Noncompact homogeneous Einstein manifolds attached to graded Lie algebras

Noncompact homogeneous Einstein manifolds attached to graded Lie algebras Hiroshi Tamaru Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, JAPAN (e-mail: tamaru@math.sci.hiroshima-u.ac.jp) Abstract. In this paper, we investigate a class of standard solvmanifolds, which are certain one-dimensional solvable extensions of the nilradicals of parabolic subalgebras of semisimple Lie algebras. We study their structures, curvatures and Einstein conditions. They are of Iwasawa-type, moreover, Carnot spaces. Our solvmanifold is Einstein if the nilradical is of two-step. New examples of Einstein solvmanifolds whose nilradicals are of three and four-step are also given. 0 Introduction A Riemannian manifold is called a solvmanifold if it admits a transitive solvable group of isometries. Up to now, all known examples of noncompact homogeneous Einstein manifolds are solvmanifolds. The study of Einstein solvmanifolds is currently very active. In particular, Heber ([11]) deeply studied Einstein solvmanifolds and obtained many essential results. One of his theorems states that the moduli space of Einstein solvmanifolds may have large dimension. Despite of that, we do not know many explicit examples of Einstein solvamnifolds; in particular, Einstein solvmanifolds whose nilradicals have the nilpotency greater than two. The purpose of this paper is to introduce and study a class of solvmanifolds, and provide new examples of Einstein spaces. Our class contains Einstein 0 Mathematics Subject Classification (2000): 53C30, 22E25, 53C25, 53C35, 1

solvmanifolds whose nilradicals are of two, three and four-step. It seems to be good to know explicit examples for the further study of Einstein solvmanifolds. Here we briefly recall some known examples of noncompact homogeneous Einstein manifolds. One strategy for finding Einstein solvmanifolds is to consider solvmanifolds with good geometric structures. Typical examples are symmetric spaces of noncompact type. Other examples are noncompact homogeneous Kähler-Einstein manifolds, which are modeled by solvable normal j-algebras (see [19]), and quaternionic Kähler solvmanifolds (see [2], [7]). The other strategy is to consider solvable extensions of good nilmanifolds. Typical examples are Damek-Ricci spaces, which are the one-dimensional solvable extensions of generalized Heisenberg groups ([5], [8], see also [3]). Other examples are given by Mori ([17]); he started from complex graded Lie algebras of second kind and considered the solvable extensions of the associated nilmanifolds. In fact, our construction is a generalization of his study. Other strategy for finding Einstein solvmanifolds can be found in recent articles, for example, see [15], [16] and [10]. In this paper we employ the second strategy. We take the nilradicals of parabolic subalgebras of semisimple Lie algebras. These nilpotent Lie algebras admit the natural inner products, and can be investigated in term of graded Lie algebras. In Section 2, we mention some properties of graded Lie algebras and define our nilmanifolds. The structures and curvatures of our nilmanifolds will be studied in Sections 3 and 4. In Section 5, we define certain one-dimensional solvable extensions of these nilmanifolds and study the curvatures of the constructed solvmanifolds. We show in Section 6 that our solvmanifold is Einstein if the nilradical is of two-step. This provides many new examples of Einstein solvmanifolds. Other new examples of Einstein solvmanifolds, whose nilradicals are of three and four-step, will be given in Section 7. Note that, if a parabolic subalgebra is maximal, then its nilradical coincides with the nilpotent part of an Iwasawa decomposition. In this case the constructed solvmanifold coincides with the rank one reduction of a symmetric space of noncompact type, which is Einstein. Therefore our class of solvmanifolds essentially contains noncompact symmetric spaces. The structures of the rank one reductions of noncompact symmetric spaces will be mentioned in Section 8. This work was partially supported by Grant-in-Aid for Young Scientists (B) 14740049, 2

