DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 81–91

HOMOGENEOUS RIEMANNIAN

MANIFOLDS WITH EINSTEIN METRICS

E. D. Rodionov

Abstract. Homogeneous Einstein manifolds (M = G/H, ρ) which have some restrictions either on the algebraic structure of G/H, or on the dimension of G/H, or on the class of G — invariant Riemannian metrics of G/H are studied in this paper.

In this paper we’ll be discussed some questions from the theory of Einstein manifolds, that is Riemannian manifolds (M, ρ) satisfying the Einstein equation Ricc(ρ) = C · ρ, where C ∈ R, ρ is the Riemannian metric tensor, Ricc is the Ricci tensor of (M, ρ).

In general case the problem of the classification of all Einstein manifolds is very difficult, it is also difficult in homogeneous case (M = G/H). Therefore different authors assume that homogeneous Riemannian manifold (G/H, ρ) has some restrictions either on the algebraic structure of G/H, or on the dimension of M = G/H and others.

The main purpose of this paper is to present some theorems of such sort.

Theorem 1 [Ro12]. Let G/H be a compact simply-connected homogeneous space which admit a homogeneous Riemannian metric of positive sectional curvature. If (G/H, ρ) is a homogeneous Einstein manifold, then it is isometric, up to a homothety,to one of the manifolds in Table 1.1.

We note that compact simply connected homogeneous spaces admitting homogeneous Riemannian metric of positive sectional curvature have been classified by M.Berger [Ber], N.Wallach [Wal] and B.Bergery [Berg]. There are compact rank one symmetric spaces (CROSS) and also some finite list of manifolds.

1991 Mathematics Subject Classification. 53C20, 53C25, 53C30.

Key words and phrases. Riemannian manifolds, homogeneous spaces, Einstein metrics.

This paper is in final form and no version of it will be submitted for publication elsewhere.

The author is partially supported by a grant from the International Soros Science Foundation and a grant from the S.-Petersburg University.

82 E. D. RODIONOV

Table 1.1

N G/H Isotropy Einstein
representation  metrics
1.  CROSS  Irreducible  Standard [Wol]
2. \frac{SU(n+1)}{SU(n)} = S^{2n+1}  p = p_{o} ⊕ p_{1}  Metric with constant
3.  \frac{U(n+1)}{U(n)} = S^{2n+1}  p = p_{o} ⊕ p_{1}  sectional curvature [Zil]
4.  \frac{Sp(n+1)}{Sp(n)} = S^{4n+3}  p = p_{o} ⊕ p_{1}  a. Metric with constant

sectional curvature [Zil]

5. \frac{Sp(n+1)×Sp(1)}{Sp(n)×Sp(1)} = S^{4n+3} p = p_{1} ⊕ p_{2} b. Metric of the distance

sphere S^{4n+3}

6. \frac{Sp(n+1)×U(1)}{Sp(n)×U(1)} = S^{4n+3} p = p_{o} ⊕ p_{1} ⊕ p_{2}  in P^{n+1}H [Bo-Ka]

a. Metric with constant

7. \frac{Spin(9)}{Spin(7)} = S^{15} p = p_{1} ⊕ p_{2} sectional curvature [Zil]

b. Metric of the distance sphere S^{15} in CaP^{2} [Bo-Ka]

8. \frac{SU(3)}{i_{k,l}(S^{1})} p = p_{o} ⊕ p_{1} ⊕ p_{2} ⊕ p_{3} Two invariant Einstein

metrics [Ko-Vl], [Wan]

9. \frac{Sp(n+1)}{Sp(n)×U(1)} = CP^{2n+1} p = p_{1} ⊕ p_{2} Two invariant Einstein

metrics [Zil]

10. \frac{SU(3)}{T_{max}} p = p_{1} ⊕ p_{2} ⊕ p_{3} Two invariant Einstein

metrics [Ro2],[Ro9]

11. \frac{Sp(3)}{Sp(1)×Sp(1)×Sp(1)} p = p_{1} ⊕ p_{2} ⊕ p_{3} Two invariant Einstein

metrics [Ro2],[Ro9]

Table 1.1 (continuation)

12.  \frac{F_{4}}{Spin(8)}  p = p_{1} ⊕ p_{2} ⊕ p_{3}  Two invariant Einstein

metrics [Ro2],[Ro9]

13.  \frac{SU(5)}{Sp(2)×S^{1}}  p = p_{1} ⊕ p_{2}  Two invariant Einstein

metrics [Ro4],[Ro9]

It is known that any homogeneous Einstein manifold M^{n} of dimension n ≤ 4 is a Riemannian symmetric space. On the other hand, in the case n ≥ 5 the problem of the complete classification of homogeneous Einstein manifolds still remains open [Bes].

