consideration to homogeneous spaces of reductive groups, we might loose some weakly symmetric spaces. To avoid this unhappy event, we prove the following lemma.

Lemma 1. Let M = G/K be a homogeneous space of a reductive group G and F := G′K ≠ G. Then M̃ decomposes, as a Riemannian manifold, into a product M̃ = M_{s} × M_{0}, where M_{s} is a simply connected covering of the homogeneous space G′/(G′ ∩ K) and M_{0} is a locally euclidian symmetric manifold.

Proof. Let μ be a G-invariant Riemannian metric on M, determined by an element b ∈ 𝔹(𝔪). Denote by 𝔪_{1} an orthogonal complement of 𝔣/𝔨 in 𝔪. Evidently, 𝔪_{1} ⊂ 𝔷(𝔤). There is an equality of Riemannian manifolds M̃ = M_{s} × M_{0}, where M_{0} is a locally euclidian symmetric Riemannian manifold and M_{s} is a simply connected covering of F/K endowing with a F - invariant Riemannian metric determined by a scalar product b|_{𝔣/𝔨}. To conclude the proof, note that because F = G′K, we have F/K = G′/(G′ ∩ K). □

Corollary. A Riemannian manifold M̃ is symmetric if and only if M_{s} is symmetric; moreover, if M_{s} is symmetric, then Isom(M_{s}) is semisimple and there is an equality Isom(M̃) = Isom(M_{s}) × Isom(M_{0}).

Proof. The product of two Riemannian manifolds is symmetric if and only if each of them is symmetric, hence the first statement is true.

Suppose M_{s} is symmetric. Assume that Isom(M_{s}) is not semisimple, i.e., M_{s} is locally isomorphic to the product M_{1}×ℝ^{n}, where M_{1} is a symmetric homogeneous space of a semisimple group Isom(M_{1}) and n ≥ 1. According to the conditions of the lemma, the group G̃′ acts on M_{s} transitively. Hence, G̃′ has a non-trivial connected centre. This leads to a contradiction, because G′ is semisimple. Since M_{s} is a symmetric homogeneous space os a semisimple group, each isometry of M̃ preserves M_{s} and M_{0}, see [13]. □

From now on, we assume that G is connected and reductive, G/K is simply connected, and G = G′K.

Let G = G_{1} × G_{2}, K = K_{1} × K_{2}, where K_{i} is a compact subgroup of G_{i}. Then one says that the homogeneous space M = G/K is reducible and writes (G, K) = (G_{1}, K_{1}) × (G_{2}, K_{2}). Definition 5. An affine complex F -variety X is called spherical if a Borel subgroup B(F) ⊂ F has an open orbit in X. If F/H is spherical, then the pair (F, H) and the subgroup H are also called spherical.

As was proved in [7], the complexification of a weakly symmetric homogeneous space is a spherical homogeneous space. Furthermore, a real space is irreducible if and only if the complexification is irreducible, see [6]. Affine spherical homogeneous spaces are classified in [11], [3], [6], and their real forms are found in [6].

Theorem 1. Suppose (G, K) = (G_{1}, K_{1}) × (G_{2}, K_{2}) is a weakly symmetric pair, and set M_{i} = G_{i}/K_{i}. Then, regardless of the choice of a G-invariant symmetric metric on M = G/K, there is a decomposition M = M_{1} × M_{2} in the sense of Riemannian manifolds.

If N_{G_{1}}(K_{1})^{0} = K_{1} or N_{G_{2}}(K_{2})^{0} = K_{2}, then it is true for any G-invariant metric. In the general case this statement is more complicated and will be proved below. As a corollary of Theorem 1 we have: each symmetric metric on M is a product of symmetric metrics on M_{1} and M_{2}. In particular, a symmetric metric on M exists if and only if such metrics exist on both M_{1} and M_{2}. Thus, in order to classify all weakly symmetric non-symmetric Riemannian manifolds, it suffices to consider only irreducible weakly symmetric homogeneous spaces.

It will be proved in the next section that there is no symmetric metric on a non-compact irreducible non-symmetric homogeneous space.

Denote by Z(G) the centre of G. Set K_{r} := K/(K ∩ Z(G)). A homogeneous space G′/K_{r} is called the central reduction of G/K.

Recall the definition of a principal pair given in [3, n. 5]. Definition 6. A pair (G, K) is called principal, if Z(K)^{0} = Z = (G^{1} ∩ Z) × ... × (G^{m} ∩ Z), where G^{i} are the simple factors of G and Z(K) is the centre of K.

A complexification of a principal weakly symmetric pair is a principal spherical pair. All principal spherical pairs are classified by Brion, see [11, p. 190].

In Table 1, we give the list of all simply connected compact irreducible weakly symmetric homogeneous spaces of reductive Lie groups whose central reductions are principal irreducible homogeneous spaces.

If a homogeneous space M in Table 1 admits a G-invariant symmetric metric, then we also indicate P, the identity component of the isometry group, and Q, the stabiliser in P of a point in M. The pair (P, Q) is called a symmetric extension of (G, K).

The dimensions of the cones of K-invariant and Q-invariant (in brackets) positive-definite scalar products on the space 𝔪 = 𝔤/𝔨, i.e., the dimensions of the sets of weakly symmetric and symmetric metrics are given in the fifth column of Table 1.

Note that if a truncation G′/(G′ ∩K) (the definition is given in the next section) of a weakly symmetric homogeneous space G/K of a non-semisimple group G is still weakly symmetric, then only one of these two homogeneous spaces is listed in Table 1, usually the former. In rows 5, 7, 17, 23b and 24 the group Isom(G/K)^{0} is equal not to P but to its quotient by some central subgroup.

