ON THE “STANDARD” CONDITION FOR NONCOMPACT HOMOGENEOUS EINSTEIN SPACES

DOROTHEE SCHUETH

Abstract. A nonflat Einstein solvmanifold (S, g) is said to be of standard type if in the associated metric Lie algebra 𝔰, the orthogonal complement 𝔞 of the derived algebra is abelian. It is an open question whether the standard condition is automatically satisfied for all nonflat Einstein solvmanifolds. We derive certain properties of the metric Lie algebra 𝔰 of a nonflat Einstein solvmanifold (S, g) under the assumption dim[𝔞, 𝔞] ≤ 1. In particular, we obtain some new sufficient conditions which imply standard type.

1. Introduction

The classification of noncompact homogeneous Einstein spaces is an important open problem. Since Ricci flat homogeneous spaces are flat [2], the interesting case is that of negative Einstein constant; equivalently: the nonflat noncompact case. A conjecture by D.V. Alekseevskii [1] says that for any nonflat homogeneous Einstein space G/K the isotropy group K must be a maximal compact subgroup of G. In case G is a linear group, the latter property implies that G/K is isometric to a Riemannian solvmanifold (S, g) where S is a simply connected solvable Lie group and g a left invariant metric on S. Alekseevskii’s conjecture is open so far; however, Yu.G. Nikonorov has recently proved some interesting partial results which support it [10].

All known examples of noncompact homogeneous Einstein manifolds do fall into the class of Riemannian solvmanifolds; even more, they all are of “standard type”; i.e., they satisfy the condition that in the associated metric Lie algebra 𝔰, the orthogonal complement 𝔞 := [𝔰, 𝔰]^{⊥} of the derived algebra is abelian. Classical examples are the symmetric spaces of noncompact type and the classical bounded homogeneous domains. For other examples and for classifications in certain cases see, e.g., [1], [5], [11], [9], [6], [8], [7].

J. Heber’s groundbreaking habilitation [8] contains several deep results concerning the structure of the moduli space of Einstein solvmanifolds. One of his results says that if a solvable Lie group carries a standard left invariant Einstein metric then it cannot carry a nonstandard one. Moreover, a given solvable Lie group can carry at most one standard Einstein metric up to isometry and scaling. Heber also shows that the standard Einstein manifolds correspond to an open subspace in the moduli space of all Einstein solvmanifolds. On this open subset, he obtains additional structural results which even allow him to outline a program for classifying all standard Einstein solvmanifolds.

In view of these results, it is particularly intriguing that the question whether nonstandard Einstein solvmanifolds exist at all is still open. A solution of this question would have very interesting implications. A general nonexistence theorem for nonstandard Einstein solvmanifolds would imply that those of Heber’s results which a priori hold only in the “standard” case actually hold for all Einstein solvmanifolds. On the other hand, a construction of counterexamples would reveal the existence of a completely new type of Einstein spaces.

Key words and phrases. Homogeneous Einstein spaces, Riemannian solvmanifolds.

2000 Mathematics Subject Classification. 53C25, 53C30.

There are only some partial results in the direction of a nonexistence theorem. In [8], Lemma 4.7, Heber lists several sufficient conditions each of which does imply standard type: Nonpositive sectional curvature [3], nonnegative Killing form (e.g., in the case of purely real roots) or Killing form of index at most one [8]; moreover, the standard condition is trivially satisfied if 𝔞 is of dimension one.

The purpose of the present paper is to extend this list of sufficient conditions. We consider the case of Einstein solvmanifolds (S, g) with the property that dim[𝔞, 𝔞] = 1 (in some sense the least violent violation of the standard condition). Although we are not able to rule out the possible existence of Einstein manifolds with this property, we obtain the following partial results:

Theorem 3.1. In any nonflat Einstein solvmanifold (S, g) with dim[𝔞, 𝔞] = 1, the orthogonal projection of [𝔞, 𝔞] to the center of 𝔫 := [𝔰, 𝔰] is zero. Moreover, there exists a subspace 𝔞′ ⊂ 𝔞 of codimension two in 𝔞 such that [𝔞′, 𝔞] = 0.

Corollary 3.2. Let (S, g) be a nonflat Einstein solvmanifold. If dim[𝔞, 𝔞] ≤ 1 (e.g., if 𝔫 is of codimension two in 𝔰) and if 𝔫 is abelian, then (S, g) is of standard type.