Theorem 2 ([Ro 1], [Ro 5], [Ro 10], [Ro 11]). Let G be a compact, connected, simply connected and semisimple Lie group which acts almost effectively on the five-dimensional simply connected homogeneous space M^{5} = G/H, where H is a connected closed subgroup of G. If (G/H, ρ) is a homogeneous Einstein manifold, then it is isometric, up to a homothety, to one of the manifolds in Table 2.1.

Table 2.1

N  M^{5}  Invariant Einstein metrics
1.  \frac{SU(3)}{i(SU(2))} = S^{5},  Normal homogeneous Riemannian
where i is the  metric with constant
standard embedding  sectional curvature
2.  \frac{SU(3)}{SO(3)}, an irreducible  Symmetric

symmetric space of type AI

3.  \frac{SU(2)×SU(2)}{SU(2)} × \frac{SU(2)}{S^{1}} =  Unique invariant
= S^{3} × S^{2}  Einstein metric

Table 2.1 (continuation)

4.  \frac{SU(2)×SU(2)}{i(S^{1})} = M_{i}^{5}  For any nondiagonal

embedding there exists

exactly one invariant

Einstein metric, and for

a diagonal embedding

at least one invariant

Einstein metric

We note that this theorem allows us to give an answer on a problem from A.Besse book [Bes] for different embeddings.

Problem. Given a compact 1 – connected homogeneous space M = G/H is the set of G-invariant Einstein metrics on M finite (up to homotheties)?

Indeed, every space M_{i}^{5} is diffeomorphic to the Riemannian product S^{3} × S^{2} and our family of homogeneous Einstein metrics on S^{3} ×S^{2} is non-homothetic and non-finite.

We also note that D.V.Alekseevskii, Isabel Dotti Miatello, and Sergio Ferrara have obtained a classification of homogeneous Einstein 5-manifolds as well as a number of new results about Einstein manifolds of positive Ricci curvature.

In dimension 6, we have obtained a classification of compact simply connected homogeneous Einstein manifolds. There are symmetric spaces, some spaces which are diffeomorphic to the symmetric spaces, and also the space SU(3)/T_{max} with two invariant Einstein metrics (see Theorem 1.).

Now we can suppouse, without the loss of generality, that G is a compact semisimple Lie group and H is a compact subgroup of G.

Definition. A G – homogeneous Riemannian manifold (G/H, ρ_{B}) (with compact semisimple G and compact H) is called standard homogeneous Riemannian manifold if its metric ρ_{B} is obtained from the mines Killing’s form B of the Lie algebra g under the natural projection π : G ↦ G/H.

Let us consider the reductive decomposotion g = h⊥_{B}p for G/H. Then the Ricci curvature of (G/H, ρ_{B}) is given by the formula [Wa-Zi]:

ricc(X) = \frac{1}{4} + \frac{1}{2} · B(C_{χ,B}(X), X),

where χ is the isotropy representation of H in gl(p), and C_{χ,B} is the Casimir operator of the isotropy representation χ .

Finally, from this formula one may obtain a formula for the calculation of the Ricci curvature of standard Riemannian manifolds with the help of the marked Dynkin’s diagrams. Using this algorithm we proved the theorems:

Theorem 3 ([Ro 8], [Ro 12]). Let (G/H, ρ_{B}) be a 1-connected compact standard homogeneous Riemannian manifold, G is a compact connected semisimple Lie group acting effectively on M = G/H, H is a closed simple subgroup of G. Then (G/H, ρ_{B}) is Einstein manifold if and only if the Lie algebras (g, h) of the Lie groups (G, H) are contained in the following list:

(1) (g, h) = (h ⊕ ... ⊕ h, △h), where △h is the diagonal inclusion;

(2) g is a simple Lie algebra and the pair (g, h) is one from the Einstein pairs of Cartan - Manturov - Wolf - Wang - Ziller.