The Ministry of Education, Culture, Sports, Science and Technology, Japan. 1 Preliminaries In this section we recall some notations and basic properties of Einstein solvmanifolds. We refer to [22], [11]. Let s be a solvable Lie algebra and, be a positive definite inner product on s. Throughout this paper, by a metric Lie algebra (s,, ) we also mean the corresponding solvmanifold, that is, the simply-connected Lie group S equipped with the induced leftinvariant Riemannian metric. Put n := [s, s] and a := n. Definition 1.1 A solvmanifold (s,, ) is called standard if a is abelian. A standard solvmanifold (s,, ) is said to be of Iwasawa-type if (i) ad A is symmetric for every A a, and (ii) there exists A 0 a such that every eigenvalue of ad A0 on n is a positive real number. To calculate the Ricci curvatures of our solvmanifolds, we need the following particular vectors. Definition 1.2 For a solvmanifold (s,, ), the element H 0 a given by the following is called the mean curvature vector: H 0, A = tr(ad A ) for every A a. Note that H 0 is the mean curvature vector for the embedding of N into S, where N denotes the Lie subgroup of S with Lie algebra n. Proposition 1.3 ([22]) Let (s,, ) be a solvmanifold of Iwasawa type and H 0 be the mean curvature vector. Denote by Ric and Ric n the Ricci curvatures of s and n, respectively. Then, (1) Ric(A, A ) = tr(ad A ) (ad A ) for A, A a, (2) Ric(A, X) = 0, (3) Ric(X) = Ric n (X) (tr(ad H0 )/ H 0 2 ) ad H0 X for X n. 3

Proposition 1.3 leads that, if (s,, ) is Einstein then so is (RH 0 + n,, ). For a solvmanifold (s,, ), We call dim a the algebraic rank and (RH 0 + n,, ) the rank one reduction. Note that H 0 is also the mean curvature vector of the solvmanifold (RH 0 + n,, ). Proposition 1.4 ([1]) The Ricci curvature of a nilmanifold (n,, ) is given by Ric n = (1/4) ad Ei (ad Ei ) (1/2) (ad Ei ) ad Ei, where {E i } is an orthonormal basis of n and (ad Ei ) operator defined by (ad Ei ) X, Y = X, ad Ei Y. denotes the dual of the adjoint Propositions 1.3 and 1.4 states that the Ricci curvatures of solvmanifolds of Iwasawatype can be calculated in terms of the mean curvature vectors, adjoint operators and their dual operators. Proposition 1.5 ([11]) For a standard Einstein solvmanifold, there exists λ > 0 such that the real part of every eigenvalue of ad λh0 is a positive integer. Let µ 1 < µ 2 < < µ m be the real parts of eigenvalues of ad λh0 without common divisor, and d 1, d 2,..., d m the corresponding multiplicities. We call (µ 1 < < µ m ; d 1,..., d m ) the eigenvalue type of an Einstein solvmanifold. For example, the eigenvalue types of Damek-Ricci spaces are (1 < 2; d 1, d 2 ). 2 Nilmanifolds attached to graded Lie algebras In this section we briefly recall some properties of graded Lie algebras and define the nilmanifolds associated to them. We refer to [13], [21] for graded Lie algebras. Let g be a semisimple Lie algebra. A decomposition of g into subspaces, g = k Z g k is called a gradation if [g k, g l ] g k+l holds for every k and l. A Lie algebra with gradation is called a graded Lie algebra. By definition, n := k>0 g k is a nilpotent subalgebra. 4

In the theory of graded Lie algebras, it is fundamental that there exist the characteristic element Z g and a grade-reversing Cartan involution σ. An element Z is called the characteristic element if g k = {X g [Z, X] = k X} for every k. A Cartan involution σ is said to be grade-reversing if σ(g k ) = g k for every k. Note that the characteristic element is unique. It is easy to show that σ is grade-reversing if and only if σ(z) = Z. A Cartan involution σ defines the natural inner product B σ on g by B σ (X, Y ) := B(X, σ(y )), where B denotes the Killing form of g. Now we can define our nilmanifolds: Definition 2.1 Let g = g k be a graded Lie algebra. The nilpotent Lie algebra n := k>0 g k endowed with the inner product B σ is called the nilmanifold attached to the graded Lie algebra, or shortly, nilmanifold attached to GLA. Note that k 0 g k is a parabolic subalgebra of g and n is the nilradical of it. Conversely, it is known that every parabolic subalgebra of g can be obtained in this way. Therefore, we are considering the nilradicals of all parabolic subalgebras of all semisimple Lie algebras. We also note that our nilpotent Lie algebras n are related to R-spaces. Let g = g k be a graded Lie algebra, G the centerless Lie group with Lie algebra g, and P the subgroup of G with Lie algebra k 0 g k. The quotient space G/P is an R-space (it is sometimes called a real flag manifold). By definition, n can be naturally identified with the tangent spaces of G/P. Therefore, there is a correspondence between the class of our nilmanifolds and the class of R-spaces (but this is not one-to-one). A graded Lie algebra g = g k is said to be of ν-th kind if g ν 0 and g k = 0 for every k > ν. The existence of a grade-reversing Cartan involution implies that dim g k = dim g k, therefore, a ν-th kind gradation satisfies g k = 0 for every k > ν. Note that, for a graded Lie algebra of ν-th kind, the nilpotent Lie algebra n = k>0 g k is not of ν-step in general, but at most ν-step nilpotent. We will explain this later. To state the classification of gradations, we now start from a semisimple symmetric pair of noncompact type (g, k) with Cartan involution σ. Take the Cartan decomposition g = k + p with respect to σ and a maximal abelian subspace a in p. Let be the root 5