In section 4 we prove that (P, Q) is really an extension of (G, K) and that there are no other extensions. It will be shown that having Table 1 one can decide whether a given G-invariant Riemannian metric on a simply connected irreducible weakly symmetric homogeneous space G/K of a reductive group G is symmetric or not.

It can be seen that for almost all homogeneous spaces listed in Table 1 there are G-invariant metrics on M, which are not P -invariant for any P. So these homogeneous spaces are not symmetric manifolds under proper choice of a metric. Exceptions are the cases 14 and 16.

Table 1.

M = G/K  P = Isom(M)^{0}  Q  dim 𝔹(𝔪)
1a  SU_{n}/SU_{n−1}  SO_{2n}  SO_{2n−1}  2(1)
1b  SU_{n}/SU_{n−k} × SU_{k} (n ≠ n − k)
1c  U_{2n}/U_{n} × SU_{n}
2  SU_{2n+1}/Sp_{n} · U_{1}
3 SU_{2n+1}/Sp_{n} SU_{2n+2} Sp_{n+1} 3(1)
4  U_{1} · Sp_{n}/U_{n}
5 Sp_{n}/Sp_{n−1} · U_{1} SU_{2n} SU_{2n−1} · U_{1} 2(1)
6  U_{1} · Sp_{n}/Sp_{n−1} · U_{1}
7 SO_{2n+1}/U_{n} SO_{2n+2} U_{n+1} 2(1)
8  U_{1} · SO_{2n+1}/U_{n}
9  U_{1} · SO_{4n+2}/U_{2n+1}
10  SO_{10}/Spin_{7} × SO_{2}
11  U_{1} · SO_{10}/Spin_{7} × SO_{2}
12  SO_{9}/Spin_{7}  SO_{16}  SO_{15}  2(1)
13  Spin_{8}/G_{2}  SO_{8} × SO_{8}  SO_{7} × SO_{7}  3(2)
14  Spin_{7}/G_{2}  SO_{8}  SO_{7}  1(1)
15  E_{6}/Spin_{10}
16  G_{2}/SU_{3}  SO_{7}  SO_{6}  1(1)
17  (U_{n+1} × SU_{n})/U_{n}  SU_{n+1} × SU_{n+1}  SU_{n+1}  2(1)
18 (SU_{n} × Sp_{m})/(U_{n−2} × SU_{2} × Sp_{m−1})
19a  (Sp_{n} × Sp_{l} × Sp_{m})/

(Sp_{n−1} × Sp_{1} × Sp_{l−1} × Sp_{m−1})

19b (Sp_{n} × Sp_{1} × Sp_{m})/ SO_{4n} × SO_{4m} SO_{4n−1} × SO_{4m−1} 6(2)
(Sp_{n−1} × Sp_{1} × Sp_{m−1})
19b′ (Sp_{1} × Sp_{n} × Sp_{1})/ SO_{4n} × SO_{4n} SO_{4n−1} × SO_{4n−1} 5(2)
(Sp_{1} × Sp_{n−1} × Sp_{1})
20  (Sp_{n} × Sp_{2})/(Sp_{n−2} × Sp_{2})
21 (SU_{n} × Sp_{m})/(SU_{n−2} × SU_{2} × Sp_{m−1})
22a  (Sp_{n} × Sp_{2} × Sp_{m})/

(Sp_{n−1} × Sp_{1} × Sp_{1} × Sp_{m−1})

22b  (Sp_{1} × Sp_{2} × Sp_{1})/(Sp_{1} × Sp_{1})  Sp_{2} × Sp_{2}  Sp_{2}  3(1)
23  (SO_{n+1} × SO_{n})/SO_{n}  SO_{n+1} × SO_{n+1}  SO_{n+1}  2(1)
24a (Sp_{n} × Sp_{m})/(Sp_{n−1} × Sp_{1} × Sp_{m−1})
24b  (Sp_{n} × Sp_{1})/(Sp_{n−1} × Sp_{1})  SO_{4n}  SO_{4n−1}  3(1)

3. Extensions of pairs and invariant metrics

Definition 7. Let P, Q, G, K be Lie groups, where Q ⊂ P and K ⊂ G. Then the pair (P, Q) is called an extension of (G, K), if

G ⫋ P, P = GQ, K = Q ∩ G.

If (P, Q) is a symmetric pair, we call it a symmetric extension of (G, K).

Let M = G/K be a homogeneous space. Suppose the pair (G, K) is effective, i.e. K contains no non-trivial normal subgroups of G. (Note that a pair (Isom(M), St(x)) is always effective.) For each G-invariant Riemannian metric μ on M, the pair (Isom(M), St(x)) is an extension of (G, K), if of course G ≠ Isom(M). In order to find all symmetric G-invariant Riemannian metrics on M it is necessary and sufficient to describe all symmetric extensions of (G, K). Or equivalently, to each symmetric pair find out all weakly symmetric pair which it extends. Recall that as was proved in the previous section, if M is a symmetric manifold, then Isom(M) is semisimple.

Remark 1. It can be shown, that Isom(M) is always reductive. Moreover, if M is a symmetric homogeneous space of G, then, due to results of Cartan and Helgason we have G = Isom(M)^{0}.