We note that the list of pairs of Cartan - Manturov - Wolf - Wang - Ziller is finite and containes some symmetric pairs of Cartan, compact non-symmetric strongly isotropy irreducible pairs of Manturov - Wolf, and compact non-strongly isotropy irreducible pairs of Wang - Ziller corresponding to the standard homogeneous Einstein manifolds with simple transitive group G [Bes].

Theorem 4 [Ro 12]. Let (G/H, ρ_{B}) be a compact 1-connected special reductive standard homogeneous Einstein manifold, G a connected 1-connected semisimple Lie group acting effectively on M = G/H , and H a closed subgroup of G whose Lie algebra h is a regular subalgebra of Lie algebra g. Then (M, ρ_{B}) is isometric to the direct Riemannian product of irreducible (in sence of de’Rham) standard Einstein manifolds (M, ρ_{B}) = (M_{1}, ρ_{B_{1}}) × ... × (M_{r}, ρ_{B_{r}}), where M_{i} = G_{i}/H_{i}, and also:

(1) Einstein constants C_{1}, ..., C_{r} of M_{1}, ..., M_{r} coincide;

(2) g_{i} are simple algebras;

(3) rkg_{i} = rkh_{i};

(4) the pair (g_{i}, h_{i}) is one from the Einstein pairs of Manturov - Wolf - Wang - Ziller satisfying the condition (3).

Moreover, with the help of this algorithm we have constructed new examples of homogeneous Einstein manifolds. It is interesting that these examples correspond to the solutions of some Diophantine equations.

We consider the embedding

H = K × L ⊂ (K × ... × K) × L ⊂ K × ... × K × G_{o} = G

where the first embedding is of the form diag × id ( K is taken t times) and the second is of the form id × ... × id × π , where π : K × L ⊂ G_{o} is some embedding; G_{o}, K, L are compact connected simple Lie groups.

Theorem 5 ([Ro 8], [Ro 12]). Let (g_{o}, k⊕l) be a compact irreducible symmetric or compact nonsymmetric strictly isotropically - irreducible pair. Then the space (G/H, ρ_{B}) will be an Einstein manifold if and only if the Lie algebras g_{o}, k, l appear in the list presented in Table 5.1, and also in the first two cases the embeddings

π : so(n) ⊕ so(m) ⊂ so(n + m),
π : sp(n) ⊕ sp(m) ⊂ sp(n + m)

must be the standard embeddings whereas in the last two they must be nonstandard embeddings.

We also note that in the orthogonal and symplectic cases we have the following nontrivial decisions of Einstein equations correspondingly:

(n, m, t) = (t^{2} − 4t + 6, t − 2, t) (t ∈ N) (SO)

(n, m, t) = (2s^{2} − 1, s, 2s) (s ∈ N) (SP)

Table 5.1

g_{0}  k  l  Einstein equations
so(n + m) so(n)  so(m)  n^{2} + (t − 5)n + 6 − 2t
= m[m + (n − 2)(t − 1)]
sp(n + m) sp(n)  sp(m)  2n^{2} + (5 − t)n + 3 − t
= 2m[(t − 1)(n + 1) + m]
so(4n)  sp(n)  sp(1)  The metric is Einstein
when t = 11 and n = 8
e_{6}  g_{2}  su(3)  The metric is Einstein
when t = 2

Theorem 6 ([Ro 8], [Ro 12]). Let

H = SO(k) × SO(n) × SO(m) ⊂
⊂ SO(k) × [SO(n) × SO(n)] × SO(m) ⊂ SO(k + n) × SO(n + m) = G

where the first embedding is of the form id × diag × id and the second is of the form [(ρ_{k} ⊗ id) ⊕ (id ⊗ ρ_{n})] × [(ρ_{n} ⊗ id) ⊕ (id ⊗ ρ_{m})]. Then (G/H, ρ_{B}) is Einstein manifold if and only if k = m, n = 2m + 1.

These examples show that the case of a semisimple isotropy group is less trivial and leads to new Einstein manifolds.