system of g with respect to a. Recall that we say α a {0} is a root if 0 g α := {X g [H, X] = α(h) X for every H a}. Note that g can be decomposed into the direct sum of root spaces: g = g 0 + α g α. Let Π := {α 1,..., α r } be a set of simple roots of and define C := {H a α i (H) Z 0 for every α i Π} = {c 1 H 1 + + c r H r c 1,..., c r Z 0 }, where {H 1,..., H r } is the dual basis of Π, that is, α i (H j ) = δ ij. Then, every element Z C gives a gradation, by putting g k := g α. α(z)=k On the other hand, every gradation of g can be obtained in this way up to conjugation (see [14], [21]). We summarize this as follows. Theorem 2.2 ([14], [21]) There is a bijective correspondence between the set of the isomorphism classes of gradations on g and the set of Aut(DD)-orbits of C, where Aut(DD) denotes the automorphism group of the Dynkin diagram of. It is easy to see that the gradation given by Z C is of ν-th kind if and only if ν = α(z), where α denotes the highest root. Obviously n = k>0 g k is at most ν-step nilpotent. We will mention here the condition for n to be ν-step nilpotent. A graded Lie algebra g = g k is said to be of type α 0 if g 1 generates k>0 g k. This notation was introduced by Kaneyuki and Asano ([14]), and they showed that a graded Lie algebra g = g k is of type α 0 if and only if Z {c 1 H 1 + + c r H r c 1,..., c r {0, 1}} up to conjugation. This property is related to the structure of the attached nilmanifold. Proposition 2.3 Let (n, B σ ) be the nilmanifold attached to a graded Lie algebra g = gk of ν-th kind. Then, (1) there exists a type α 0 gradation g = g k which satisfies n = k>0 g k, (2) n is of ν-step nilpotent if and only if the gradation is of type α 0. 6

Proof. (1): Denote the characteristic element of the gradation g = g k by Z = c i1 H i 1 + + c is H is with c i1,..., c is > 0. Then the gradation given by Z = H i 1 + + H is satisfies the condition. (2): Assume g = g k is of type α 0. Since g 1 generates g ν, the lower central series satisfies C ν (n) 0. Thus n is of ν-step nilpotent. To show the converse, assume g = g k is not of type α 0. In this case one has Z = c i1 H i 1 + +c is H is and some coefficients are greater than 1. As in the proof for (1), the gradation g = g k given by H i 1 + + H is is of type α 0 and satisfies n = k>0 g k. The assumption implies that g = g k is of ν -kind with ν < ν. Therefore n is ν -step nilpotent and not ν-step nilpotent. This proposition means that, to study the nilmanifold attached to GLA, it is enough to consider gradations of type α 0. 3 Ricci curvatures of the attached nilmanifolds In this section we calculate the Ricci curvatures of the nilmanifolds attached to GLA. The Ricci curvatures can be written by using the bracket products and the grade-reversing Cartan involutions. Lemma 3.1 The nilmanifolds (n, B σ ) attached to GLA satisfy that (1) B σ ([U, V ], W ) = B σ (V, [σu, W ]) for every U, V, W n, (2) (ad U ) V = [V, σu] n for every U, V n. These formulae can be checked easily but quite useful for calculating the Ricci curvatures. Definition 3.2 Let {E (k) i the k-th mean curvature vector. } be an orthonormal basis of g k. We call Z k := [σ(e (k) i ), E (k) i ] It is easy to show that Z k is independent of the choice of orthonormal basis. Furthermore, Lemma 3.1 leads that H 0 := Z 1 + +Z ν coincides with the mean curvature vector of the solvmanifold (a + n, B σ ). 7