If N is a normal subgroup of G and N ⊂ K, then G/K = (G/N)/(K/N) and N acts on M trivially. Therefore, we may consider in the sequel only effective pairs. Recall that a G-invariant Riemannian metric on M is determined by an element of the set 𝔹(𝔪), i.e., by a K-invariant scalar product on 𝔪. Note that the normaliser N_{G}(K) naturally acts on 𝔪 = 𝔤/𝔨.

Lemma 2. Let G/K be a weakly symmetric homogeneous space. Then each K-invariant scalar product on 𝔪 is also N_{G}(K)-invariant.

Proof. The orbits of K and N(K) in 𝔪 coincide, see [5]. Hence, these groups have the same invariants in ℝ[𝔪]. In particular, S^{2}(𝔪)^{K} = S^{2}(𝔪)^{N(K)}. □

Corollary 1. The set 𝔹(𝔪) depends only on the pair (𝔤, 𝔨), i.e., the set of weakly symmetric invariant Riemannian metrics on a simply connected homogeneous space G/K coincides with the analogous set on any homogeneous space locally isomorphic to G/K.

Suppose (G, K) is a weakly symmetric pair such that N(K)^{0} ≠ K. Then G/K = (N × G)/N(K), where N = N(K)/K acts on G/K by right multiplications.

Corollary 2 (Of Lemma 2). Each G-invariant Riemannian metric on G/K is also (N × G)- invariant.

Let M = G/K be a weakly symmetric homogeneous space. Suppose G has a non-trivial connected centre, i.e., G′ ≠ G. A pair (G′, G′ ∩ K) is called a truncated weakly symmetric. Denote by K” a group G′ ∩ K. Note that a truncated weakly symmetric pair might be or might be not weakly symmetric. For example, a pair (SU_{n+1}, SU_{n}) is a truncation of (U_{n+1}, U_{n}) and a non-weakly symmetric pair (SU_{2n}, SU_{n} × SU_{n}) is a truncation of a weakly symmetric pair (U_{2n}, U_{n} × SU_{n}).

Corollary 3 (Of Lemma 2). If (G′, K”) is a weakly symmetric pair, then the sets of G- and G′-invariant Riemannian metrics on M coincide.

The last statement means that the homogeneous spaces G/K and G′/K” correspond exactly to the one and the same weakly symmetric Riemannian manifold.

As proved in [7], N is an Abelian group. In particular, replacing G by G × N is, in a sense, a process inverse to a truncation.

Lemma 3. Let (G, K) be an irreducible weakly symmetric pair, where 𝔷(𝔤) ≠ 0. Suppose the corresponding truncated pair is reducible, i.e., (G′, G′ ∩ K) = (G_{1}, K_{1}) × (G_{2}, K_{2}). Then there are weakly symmetric pairs (Ǧ_{1}, Ǩ_{1}) and (Ǧ_{2}, Ǩ_{2}) such that their product is an extension of (G, K) and the sets of G- and (Ǧ_{1} × Ǧ_{2})-invariant Riemannian metrics on G/K coincide.

Proof. Set (Ǧ_{1}, Ǩ_{2}) := (G/G_{2}, K/(K ∩G_{2})) and (Ǧ_{2}, Ǩ_{2}) := (G/G_{1}, K/(K ∩G_{1})). Note that there is a decomposition Ǧ_{1} × Ǧ_{2} = G′(Ǩ_{1} × Ǩ_{2}). The group G is embedded in a natural way into Ǧ_{1} × Ǧ_{2}, here the centre of G is embedded diagonally into the product of centres of G/G_{2} and G/G_{1}. Evidently, (G/G_{2}) × (G/G_{1}) = G(Ǩ_{1} × Ǩ_{2}), hence, the product (Ǧ_{1}, Ǩ_{2}) × (Ǧ_{2}, Ǩ_{2}) is really an extension of (G, K).

The pair (G/G_{2}×G/G_{1}, K) is weakly symmetric, because its central reduction coincides with the central reduction of (G, K). Evidently, K/(K ∩G_{2})×K/(K ∩G_{1}) ⊂ N(K) (here we consider the normaliser in Ǧ_{1} × Ǧ_{2}). In particular, the orbits of K and Ǩ_{1} × Ǩ_{2} in 𝔪 = 𝔤_{1}/𝔨_{1} ⊕ 𝔤_{1}/𝔨_{2} are the same. To conclude the proof, note that T_{eK}(G/K) is isomorphic to 𝔪 as a K-, N(K)- and, hence, Ǩ_{1} × Ǩ_{2}-module. □

Corollary. Every symmetric metric on G/K is a product of symmetric metrics on G_{1}/K_{1} and G_{2}/K_{2}.

In what follows we will consider only those weakly symmetric spaces G/K, whose truncated pairs (G′, K”) are irreducible. As was already shown, (G′, N_{G′}(K”)) is a weakly symmetric pair, moreover, it is irreducible if (G′, K”) is irreducible. In particular, we can assume that the list of the irreducible truncated pairs is known.

Recall that a weakly symmetric space G/K under consideration is also a homogeneous space of semisimple group G′.

Theorem 2. An irreducible (as a homogeneous space) non-symmetric non-compact homogeneous space of a semisimple group G is not a symmetric Riemannian manifold regardless of the choice of a G-invariant metric.