These results may also be viewed as practical hints about where not to search for possible counterexamples to the standard condition.

2. Preliminaries

Throughout this paper, we consider a solvable metric Lie algebra (𝔰, ⟨ , ⟩); that is, 𝔰 is a solvable Lie algebra and ⟨ , ⟩ is a euclidean inner product on 𝔰. Let S denote the simply connected Lie group with Lie algebra 𝔰, endowed with the left invariant metric g whose restriction to 𝔰 = T_{e} S is ⟨ , ⟩. We identify elements of 𝔰 with left invariant vector fields on S.

Notation and Remarks 2.1.

(i) Let 𝔫 := [𝔰, 𝔰] be the derived algebra; note that 𝔫 is nilpotent. By 𝔞 := 𝔫^{⊥} we denote the orthogonal complement of 𝔫 in 𝔰. We do not assume 𝔞 to be abelian.

(ii) For A ∈ 𝔞, we let F_{A} := ad_{A}|_{𝔫} : 𝔫 → 𝔫.

(iii) For X ∈ 𝔫, we define a skewsymmetric map j_{X} : 𝔞 → 𝔞 by requiring

⟨j_{X}A, A′⟩ = ⟨X, [A, A′]⟩

for all A, A′ ∈ 𝔞. (iv) The mean curvature vector H ∈ 𝔞 is given by ⟨H, U⟩ = tr ad_{U} for all U ∈ 𝔰.

(v) The Killing form B : 𝔰 × 𝔰 → ℝ is defined by B(U, V) := tr(ad_{U} ∘ ad_{V}) for all U, V ∈ 𝔰.

(vi) We choose an orthonormal basis {E_{1}, …, E_{n}} of 𝔰.

The Ricci tensor associated with the left invariant metric g on S, viewed as a symmetric bilinear map ric : 𝔰 × 𝔰 → ℝ, is given by the following formula (and its polarized version):

Lemma 2.2 ([4], 7.38). Let U ∈ 𝔰. Then

ric(U, U) = −⟨[H, U], U⟩ − 1/2 B(U, U) − 1/2 ∥ad_{U}∥^{2} + 1/4 ∑_{i,j}⟨[E_{i} , E_{j}], U⟩^{2}.

Here ∥ad_{U}∥^{2} denotes the usual Euclidean squared norm tr(^{t}ad_{U} ∘ ad_{U}).

We will also need the following Lemma by J. Heber and its corollary below, which is also implicitly contained in [8].

Lemma 2.3 ([8], 4.1). Assume that ⟨ , ⟩ is an Einstein metric on 𝔰; that is, ric = c · ⟨ , ⟩ for some c ∈ ℝ. Then

tr(ad_{φ(H)} ∘ ^{t}φ) + 1/2 ∑_{i} B(^{t}φφ(E_{i}), E_{i}) ≤ 0

for every derivation φ of 𝔰, and equality holds if and only if ^{t}φ is also a derivation of 𝔰.

Corollary 2.4. If ⟨ , ⟩ is Einstein and φ is a derivation of 𝔰 that vanishes on 𝔞, then ^{t}φ is also a derivation of 𝔰.

Proof. We have φ(H) = 0 since H ∈ 𝔞; thus the first term in the inequality of Lemma 2.3 vanishes. We claim that the second term vanishes, too. In fact, by φ|_{𝔞} = 0 we can assume that the E_{i} run through an orthonormal basis of 𝔫 alone; moreover, we need to consider only the traces of the operators ad_{^{t}φφ(E_{i})} ∘ ad_{E_{i}} restricted to 𝔫 since Im(ad_{𝔰}) = 𝔫. But if E_{i} ∈ 𝔫 then ad_{E_{i}}|_{𝔫} is a nilpotent derivation of the nilpotent Lie algebra 𝔫, and ad_{^{t}φφ(E_{i})}|_{n} is a derivation of 𝔫; by considering the grading of 𝔫 given, e.g., by the lower central series, one easily concludes that the trace of the composition of these two derivations vanishes. □