Lemma 3.3 Let {E (k) i } be an orthonormal basis of g k and define the operators A k and B k on n by A k (U) := [E (k) i, [σe (k) i, U]], B k (U) := [σe (k) i, [E (k) i, U]]. Then, for every U l g l, we have (1) A k (U l ) = B k (U l ) [Z k, U l ], (2) B k (U l ) = A k+l (U l ) = B k+l (U l ) [Z k+l, U l ]. Proof. (1) is a direct consequence of the Jacobi identity. To show (2), we need an orthonormal basis {E (k+l) j } of g k+l. It follows from [E (k) i, U l ] g k+l that B m (U l ) = i = i = i = j [σe (k) i, [E (k) i, U l ]] j j [σe i, [E (k) i [σ( E (k) i, U l ], E (k+l) j E (k+l) j ], [σu l, E (k+l) j [σ([σu l, E (k+l) j ]), E (k+l) j ] ] E (k) i ), E (k+l) j ] = A l+m (U l ). This concludes the lemma. By definition, B k (U l ) = 0 if k + l > ν. Therefore one can determine A k and B k by these recursion formulae. These operators play important roles for calculating the Ricci curvatures of our nilmanifolds. Theorem 3.4 Let (n, B σ ) be the nilmanifold attached to GLA and Ric n curvature. Then, for every U l g l, one has be the Ricci (1) Ric n (U l ) = (1/4) m<l A m (U l ) + (1/2) m>l A m (U l ), (2) Ric n (U 1 ) = (1/2) (1 k)[z k, U 1 ], (3) Ric n (U 2 ) = (1/4) (2 k)[z k, U 2 ]. 8

Proof. Let {E i } be an orthonormal basis of g k. Direct calculations lead that adei (ad Ei ) (U l ) = A m for m < l, [E i, [σe i, U l ] n ] = 0 for m l. Furthermore, one can see that (adei ) ad Ei (U l ) = B m (U l ) = A l+m (U l ). Thus Proposition 1.4 concludes (1). Lemma 3.3 as follows: One can calculate A k (U 1 ) and A k (U 2 ) by using A k (U 1 ) = B k (U 1 ) [Z k, U 1 ] = (1/2)B k+1 (U 1 ) [Z k, U 1 ] [Z k+1, U 1 ] = = [Z k + Z k+1 + + Z ν, U 1 ], A k (U 2 ) = [Z k, U 2 ] [Z k+2, U 2 ] [Z k+4, U 2 ]. By substituting these formulae for the equation (1), we conclude (2) and (3). 4 Mean curvature vectors As we saw in the previous section, one needs to know the mean curvature vectors Z 1,..., Z ν for calculating the Ricci curvature of the nilmanifolds attached to GLA. In this section we investigate these vectors. We call H α a the root vector of a root α if B σ (H α, H) = α(h) for every H a. Our root vectors are the negative of the usual ones, since B σ = B on a a. We employ this definition for convenience. Let k := {α α(z) = k}. Lemma 4.1 The k-th mean curvature vector Z k satisfies (1) Z k = α k (dim g α )H α, and (2) B σ (Z k, Z) = k dim g k. Proof. Let E g α be a unit vector. Then, Lemma 3.1 follows that H α = [σe, E], which concludes (1). To show (2) we need {E i }, an orthonormal basis of g k. One can see that B σ (Z k, Z) = B σ ( [σe i, E i ], Z) = B σ (E i, [Z, E i ]) = B σ (E i, ke i ) = k dim g k. 9