Proof. Assume that M = G/K is symmetric. Decompose it into a product of irreducible Riemannian manifolds. Let M_{n} be a product of all non-compact factors, i.e., a symmetric space of negative curvature. Suppose a semisimple group H ⊂ Isom(M_{n}) acts transitively on M_{n}. By the Karpelevich theorem, see [1, Theorem 1], there is a Cartan involution σ of Isom(M_{n}) such that σ(H) = H and σ|_{𝔥} is a Cartan involution of 𝔥. In particular, 𝔥 = 𝔨_{1} ⊕ 𝔪_{1}, 𝔪_{1} ⊂ 𝔪. But then 𝔪_{1} = 𝔪. Because [𝔪, 𝔪] ⊕ 𝔪 = 𝔦𝔰𝔬𝔪(M_{n}), we have 𝔥 = 𝔦𝔰𝔬𝔪(M_{n}). Hence, any connected semisimple subgroup of Isom(M_{n}) acting transitively on M_{n} coincides with Isom(M_{n})^{0}. The group Isom(M_{n}) can not act non-trivially on a compact or locally euclidian symmetric Riemannian manifold. This means that G contains Isom(M_{n})^{0} as a factor. Thus, if M = M_{n}, then the homogeneous space G/K is symmetric and if M ≠ M_{n}, then G/K is reducible (as a homogeneous space). □

From now on, assume that G/K is compact or, equivalently, that G is compact. It is well known, that the isometry group of a compact Riemannian manifold is compact.

Definition 7. Let 𝔭, 𝔮, 𝔤, 𝔨 be Lie algebras, where 𝔮 ⊂ 𝔭 and 𝔨 ⊂ 𝔤. A pair (𝔭, 𝔮) is called an extension of (𝔤, 𝔨), if

𝔤 ⫋ 𝔭, 𝔭 = 𝔤 + 𝔮, 𝔨 = 𝔮 ∩ 𝔤.

If 𝔮 = 𝔭^{σ}, where σ is an involution of 𝔭, then we call (𝔭, 𝔮) a symmetric extension of (𝔤, 𝔨).

Let us call a pair of algebras (𝔤, 𝔨), corresponding to a weakly symmetric pair (G, K), weakly symmetric.

If G is embedded into P, then (P, Q) is a symmetric extension of (G, K) if and only if (𝔭, 𝔮) is a symmetric extension of (𝔤, 𝔨). Therefore, to solve our problem we have to describe only the symmetric extensions of weakly symmetric pairs of algebras.

We shall say that (𝔤, 𝔥) is effective if 𝔥 does not contains any ideals of 𝔤. If (G, H) is locally effective then (𝔤, 𝔥) is effective.

We will deal with complex Lie algebras for the convenience of the reader. Let 𝔤_{ℂ}, 𝔨_{ℂ}, 𝔭_{ℂ}, 𝔮_{ℂ} be the complexifications of 𝔤, 𝔨, 𝔭, 𝔮, respectively. A pair (𝔭, 𝔮) is a symmetric extension of (𝔤, 𝔨) if and only if (𝔭_{ℂ}, 𝔮_{ℂ}) is a symmetric extension of (𝔤_{ℂ}, 𝔨_{ℂ}).

4. Symmetric extensions of spherical pairs

4.0. In this section all algebras are complex and reductive and all pairs of Lie algebras are effective.

We introduce an addition on the set of pairs of algebras:

(𝔤^{1}, 𝔥^{1}) ⊕ (𝔤^{2}, 𝔥^{2}) = (𝔤^{1} ⊕ 𝔤^{2}, 𝔥^{1} ⊕ 𝔥^{2}).

A pair (𝔤, 𝔥) is said to be irreducible if it can not be realised as a sum of two pairs.

Let (𝔤, 𝔥) be a spherical pair. Then we call (𝔤′, 𝔤′ ∩ 𝔥) a truncated spherical pair. Denote by 𝔥” an algebra 𝔤′ ∩ 𝔥.

In what follows we assume that (𝔤 = ⨁ 𝔤_{j}, 𝔥) is an irreducible spherical or truncated spherical pair, 𝔤 is a reductive algebra, 𝔤_{j} is a simple ideal of 𝔤, 𝔥_{j} is a projection of 𝔥 on 𝔤_{j}, 𝔥_{i,j} is a projection of 𝔥 on an ideal 𝔤_{i} ⊕ 𝔤_{i}.

A pair (𝔭, 𝔮) = ⨁(𝔭_{i}, 𝔮_{i}) is symmetric, (𝔭_{i}, 𝔮_{i}) is an irreducible symmetric pair, 𝔭 is a semisimple Lie algebra; π_{i} is a projection operator on 𝔭_{i}.

We call a pair of algebras (𝔤, 𝔥) simple (semisimple), if 𝔤 is simple (semisimple).

The purpose of this section is to describe all symmetric extensions of reductive spherical pairs. Along with other things, we use the fact that any extension of a reductive spherical pair is also an extension of the corresponding semisimple truncated spherical pair.

Note that (𝔤, 0) is an extension of truncated spherical pair only if 𝔤 = 𝔰𝔩_{2}.

For the clarity of our presentation all irreducible non-symmetric spherical pairs in which 𝔤 is not simple are listed in Table 2.

In each case 𝔤 equals the direct sum of algebras in the upper row, 𝔥 equals the direct sum of algebras in the lower row; the embeddings are denoted by line segments. Case 9 is a scheme of a non-principal spherical pair (see [3]). As proved in Theorem 3, non-principal pairs have no symmetric extensions, therefore, no details concerning their structure will be given. Notice only that 𝔷(𝔥) ≠ 0 for a non-principal pair (𝔤, 𝔥).