3. Implications in case dim[𝔞, 𝔞] = 1

As mentioned in the Introduction, all known examples of solvable metric Lie algebras (𝔰, ⟨ , ⟩), where ⟨ , ⟩ is Einstein and the corresponding solvmanifold (S, g) is nonflat, are of standard type; that is, the orthogonal complement 𝔞 of the derived algebra 𝔫 is abelian. It is an open question whether this condition is automatically satisfied in the above situation. In the following, we derive some consequences of the specialized assumption that [𝔞, 𝔞] is of dimension one and ⟨ , ⟩ is Einstein with (S, g) nonflat. Although this will not answer the question whether this constellation is at all possible, we are at least able to exclude certain types of possible counterexamples to the standard condition. This provides an addition to Heber’s earlier list of conditions which imply standard type (see [8], Lemma 4.7). Our main result is the following.

Theorem 3.1. Assume that the solvmanifold (S, g) associated with the solvable metric Lie algebra (𝔰, ⟨ , ⟩) is Einstein and nonflat. Furthermore, suppose that dim[𝔞, 𝔞] = 1, where 𝔞 = 𝔫^{⊥}, 𝔫 = [𝔰, 𝔰]. Then the following holds:

(i) There is a subspace 𝔞′ ⊂ 𝔞 of codimension two in 𝔞 such that [𝔞, 𝔞′] = 0.

(ii) [𝔞, 𝔞] is orthogonal to the center 𝔷_{𝔫} of 𝔫.

Corollary 3.2. Assume that (S, g) is Einstein and nonflat. If dim[𝔞, 𝔞] ≤ 1 (for example, if 𝔫 has codimension two in 𝔰) and if 𝔫 is abelian, then (S, g) is of standard type.

The remainder of this section is devoted to proving the above theorem. In the following, we will always assume that (S, g) is Einstein and nonflat. Note that the Einstein constant c is necessarily negative here because, as is well known, homogeneous Einstein manifolds with positive or vanishing Einstein constant are compact or flat, respectively (the first assertion follows from the Bonnet-Myers Theorem, the second was shown in [2]).

We continue to use Notation 2.1. The following lemma holds regardless of dim[𝔞, 𝔞].

Lemma 3.3. We have ⋂_{A∈𝔞} ker F_{A} = {0}.

Proof. Let X ∈ 𝔫 be in the kernel of F_{A} for all A ∈ 𝔞. Then ad_{X} vanishes on 𝔞. By Corollary 2.4, ^{t}ad_{X} is also a derivation of 𝔰 and hence (by restriction) of 𝔫. Considering the lower central series of 𝔫 once more, one easily sees that the transpose of an inner derivation ad_{X} of 𝔫 cannot be a derivation unless ad_{X} vanishes. So X lies in the center of 𝔫 and also of 𝔰. In the formula for ric(X, X) of Lemma 2.2, the first three terms now vanish since ad_{X} = 0 on 𝔰. The fourth term is nonnegative, thus ric(X, X) ≥ 0. But this contradicts c < 0. □

Proof of Theorem 3.1(i). Let dim[𝔞, 𝔞] = 1 and suppose that [𝔞, 𝔞′] were nontrivial for each subspace 𝔞′ of codimension two in 𝔞. Let X ∈ 𝔫 be a vector which generates [𝔞, 𝔞], and let A ∈ 𝔞 be an eigenvector of j_{X}^{2}. Define 𝔞′ as the orthogonal complement in 𝔞 of span{A, j_{X}A}. Then 𝔞′ commutes with A and j_{X}A. Since 𝔞′ is of codimension at most two, [𝔞, 𝔞′] is nonzero by assumption. Hence there exist vectors B, C ∈ 𝔞′ such that [B, C] = X. The Jacobi identity now implies F_{A}X = [A, [B, C]] = [[A, B], C] + [B, [A, C]] = 0. Since the eigenvectors of j_{X}^{2} span the whole space 𝔞, we have shown that F_{A}X = 0 for all A ∈ 𝔞. But this contradicts Lemma 3.3 because of X ≠ 0. □

Notation and Remarks 3.4. In the following, we always suppose that dim[𝔞, 𝔞] = 1.

(i) Let 𝔞′ ⊂ 𝔞 be as in Theorem 3.1(i), and let {A_{1}, A_{2}} ⊂ 𝔞 be an orthonormal basis of the orthogonal complement of 𝔞′ in 𝔞.