This concludes (2). We now determine the mean curvature vectors of our nilmanifolds. Recall that we can assume, without loss of generality, that Z is a linear combination of {H i }, the dual basis of the simple roots Π = {α i }. We also recall that the Weyl group is generated by {s 1,..., s r }, where s j denotes the reflection of a with respect to the hyperplane {H a α j (H) = 0}. Theorem 4.2 Let (n, B σ ) be the nilmanifold attached to GLA and assume that the characteristic element is Z = c i1 H i 1 + c i2 H i 2 + + c is H is with c i1,..., c is > 0. Then, (1) Z k = z i1 H i 1 + + z is H is with z ij = β k (dim g β )B σ (H αij, H β ), for every k, and (2) H 0 = h i1 H i 1 + + h is H is with h ij = β i j (dim g β )B σ (H αij, H β ), where i j := Proof. {β β(z) = β(h i j ) > 0}. Furthermore, h ij > 0. (1): First of all we show that Z k V := RH i 1 + RH i 2 + + RH is. Since the orthogonal complement of V in a is spanned by root vectors {H αj j i 1,..., i s }, it is sufficient to show that B σ (Z k, H αj ) = 0 for every j i 1,..., i s. By the definition of Weyl group, this condition is equivalent to s j (Z k ) = Z k. Take j i 1,..., i s. Because s j acts trivially on V, one has s j (Z) = Z and hence s j ( k ) = k. This implies that s j (Z k ) = s j ( β k (dim g β )H β ) = β k (dim g β )s j (H β ) = β k (dim g sj (β))h sj (β) = Z k. Note that the multiplicities of roots, dim g β, are invariant under the action of the Weyl group. Therefore we have B σ (Z k, H αj ) = 0, and hence Z k V. Each coefficient of Z k = z i1 H i 1 + + z is H is z ij = α ij (Z k ) = α ij ( can be determined by β k (dim g β )H β ) = β k (dim g β )B σ (H αij, H β ). (2): Let := {β β(z) > 0}. Because of (1), one can write H 0 = h i1 H i 1 + + h is H is with h ij = β (dim g β )B σ (H αij, H β ). We here claim that s ij preserves i j. For every β i j, one can write β = n k α k so that n it formula of the reflection s ij (β) = β 2 α i j, β β, β α i j > 0 for some t j. The 10

leads that the α it -coefficient of s ij (β) is also n it. Hence, s ij (β)(z) n it > 0 and s ij (β)(z) n it + β(h i j ) > β(h i j ). This completes the claim. Thus, the similar argument as (1) leads that (dim g β )B σ (H αij, H β ) = 0, β i j which concludes the first part of (2). Now we show that h ij > 0. There might exist β i j such that B σ (H αij, H β ) < 0. But, for such β, one has s ij (β) i j and B σ (H αij, s ij (H β )) = B σ (H αij, H β ). Therefore every negative term cancels with another term. Hence, we got h ij = (dim g β )B σ (H αij, H β ), where β run through {β i j B σ (H αij, H β ) > 0, s ij (β) i j }. Since this set contains α ij, one has h ij > 0. Therefore one can determine Z k and H 0 by counting roots and their multiplicities. Of course, it is not easy. We will mainly consider the following easy case. Corollary 4.3 The k-th mean curvature vector Z k is parallel to Z if (i) Z = H i, or (ii) Z = H i + H j and a Dynkin diagram automorphism interchanges α i and α j. Proof. If the condition (i) holds, then Z k RH i = RZ. Assume the condition (ii) holds. In this case Z k can be written by Z k = z i H i + z j H j, where z i = (dim g β )B σ (H αi, H β ), z j = (dim g β )B σ (H αj, H β ), β k β k Let f be a Dynkin diagram automorphism which interchanges α i and α j. One knows that f interchanges H i and H j, f(z) = Z and f( k ) = k. Thus, the same argument as the proof of Theorem 4.2 (1) implies that z i = z j. This observation is useful for finding explicit examples of Einstein solvmanifolds. 5 Curvature properties of the attached solvmanifolds We now consider the one-dimensional solvable extensions of the nilmanifolds attached to GLA, and investigate their curvature properties and Einstein conditions. Let a H 0 = Z 1 + + Z ν be the mean curvature vector of (a + n, B σ ) and define s := RH 0 + n,, c := cb σ RH0 RH 0 + B σ n n, 11