Table 2.

\begin{matrix}1) [figure: _{𝔰𝔩_{n}}; _{𝔰𝔩_{n+1}}; _{𝔰𝔩_{n}}; ℂ] & 2) [figure: _{𝔰𝔩_{n}}; _{𝔰𝔭_{2m}}; _{𝔤𝔩_{n−2}}; _{𝔰𝔩_{2}}; _{𝔰𝔭_{2m−2}}] & 3) [figure: _{𝔰𝔭_{2n}}; _{𝔰𝔭_{2l}}; _{𝔰𝔭_{2m}}; _{𝔰𝔭_{2n−2}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2l−2}}; _{𝔰𝔭_{2m−2}}] \\ 4) [figure: _{𝔰𝔭_{2n}}; _{𝔰𝔭_{4}}; _{𝔰𝔭_{2n−4}}; _{𝔰𝔭_{4}}] & 5) [figure: _{𝔰𝔩_{n}}; _{𝔰𝔭_{2m}}; _{𝔰𝔩_{n−2}}; _{𝔰𝔩_{2}}; _{𝔰𝔭_{2m−2}}] & 6) [figure: _{𝔰𝔭_{2n}}; _{𝔰𝔭_{4}}; _{𝔰𝔭_{2m}}; _{𝔰𝔭_{2n−2}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2m−2}}] \\ 7) [figure: _{𝔰𝔬_{n}}; _{𝔰𝔬_{n+1}}; _{𝔰𝔬_{n}}] & 8) [figure: _{𝔰𝔭_{2n}}; _{𝔰𝔭_{2m}}; _{𝔰𝔭_{2n−2}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2m−2}}] & 9) [figure: 𝔤_{1}; ...; 𝔤_{n}; 𝔷(𝔥); 𝔥′_{1}; ...; 𝔥′_{2}] \end{matrix}

4.1. The decompositions of reductive Lie algebras into sums of two reductive subalgebras are described in [4]. The decompositions of simple algebras are listed in Table 3. Analysing it, we will describe all symmetric extensions of spherical pairs. Decompositions of non-simple algebras are more intricate. The dealing with all decomposition would only made our paper more complicated.

Table 3.

𝔤  𝔤^{1}  φ^{1}  𝔤^{2}  φ^{2}  𝔲
𝔰𝔩_{2n}  𝔰𝔭_{2n}  ϖ_{1}  𝔰𝔩_{2n−1}  ϖ_{1} + I1  𝔰𝔭_{2n−2}
𝔰𝔩_{2n−1} ⊕ ℂ  𝔰𝔭_{2n−2} ⊕ ℂ
𝔰𝔬_{7} G_{2} ϖ_{1} 𝔰𝔬_{5} ϖ_{1} + I1 + I1 𝔰𝔩_{2}
𝔰𝔬_{5} ⊕ ℂ  𝔰𝔩_{2} ⊕ ℂ
𝔰𝔬_{6}  ϖ_{1} + I1  𝔰𝔩_{3}
𝔰𝔬_{2n+4} 𝔰𝔬_{2n+3} ϖ_{1} + I1  𝔰𝔩_{n+2}  ϖ_{1} + ϖ_{n+1}  𝔰𝔩_{n+1}
𝔰𝔩_{n+2} ⊕ ℂ  𝔰𝔩_{n+1} ⊕ ℂ
𝔰𝔬_{4n} 𝔰𝔬_{4n−1} ϖ_{1} + I1 𝔰𝔭_{2n} ϖ_{1} + ϖ_{1} 𝔰𝔭_{2n−2}
𝔰𝔭_{2n} ⊕ ℂ  𝔰𝔭_{2n−2} ⊕ ℂ
𝔰𝔭_{2n} ⊕ 𝔰𝔩_{2}  𝔰𝔭_{2n−2} ⊕ 𝔰𝔩_{2}
𝔰𝔬_{16}  𝔰𝔬_{15} ϖ_{1} + I1  𝔰𝔬_{9}  ϖ_{4}  𝔰𝔬_{7}
𝔰𝔬_{8}  𝔰𝔬_{7}  ϖ_{3}  𝔰𝔬_{7}  ϖ_{1} + I1  G_{2}

Here 𝔤^{1}, 𝔤^{2} are subalgebras of 𝔤, 𝔤 = 𝔤^{1} + 𝔤^{2}, 𝔲 = 𝔤^{1} ∩ 𝔤^{2}. In all cases n > 1, φ^{1} and φ^{2} are the restrictions of the defining representation of 𝔤 on 𝔤^{1} and 𝔤^{2} (highest weights are indicated), ϖ_{m} are the fundamental weights, I1 is a trivial representation.

Notice that in Table 3 all algebras 𝔤^{1} are simple; if 𝔤^{2} is not simple, then 𝔤^{2} = 𝔯 ⊕ 𝔞, where 𝔯 is simple, 𝔞 = ℂ or 𝔞 = 𝔰𝔩_{2}.

Note also that a pair (𝔤, 𝔲) is spherical or truncated spherical only in one case, namely then 𝔤 = 𝔰𝔬_{8}, 𝔲 = G_{2}.

Table 3 gives us all simple symmetric extensions. One has to choose those rows in which (𝔤, 𝔤_{1}) (or (𝔤, 𝔤_{2})) is symmetric and another pair (𝔤_{1}, 𝔲) (or, respectively (𝔤_{1}, 𝔲)) is spherical. These extensions are listed in rows 1a, 3, 5, 7, 12, 14, 16 and 24b of Table 1. 4.2. Here we describe non-simple symmetric extensions.