(ii) Let X := [A_{1}, A_{2}] ∈ 𝔫. Note that X generates [𝔞, 𝔞]; in particular, X ≠ 0. Denote

by X^{𝔷} the orthogonal projection of X to the center 𝔷_{𝔫} of 𝔫. (iii) Let 𝔴 ⊂ 𝔫 be the smallest subspace which contains X^{𝔷} and is invariant under F_{A}

for each A ∈ 𝔞. Note 𝔴 ⊂ 𝔷_{𝔫} because 𝔷_{𝔫} is invariant under the derivations F_{A} of 𝔫. (iv) Let 𝔲 ⊂ 𝔴 be the orthogonal complement of span{X^{𝔷}} in 𝔴, and let m := dim 𝔲.

(v) For A ∈ 𝔞, we denote by D_{A} : 𝔫 → 𝔫 the symmetric part of F_{A}. By D_{A}^{𝔲} we denote the restriction D_{A}|_{𝔲}, followed by orthogonal projection to 𝔲.

(vi) Let a_{i} := tr F_{i} ∈ ℝ for i = 1, 2. Note that a_{i} = ⟨H, A_{i}⟩.

Together with Lemma 2.2 and Theorem 3.1(i), the following Lemma is the key of the proof of Theorem 3.1(ii).

Lemma 3.5.

(i) F_{A′}(𝔴) = 0 for all A′ ∈ 𝔞′.

(ii) (^{t}F_{A_{i}} + a_{i} Id)X^{𝔷} ⊥ 𝔷_{𝔫} for i = 1, 2.

(iii) 𝔲 is invariant under F_{A_{i}} for i = 1, 2.

(iv) If X^{𝔷} ≠ 0 then dim 𝔴 ≥ 2; in particular, m ≥ 1.

Proof. (i) Let A′ ∈ 𝔞′. By Corollary 2.4, ^{t}ad_{A′} is a derivation. Since ^{t}ad_{A′} vanishes on 𝔞, it follows that ^{t}ad_{A′} commutes with ad_{A} for each A ∈ 𝔞. In particular, ad_{A′} is a normal derivation; thus F_{A′} is a normal derivation of 𝔫. Moreover, the Jacobi identity implies F_{A′}(X) = 0 since [A_{1}, A_{2}] = X and A′ commutes with A_{1} and A_{2}. Being a normal derivation, F_{A′} must therefore annihilate the 𝔷_{𝔫}-component X^{𝔷} of X as well. It follows that F_{A′} annihilates 𝔴 since it commutes with each F_{A}.

(ii) Let Z ∈ 𝔷_{𝔫} be arbitrary. Since the Killing form B vanishes on 𝔰 × 𝔫 and since 𝔞 ⊥ [𝔰, 𝔰], it follows from Lemma 2.2 that

2 ric(A_{2}, Z) = −⟨[H, A_{2}], Z⟩ − ⟨ad_{A_{2}}, ad_{Z}⟩
= −⟨[a_{1}A_{1}, A_{2}], Z⟩ − ⟨[A_{1}, A_{2}], F_{A_{1}}Z⟩ = −⟨a_{1}X, Z⟩ − ⟨^{t}F_{A_{1}}X, Z⟩.

But ric(A_{2}, Z) must vanish by the Einstein condition. This implies the statement for i = 1; in case i = 2 it is shown analogously.

(iii) Since 𝔴 is invariant under F_{A_{i}} by definition, we only need to check that F_{A_{i}}(𝔲) is orthogonal to X^{𝔷}. But (ii) already implies that ^{t}F_{A_{i}}(X^{𝔷}) is orthogonal to 𝔲.

(iv) Lemma 2.2 implies

ric(X^{𝔷}, X^{𝔷}) = −⟨[H, X^{𝔷}], X^{𝔷}⟩ − 1/2 ∥ad_{X^{𝔷}}∥^{2} + α,

where α ≥ 1/2 ∑_{i=1}^{2} |^{t}F_{A_{i}}(X^{𝔷})|^{2}. In fact, for an adapted choice of the orthonormal basis {E_{1}, …, E_{n}}, the latter expression corresponds to those summands in the last term of the sum in Lemma 2.2 where E_{i} ∈ span{A_{1}, A_{2}} and E_{j} ∈ 𝔫 or vice versa. If X^{𝔷} ≠ 0 then ric(X^{𝔷}, X^{𝔷}) < 0 since the Einstein constant c is negative. Using the above statements (i) and (ii), we derive