where c is an arbitrary positive number. A solvmanifold (s = a + n,, ) is called a Carnot space if dim a = 1 and there exists A 0 a such that every eigenvalue of ad A0 on n is a positive integer (see [18]). We call a Carnot space is of (k + 1)-step if k is the maximum of the eigenvalues of ad A0. Then, Theorem 4.2 (2) directly leads that Proposition 5.1 The solvmanifolds (s,, c ) are of Iwasawa-type. are Carnot spaces. Furthermore, they Now we calculate the Ricci curvatures of our solvmanifolds. Proposition 5.2 Denote the Ricci curvatures of (s,, c ) and (n, B σ ) by Ric c and Ric n, respectively. Then, (1) Ric c (H 0 ) = (1/2c)H 0, (2) Ric c (U) = Ric n (U) (1/c)[H 0, U] for every U n. Proof. The solvmanifolds (s,, c ) are of Iwasawa-type. Hence, by applying Proposition 1.3, we get Ric c (H 0 ) = (tr(ad H0 ) 2 / H 0 2 ) H 0, Ric c (U) = Ric n (U) (tr(ad H0 )/ H 0 2 ) [H 0, U] for U n. By the definition of the inner product, we have H 0 2 = c B σ (H 0, H 0 ) = c tr(ad g H 0 ) 2, where ad g denotes the adjoint operator of g. Theorem 4.2 implies that [H 0, g 0 ] = 0, therefore one has tr(ad g H 0 ) 2 = tr(ad g H 0 σ(n) ) 2 + tr(ad g H 0 g0 ) 2 + tr(ad g H 0 n ) 2 = 2 tr(ad H0 ) 2. This concludes (1). For proving (2), let {E i } be an orthonormal basis of n. The definition of H 0 leads that c tr(ad H0 ) = c B σ (E i, [H 0, E i ]) = c B σ ([σ(e i ), E i ], H 0 ) = c B σ (H 0, H 0 ) = H 0, H 0 = H 0 2. This implies (2). 12

Corollary 5.3 Assume that the nilmanifold attached to GLA is abelian. Then, for every c > 0, the solvmanifold (s,, c ) is Einstein with Einstein constant (1/2c). Proof. Since n is abelian, we have Ric n = 0. Furthermore, the characteristic element is Z = H i for some i. Therefore Lemma 4.1 (2) follows that H 0 = Z 1 = (dim g 1 /B σ (Z, Z))Z = (1/2)Z. The corollary can be shown by substituting this equation for Proposition 5.2. In this case the Einstein space (s,, c ) is obviously isometric to the real hyperbolic space. Note that, in this case, the corresponding R-space is a symmetric R-space. Theorem 5.4 Let (n,, ) be the nilmanifold attached to a graded Lie algebra of type α 0 and assume n is not abelian. (1) If (s,, c ) is Einstein, then c = 2 and kz k is parallel to Z. (2) If c = 2 and kz k is parallel to Z, then Ric RH0 +g 1 +g 2 = (1/4) id. Proof. (1): Assume (s,, c ) is Einstein. Proposition 5.2 (1) implies that the Einstein constant must be (1/2c). Therefore, for every U 1 g 1, (1/2c)U 1 = Ric c (U 1 ) = Ric n (U 1 ) (1/c)[H 0, U 1 ] = (1/2) (1 k)[z k, U 1 ] (1/c) [Z k, U 1 ] = ( 1 2 k 2 1 c )[Z k, U 1 ]. On the other hand, (1/2c)U 1 = (1/2c)[Z, U 1 ]. Therefore one has 1 2c ad Z = ( 1 2 k 2 1 c )ad Z k on g 1. Since the gradation is of type α 0, n is generated by g 1 and hence these operators are equal on n. Furthermore, it is easy to see that they coincides also on σ(n). Combining these with [Z, g 0 ] = [Z k, g 0 ] = 0 and the semisimplicity of g, one conclude that 1 2c Z = ( 1 2 k 2 1 c )Z k. 13