Lemma 4. Let a symmetric pair (𝔮 ⊕ 𝔮, 𝔮) be an extension of a spherical pair (𝔤, 𝔥), where 𝔮 is a simple Lie algebra. Then (𝔤, 𝔥) = (𝔮 ⊕ 𝔥, 𝔥) and the schemes of the embeddings 𝔤 ⊂ (𝔮 ⊕ 𝔮), 𝔥 ⊂ 𝔤 have the following form.

𝔮  𝔮 𝔮  𝔥  [figure: 𝔮; 𝔥; 𝔥]

Moreover, for any proper subalgebra 𝔣 ⊂ 𝔥 that does not contain 𝔥′, a pair (𝔤, 𝔣) is not spherical or either truncated spherical.

Proof. Denote by 𝔮_{1} and 𝔮_{2} the projections of 𝔤 on the first and second summand of 𝔭 = 𝔮⊕𝔮, respectively. Evidently, 𝔮_{1} + 𝔮_{2} = 𝔮. The pair (𝔮_{1} ⊕ 𝔮_{2}, 𝔮_{1} ∩ 𝔮_{2}) is an extension of (𝔤, 𝔥), hence it is spherical.

Assume that neither 𝔮_{1} nor 𝔮_{2} is equal to 𝔮, i.e., there is a non-trivial decomposition of 𝔮. It can be seen form Table 3 that 𝔮_{1} is a simple algebra and 𝔮_{2} is a sum of at most two ideals (or vise versa). We also can say that 𝔮 = 𝔮_{1} + 𝔮′_{2}. As a corollary of the classification of the spherical pairs, see Table 2, we have 𝔮′_{2} ⊂ 𝔮_{1}. But in that case 𝔮 = 𝔮_{1} + 𝔮′_{2} = 𝔮_{1}. If 𝔤 = 𝔮, then 𝔥 = 0, but it is impossible because (𝔤, 𝔥) is spherical.

Thus, we have proved that 𝔤 contains one of the summands 𝔮. Hence, the structure of (𝔤, 𝔥) and the embedding schemes described in the lemma settings are entirely determined. The second statement of the lemma can be deduced from the classification of the spherical pair. One can easily check it using Table 2. □

The rows 17, 22b and 23 of Table 1 correspond to a symmetric extensions of a type described in Lemma 4.

Lemma 5. If a simple symmetric pair (𝔭, 𝔮) is an extension of a semisimple truncated spherical (or spherical) pair (𝔤, 𝔥), where 𝔤 is not simple, then

\{ \begin{matrix}(𝔭, 𝔮) = (𝔰𝔬_{4n}, 𝔰𝔬_{4n−1}), \\ (𝔤, 𝔥) = (𝔰𝔭_{2n} ⊕ 𝔰𝔩_{2}, 𝔰𝔭_{2n−2} ⊕ 𝔰𝔩_{2}). \end{matrix}

Moreover, a pair (𝔤, 𝔣) is truncated spherical (or spherical) for neither proper subalgebra 𝔣 ⊂ 𝔥.

Proof. The first statement can be deduced from Table 3 and the second one from Table 2. □ Lemma 6. If a non-simple symmetric pair (𝔭, 𝔮) is an extension of a simple spherical pair (𝔤, 𝔥), then

\{ \begin{matrix}(𝔭, 𝔮) = (𝔰𝔬_{8} ⊕ 𝔰𝔬_{8}, 𝔰𝔬_{7} ⊕ 𝔰𝔬_{7}), \\ (𝔤, 𝔥) = (𝔰𝔬_{8}, G_{2}). \end{matrix}

Proof. Suppose (𝔭, 𝔮) is a symmetric extension of (𝔤, 𝔥). Due to Lemma 4, (𝔭, 𝔮) does not contain summands of the type (𝔮 ⊕ 𝔮, 𝔮). Let 𝔭_{1} and 𝔭_{2} be a simple ideals of 𝔭. Then there is a decomposition 𝔤 = 𝔩_{1} + 𝔩_{2}, where 𝔩_{1} = π_{1}(𝔤) ∩ 𝔮_{1}, 𝔩_{2} = π_{2}(𝔤) ∩ 𝔮_{2}. Because 𝔥 ⊂ (𝔩_{1} ∩ 𝔩_{2}), t pair (𝔤, 𝔲 = 𝔩_{1} ∩ 𝔩_{2}) is spherical. But it can happen only if 𝔤 = 𝔰𝔬_{8}, 𝔥 = G_{2}.

The pair (𝔰𝔬_{8}, G_{2}) has a symmetric extension, namely it is the pair (𝔰𝔬_{8}, 𝔰𝔬_{8}^{σ_{1}}) ⊕ (𝔰𝔬_{8}, 𝔰𝔬_{8}^{σ_{2}}), where σ_{1} and σ_{2} are different non-conjugated outer involutions of 𝔰𝔬_{8} such that 𝔰𝔬_{8}^{σ_{1}} = 𝔰𝔬_{8}^{σ_{2}} = 𝔰𝔬_{7}. □

Theorem 3. There are only five symmetric extensions of irreducible spherical pair that are not described above, namely:

[figure: _{𝔰𝔬_{n+1}}; _{𝔰𝔬_{n+1}}; _{𝔰𝔬_{n+1}}; _{𝔰𝔬_{n+1}}; _{𝔰𝔬_{n}}; _{𝔰𝔬_{n}}; _{𝔰𝔩_{n+1}}; _{𝔰𝔩_{n+1}}; _{𝔰𝔩_{n+1}}; _{𝔰𝔩_{n+1}}; _{𝔰𝔩_{n}}; _{𝔰𝔩_{n}}; _{𝔰𝔭_{4}}; _{𝔰𝔭_{4}}; _{𝔰𝔭_{4}}; _{𝔰𝔭_{4}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2}}]