0 > ∑_{i=1}^{2}(−a_{i}⟨F_{A_{i}}X^{𝔷}, X^{𝔷}⟩ − \frac{1}{2} |F_{A_{i}}(X^{𝔷})|^{2}) + α
≥ ∑_{i=1}^{2}(a_{i}^{2}|X^{𝔷}|^{2} − \frac{1}{2}|F_{A_{i}}(X^{𝔷})|^{2} + \frac{1}{2}|^{t}F_{A_{i}}(X^{𝔷})|^{2}).

Now if dim 𝔴 ≥ 2 were false, then X^{𝔷} would be an eigenvector of each F_{A_{i}}. In particular, we would have |^{t}F_{A_{i}}(X^{𝔷})|^{2} ≥ |F_{A_{i}}(X^{𝔷})|^{2}. Thus the right hand side in the above inequality would be nonnegative, which is a contradiction. □

Proof of Theorem 3.1(ii). Suppose that statement (ii) of the Theorem were false; that is, X^{𝔷} ≠ 0. We will derive a contradiction from this using 2.2 and 3.5.

Let i ∈ {1, 2}. In the expression for ric(A_{i}, A_{i}) from Lemma 2.2, the first and fourth terms vanish. Therefore,

ric(A_{i}, A_{i}) = −1/2 B(A_{i}, A_{i}) − 1/2 ∥ad_{A_{i}}∥^{2} = −1/2 ⟨ad_{A_{i}} , ^{t}ad_{A_{i}} + ad_{A_{i}}⟩
= −⟨F_{A_{i}} , D_{A_{i}}⟩ − 1/2 ∥ad_{A_{i}}|_{𝔞}∥^{2} = −∥D_{A_{i}}∥^{2} − 1/2 |X|^{2}.

Recall that ⟨D_{A_{i}}X^{𝔷}, X^{𝔷}⟩ = ⟨^{t}F_{A_{i}}X^{𝔷}, X^{𝔷}⟩ = −a_{i}|X^{𝔷}|^{2} by Lemma 3.5(ii). Therefore, ∥D_{A_{i}}∥^{2} ≥ a_{i}^{2} + ∥D_{A_{i}}^{𝔲}∥^{2} ≥ a_{i}^{2} + (tr D_{A_{i}}^{𝔲})^{2}/m. Summing over i = 1, 2, we conclude that

(1)  2mc ≤ −m(a_{1}^{2} + a_{2}^{2}) − (tr D_{A_{1}}^{𝔲})^{2} − (tr D_{A_{2}}^{𝔲})^{2} − m|X|^{2}.

On the other hand, consider ric(Z, Z) for Z ∈ 𝔲. Just as we did in the above proof of Lemma 3.5(iv) for X^{𝔷}, we see that

ric(Z, Z) ≥ ∑_{i=1}^{2}(−a_{i}⟨F_{A_{i}}Z, Z⟩ − \frac{1}{2}|F_{A_{i}}Z|^{2} + \frac{1}{2}|^{t}F_{A_{i}}Z|^{2}).

Now let Z run through an orthonormal basis of 𝔲 and consider the sum of the corresponding inequalities. Then the sum involving the last two of the three terms on the right hand side is nonnegative because 𝔲 is invariant under F_{A_{i}} by Lemma 3.5(iii). Therefore we obtain

mc ≥ − ∑_{i=1}^{2} a_{i} tr D_{A_{i}}^{𝔲} ,

which by the Cauchy-Schwarz inequality implies

(2)  2mc ≥ −a_{1}^{2} − a_{2}^{2} − (tr D_{A_{1}}^{𝔲})^{2} − (tr D_{A_{2}}^{𝔲})^{2}.

From the inequalities (1) and (2) we conclude

0 ≥ (m − 1)(a_{1}^{2} + a_{2}^{2}) + m|X|^{2},

which is impossible since m ≥ 1 by Lemma 3.5(iv), and X ≠ 0. □

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Mathematisches Institut, Universität Bonn, Beringstr. 6, D-53115 Bonn, Germany

 E-mail address: schueth@math.uni-bonn.de