Definition. Periodic space with multiplication of order k is a manifold M with a multiplication μ : M ×M → M, (x, y) ↦ x·y such that the mappings s_{x} : M → M, y ↦ s_{x}(y) = x · y satisfy the following axioms:

(1) s_{x}(x) = x ∀x ∈ M; (2) s_{x} ∘ s_{y} = s_{z} ∘ s_{x}, where z = s_{x}(y) ∀x, y, z ∈ M;

(3) for every point x ∈ M the symmetry s_{x} is a diffeomorphism and endomorphism id − (ds_{x})_{x} of the tangent space T_{x}M is non-degenerated;

(4) there exists a number k ∈ Z^{+} such that for all x, y ∈ M we have (s_{x})^{k}(y) = = y and k is a minimal number with this property.

Theorem 7 ([Ro 3], [Ro 7]). Let (G/H, ρ_{B}) be a simply connected compact standard periodic Riemannian manifold, G a compact connected and simply connected semisimple Lie group acting effectively on M = G/H, and H a closed subgroup of G. Then (G/H, ρ_{B}) is an Einstein manifold if and only if it is either isometric to a compact symmetric space or isometric to the direct product of compact standard periodic Riemannian manifolds: (G/H, ρ_{B}) = (G_{1}/H_{1}, ρ_{B_{1}}) × ... × (G_{n}/H_{n}, ρ_{B_{n}}), each of which has the property that the Lie algebras (g_{i}, h_{i}), i ∈ {1, ..., n}, of the Lie groups (G_{i}, H_{i}) occur in Table 7.1, and the Einstein constants C_{1}, ..., C_{n} of these spaces coincide.

Table 7.1. Application. t_{max} is the maximal Abelian subalgebra of the corresponding simple algebra g_{i}.

No.  g_{i}  h_{i}  C_{i}
1  g_{o} ⊕ ... ⊕ g_{o}  △g_{o}  1/4 + 1/(2n)
n times  g_{o} simple
2  su(nk)  n · su(k) ⊕ R^{n−1}  1/4 + 1/(2n)
2 ≤ k, 3 ≤ n
3  sp(3n − 1)  sp(n) ⊕ R ⊕ su(2n − 1)  5/12

1 ≤ n

4  so(3n + 2)  so(n) ⊕ R ⊕ su(n + 1)  5/12

3 ≤ n

5  e_{6}  so(8) ⊕ R^{2}  5/12
6  su(n)  t_{max}  1/4 + 1/(2n)

3 ≤ n

7  so(2n)  t_{max}  1/4 + 1/(4n − 4)

3 ≤ n

8  e_{6}  t_{max}  7/24
9  e_{7}  t_{max}  5/18
10  e_{8}  t_{max}  4/15
11  sp(2)  su(2)  9/20

Table 7.1. (continuation)

No.  g_{i}  h_{i}  C_{i}^{12  g_{2}  su(3)  5/12}^{13  f_{4}  su(3) ⊕ su(3)  5/12}^{14  e_{6}  3su(3)  5/12}^{15  e_{7}  su(6) ⊕ su(3)  5/12}^{16  e_{8}  su(9)  5/12}^{17  e_{8}  e_{6} ⊕ su(3)  5/12}^{18  e_{8}  2su(5)  7/20}

Further we shall establish a connection between standard homogeneous Einstein manifolds and Einstein triple algebras. Before this:

Definition. Lie triple algebra is an anticommutative algebra (A, ∘) over field F with trilinear operation A × A × A ∋ (X, Y, Z) ↦ [X, Y, Z] ∈ A and axioms:

(1) [X, X, Y] = 0;

(2) σ{[X, Y, Z] − (X ∘ Y) ∘ Z} = 0;
(3) σ{[X ∘ Y, Z, U]} = 0;
(4) [X, Y, U ∘ V] = [X, Y, U] ∘ V + U ∘ [X, Y, V];
(5) [U, V, [X, Y, Z]] = [[U, V, X], Y, Z] + [X, [U, V, Y], Z] + [X, Y, [U, V, Z]],

where σ denotes cyclic sum over X, Y, Z ∈ A.

Let G/H be a reductive homogeneous space with the reductive decomposition g = h ⊕ m. Let us define two operations on m by the rule:

X ∗ Y = [X, Y]_{m}, I(X, Y, Z) = [[Y, X]_{h}, Z]

for X, Y, Z ∈ m.