[figure: _{𝔰𝔬_{4n−1}}; _{𝔰𝔬_{4m−1}}; _{𝔰𝔬_{4n}}; _{𝔰𝔬_{4m}}; _{𝔰𝔭_{2n}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2m}}; _{𝔰𝔭_{2n−2}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2m−2}}; _{𝔰𝔬_{4n−1}}; _{𝔰𝔬_{4n−1}}; _{𝔰𝔬_{4n}}; _{𝔰𝔬_{4n}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2n}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2n−2}}; 𝔰𝔭_{2}.]

(In each case 𝔮 equals the direct sum of the upper algebras, 𝔭 equals the direct sum of the algebras in the second row, 𝔤 – in the third row, 𝔥 – in the lower row.)

Proof. Suppose (𝔭, 𝔮) is a symmetric extension of (𝔤, 𝔥). Recall that it is also an extension of a corresponding irreducible truncated spherical pair (𝔤′, 𝔥”), where 𝔥” = 𝔤′∩𝔮. The irreducibility of (𝔤′, 𝔥”) means that for any simple ideal 𝔤_{i} there is a simple ideal 𝔤_{j} (i ≠ j) and an algebra 𝔭_{l} such that 𝔤_{i} ⊕ 𝔤_{j} is embedded into 𝔭_{l} and (𝔤_{i} ⊕ 𝔤_{j}, 𝔮_{l} ∩ (𝔤_{i} ⊕ 𝔤_{j})) is irreducible.

Due to Lemmas 4 and 5, we know that (𝔥”)_{i} and (𝔥”)_{j} are semisimple, i.e., 𝔥” is semisimple. In particular, non-principal symmetric pairs have no symmetric extensions. Since algebras 𝔥” corresponding to them have a non-trivial centres.

If 𝔭_{l} is simple, then by Lemma 5, (𝔭_{l}, 𝔮_{l}) is an extension of (𝔤_{i} ⊕ 𝔤_{j}, (𝔥”)_{i,j}). Suppose 𝔭_{l} = 𝔮_{l} ⊕ 𝔮_{l}. Then if (𝔤_{i} ⊕ 𝔤_{j}, 𝔮_{l} ∩ (𝔤_{i} ⊕ 𝔤_{j})) is irreducible, then, by Lemma 4, 𝔤_{i} = 𝔮_{l} or 𝔤_{j} = 𝔮_{l}. In any case (𝔭_{l}, 𝔮_{l}) is an extension of (𝔤_{i} ⊕ 𝔤_{j}, (𝔥”)_{i,j}).

Note that if

(𝔭, 𝔮) = (𝔭^{1}, 𝔮^{1}) ⊕ (𝔭^{2}, 𝔮^{2}),

and (𝔭^{1}, 𝔮^{1}) is an extension of (𝔤, 𝔥) (or of (𝔤′, 𝔥”)), then (𝔭^{2}, 𝔮^{2}) = 0. (This is a corollary of the fact that a symmetric pair has no symmetric extension.) Hence, if 𝔤 has only two simple ideals, then its symmetric extension is an irreducible pair. All such extensions are described above.

If (𝔤′, 𝔥”) is irreducible, then, as can be seen in Table 2, 𝔤′ is a sum of at most three simple ideals. Moreover, if it is a sum of exactly three ideals, then (𝔤′, 𝔥”) is spherical and the centre of a normaliser of 𝔥” in 𝔤′ is trivial. Hence, in the given case 𝔤 = 𝔤′ and (𝔤, 𝔥) ia a pair from the third or sixth row of Table 2. So, to conclude we have to describe symmetric extensions of these two spherical pairs.

Suppose that the embedding scheme of 𝔥 ⊂ 𝔤 has a suspended vertex, corresponding to 𝔥, i.e., there is a simple ideal 𝔤 of 𝔥, which is a subalgebra of some simple ideal of 𝔤 (without loss of generality we can assume that 𝔣 ⊂ 𝔤_{1}). Then, by Lemma 4, an ideal 𝔤_{1} is embedded into a simple symmetric pair. According to Lemma 5, we have 𝔤_{1} = 𝔰𝔭_{2n} and there is a simple ideal (𝔤_{2}) such that (𝔤_{1} ⊕ 𝔤_{2}, 𝔥_{1,2}) is a pair from the eighth row of Table 2 with m = 1. Moreover, there is a pair

(𝔭_{j}, 𝔮_{j}) = (𝔰𝔬_{4n}, 𝔰𝔬_{4n−1}),

such that 𝔤_{1} ⊕ 𝔤_{2} is embedded into 𝔭_{j}. It means that if a pair from the third row of Table 2 has a symmetric extension, then one of the numbers n, m, l is equal to 1. Moreover, the symmetric extension has to be a sum of two pais, where one of the simple ideals of 𝔤 is embedded into both summands. If this ideal is not equal to 𝔰𝔭_{2}, then the others are.

In the sixth case both n and m have to be equal to 1. Theoretically, besides the given in the theorem extension one more symmetric extension could be possible:

[figure: _{𝔰𝔬_{7}}; _{𝔰𝔬_{7}}; _{𝔰𝔬_{8}}; _{𝔰𝔬_{8}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{4}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2}}; _{𝔰𝔭_{2}}]  ,

but in this case we have not the decomposition of 𝔰𝔬_{8}⊕𝔰𝔬_{8}, because 𝔰𝔭_{4} does not have non-trivial decompositions. □

5. Extensions of reducible pairs

In this section we consider only real Lie algebras. Still we keep the notation of the previous section. Thus, (𝔭, 𝔮) = ⨁(𝔭_{i}, 𝔮_{i}) is a symmetric pair, (𝔭_{i}, 𝔮_{i}) is an irreducible symmetric pair, 𝔭 is a semisimple Lie algebra; π_{i} is a projection onto 𝔭_{i}.

Lemma 7. Suppose a real semisimple symmetric pair (𝔭, 𝔮) is an extension of a reducible spherical pair (𝔤, 𝔥) = (𝔤_{1}, 𝔥_{1}) ⊕ (𝔤_{2}, 𝔥_{2}). Then π_{i}(𝔷(𝔤_{1})) = 0 or π_{i}(𝔷(𝔤_{2})) = 0 for any i.

Proof. Assume that it is not true for some i, i.e., there are two vectors ξ_{1} ∈ 𝔷(𝔤_{1}), ξ_{2} ∈ 𝔷(𝔤_{2}) such that π_{i}(ξ_{1}) ≠ 0 and π_{i}(ξ_{2}) ≠ 0. Note that dim 𝔷(π_{i}(𝔤)) = dim 𝔷((π_{i}(𝔤) ∩ 𝔮_{i})) = 1. For a simple pair (𝔭_{i}, 𝔮_{i}) it can be seen from Table 3, for a pair (𝔮 ⊕ 𝔮, 𝔮) it is due to Lemma 4, since a centre of a spherical subalgebra in an irreducible pair is at most one-dimensional.

If (𝔭_{i}, 𝔮_{i}) is simple then denote by 𝔣 a simple ideal of 𝔤 isomorphically embedded, as can be seen from Table 3, into π_{i}(𝔤)′; and if (𝔭_{i}, 𝔮_{i}) = (𝔮 ⊕ 𝔮, 𝔮) then denote by 𝔣 a simple ideal isomorphically embedded into 𝔮 ⊕ 0.

Denote by 𝔥̂ and 𝔥̃ projections of 𝔥 and 𝔥∩π_{i}^{−1}(𝔮) onto 𝔣⊕ℝξ_{1} ⊕ℝξ_{2}, respectively. Evidently, 𝔥̂ ⊂ 𝔥̃. We can assume without loss of generality that 𝔣 ⊂ 𝔤_{1}. Then 𝔥̂ contains the vector ξ_{2}. But, on the other hand, dim 𝔷(𝔥̃) = 2 and the embedding scheme of 𝔥̃ ⊂ (𝔣 ⊕ ℝξ_{1} ⊕ ℝξ_{2}) has the following form.

[figure: 𝔣; ℝξ_{1}; ℝξ_{2}; 𝔥̃′; ℝη_{1}; ℝη_{2}]

Hence, ξ_{2} can not be contained in 𝔥̃. □

The proof of Theorem 1. Suppose an element b ∈ 𝔹(𝔪) determined a symmetric Riemannian metric on G/K. It is sufficient to show that b(𝔪_{1}, 𝔪_{2}) = 0. If N_{G_{1}}(K_{1}) = K_{1} or N_{G_{2}}(K_{2}) = K_{2}, then 𝔤_{1}/𝔨_{1} and 𝔤_{2}/𝔨_{2} have no isomorphic non-trivial K_{1} × K_{2}-submodules, hence, 𝔪_{1} and 𝔪_{2} are orthogonal with respect to any element of b ∈ 𝔹.

Consider the other case. Let (G′_{1}, K̂_{1}) be a central reduction of (G_{1}, K_{1}), i.e., K̂_{1} = K_{1}/(K_{1}∩ Z(G_{1})). Denote by N the group N_{G′_{1}}(K̂_{1})/K̂_{1}. Instead of (G_{1}, K_{1}) we can deal with its extension (N × G_{1}, N_{G′_{1}}(K̂_{1})). As was already said, the scalar product b is also N_{G′}(K̂_{1})-invariant and N is Abelian.

Thus, we can assume that the condition N_{G′_{i}}(K̂_{i}) = K̂_{i} for i = 1, 2 was satisfied initially. In particular, there is no non-zero K-invariant vector in 𝔤′_{i}/𝔨̂_{i}. Hence, b(𝔤′_{1}/𝔨̂_{1}, 𝔤′_{2}/𝔨̂_{2}) = 0. To conclude we have to verify that (𝔷(𝔤_{1}) + 𝔨)/𝔨) and (𝔷(𝔤_{2}) + 𝔨)/𝔨 are orthogonal with respect to b.

Denote by Q the stabiliser of eK in P = Isom(M)^{0}. Let 𝔪 = ⨁ 𝔪_{i} be a decomposition of 𝔪 into the sum of irreducible representations of Q. It is well known, that b(𝔪_{i}, 𝔪_{j}) = 0 for i ≠ j. Due to Lemma 7, we know that the spaces (𝔷(𝔤_{1}) + 𝔮)/𝔮 and (𝔷(𝔤_{2}) + 𝔮)/𝔮 are embedded into the sums ∑_{i∈I_{1}} 𝔪_{i} and ∑_{i∈I_{2}} 𝔪_{i}, respectively, where I_{1} ∩ I_{2} = ∅. Hence, they are orthogonal with

respect to b. □

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