It is well-known that the triple (m, ∗, I) is a Lie triple algebra and its called Lie triple algebra (L.t.a.) of the reductive homogeneous space G/H.

Theorem 8 ([Ro 6], [Ro 12]). Let (G/H, ρ_{B}) be a compact irreducible standard homogeneous Riemannian manifold. Then (G/H, ρ_{B}) is an Einstein manifold if and only if the Lie triple algebra (m, ∗, I) of G/H satisfies the identity:

X ∗ ∑_{i} I(X, X_{i}, X_{i}) = 0 ∀X ∈ m,

where {X_{i}} is a B-orthonormal basis of m.

We note that in non-irreducible case we can apply the theorem of Costantde’Rham and reduce this case to the irreducible case.

Definition. A finite-dimensional L.t.a. (A, ∗, I) over R with ∗-invariant scalar product Q is called Einstein triple algebra if for every Q-orthonormal basis {X_{i}} of A and each element X ∈ A the following identity

X ∗ ∑_{i} I(X, X_{i}, X_{i}) = 0

is true.

So we have connections between standard homogeneous Einstein manifolds, marked Dynkin’s diagrams and Einstein triple algebras.

Examples of Einstein triple algebras.

(1) If (G/H, ρ_{B}) is a compact irreducible symmetric space with H ≠ e then for the decomposition g = h⊥_{B}m one has m∗m = 0 and therefore (m, ∗, I) is an Einstein triple algebra.

(2) Let (G, ρ_{B}) be a compact simple Lie group with biinvariant Riemannian metric. Then (g, [., .]) = (m, ∗) and I ≡ 0. Hence (g, [., .], I ≡ 0) is an Einstein triple algebra and (G, ρ_{B}) is an Einstein manifold.

(3) The Lie triple algebras of the spaces Manturov-Wolf-Wang-Ziller give us another example of Einstein triple algebras.

(4) The space of Ledger-Obata (G_{o} × ... × G_{o}/△G_{o}, ρ_{B}) (with G_{o} compact simple and ρ_{B} standard) give us Lie triple algebra which is Einstein triple algebra too.

(5) Above examples of standard homogeneous Einstein manifolds give also examples of Einstein triple algebras.

We shall present a classification of the reductive homogeneous Riemannian manifolds corresponding to the special simple Lie triple algebras which were considered at first by A.Sagle [Sag]. From this classification it follows that such spaces are, in general, standard homogeneous Einstein manifolds.

Definition. A finite-dimensional Lie triple algebra (A, ∘, [., ., .])_{R} is triple S-alge-bra if the identity

[X, Y, Z] = α(X ∘ Y) ∘ Z + β(Y ∘ Z) ∘ X + γ(Z ∘ X) ∘ Y

is true for all X, Y, Z ∈ A and some constants α, β, γ ∈ R.

A.Sagle has been proved that for simple algebra (A, ∘) we have the following possibilities only:

(L) {α, β, γ} = {1, 0, 0} and (A, ∘) is a simple Lie algebra; (M) {α, β, γ} = {−1, 1, 1} and (A, ∘) is a simple Malcev algebra; (S) {α, β, γ} = {1/2, 1/4, 1/4} and (A, ∘) is a simple algebra satisfying the identity J(X, Y, Z)∘W = J(W, X, Y ∘Z)+J(W, Y, Z ∘X)+J(W, Z, X ∘Y)

where J(X, Y, Z) is the Jacobian.

We have studied homogeneous spaces G/H with Lie triple algebras (m, ∗, I) of S-type and have proved the following Theorem 9 [Ro 12]. Let M = G/H be a reductive homogeneous space whose Lie triple algebra is an algebra of types (L) or (M). Then

(1) there exists only one, up to a homothety, G-invariant Riemannian metric on M (canonical Riemannian metric);

(2) in case (L) the space (M, ρ) is isotropy irreducible and locally symmetric;

(3) in case (M) the universal covering (M_{o}, ρ_{o}) of (M, ρ) is isometric to the space Spin(7)/G_{2}.

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Altai State University Dimitrov Street 66, 656099 Barnaul-99 Russia

E–mail address : rodionov@math.mezon.altai.su