By taking the inner product with Z, it follows from Lemma 4.1 (2) that 1 2c B σ(z, Z) = ( 1 2 k 2 1 c )k dim g k = ( 1 2 1 c )k dim g k 1 4 B σ(z, Z). Let us assume c 2. Thus one has B σ (Z, Z) = 2 k dim g k. On the other hand, the definition of B σ leads that B σ (Z, Z) = 2 k 2 dim g k. This implies that dim g k = 0 for every k > 1, which means that n is abelian. This is a contradiction and hence we get that c = 2. If c = 2, then obviously kz k = (1/2)Z. (2): Assume c = 2 and kz k is parallel to Z. Lemma 4.1 (2) leads that kz k = (1/2)Z. We know that Ric c (H 0 ) = (1/4)H 0. For U 1 g 1, the calculation in (1) leads that Ric c (U 1 ) = (1/2) [k Z k, U 1 ] = (1/4)U 1. Proposition 3.4 and the similar calculations show that Ric c (U 2 ) = (1/4)U 2 for U 2 g 2. Because of this theorem, we call (s,, c ) the solvmanifold attached to GLA if c = 2. We denote it by (s,, ) and the Ricci curvature by Ric. In the case that the gradation is of second kind, the attached solvmanifold is Einstein if and only if kz k is parallel to Z. This is fundamental for the arguments of the next section. Despite of that, such characterization is not true for graded Lie algebras of ν-th kind with ν > 2. We will see examples and counter-examples in Section 7. 6 Two-step case In this section we show that every solvmanifold attached to a graded Lie algebra of second kind is Einstein. Most of those Einstein solvmanifolds have an eigenvalue type (1 < 2 ; d 1, d 2 ), but some of them have another eigenvalue type. We refer [13] for the explicit classification list of second kind graded Lie algebras. Theorem 6.1 Every solvmanifold (s,, ) attached to a graded Lie algebra of second kind is Einstein. Proof. Let g = 2 k= 2 g k be a graded Lie algebra of second kind. By Theorem 5.4, it is sufficient to show that Z 1 +2Z 2 is parallel to Z. If the characteristic element Z satisfies the assumption of Corollary 4.3, Z 1 and Z 2 are parallel to Z. Hence, so is Z 1 +2Z 2. Therefore we have only to consider the case Z = H i + H j and Dynkin diagram automorphisms can 14

not interchange α i and α j. Such situation can only occur for root systems of A r -type. Assume the root system is of A r -type and Z = H i + H j. We can normalize the inner product so that every root has length 2. We can also ignore the multiplicities of roots, since every root has the same multiplicity. Then, direct calculations lead that Z 1 = (r + 1 2j)H i + (r + 1 2i)H j, Z 2 = (r + 1 j)h i + ih j, hence we have Z 1 + 2Z 2 = (r + 1)Z. Finally we mention the eigenvalue type. If Z 1 +Z 2 is parallel to Z, then the eigenvalue type of the Einstein solvmanifold (s,, ) is (1 < 2; d 1, d 2 ). On the other hand, (s,, ) has another eigenvalue type in case that the root system is of A r -type and Z = H i + H j with i + j r + 1. In such case, one can directly show that H 0 = Z 1 + Z 2 = jh i + (r + 1 i)h j up to scalar. If i + j = r + 1 then the eigenvalue type is (1 < 2 ; d 1, d 2 ). If i + j < r + 1, the eigenvalue type is (j < r + 1 i < r + 1 i + j ; d 1, d 2, d 3 ). These arguments implies that Proposition 6.2 For every 0 < µ 1 < µ 2, there exists a standard Einstein solvmanifold (s,, ) with two-step nilpotent n, whose eigenvalue type is (µ 1 < µ 2 < µ 1 +µ 2 ; d 1, d 2, d 3 ). We see a simple example here. Example 6.3 Consider the nilpotent subalgebra of sl(n + m + l, R), 0 X Z n := 0 0 Y X M n,m(r), Y M m,l (R), Z M n,l (R). 0 0 0 Then the nilmanifold (n, B σ ) has an Einstein solvable extension. Proof. Consider the gradation of sl(n+m+l, R) of second kind, given by the above block decomposition. Then (n, B σ ) coincides with the nilmanifold attached to this gradation. 15

In this case, the root system is of A n+m+l 1 -type, and direct calculations lead that H n (m + l) 1 1 n = n 1 n + m + l m, n 1 l H m = l 1 1 n l 1 n + m + l m. (n + m) 1 l The characteristic element is Z = H n + H m. The solvmanifold RZ + n is a three-step Carnot space, but this is not what we want. Our Einstein solvable extension is RH 0 + n, where H 0 is parallel to (n + m)h n + (m + l)h n+m. This is (n + 2m + l + 1)-step Carnot space if n + m and m + l have no common devisor, whereas n is two-step nilpotent. We also note that, in case n = l = 1, this nilmanifold is nothing but the Heisenberg Lie algebra, and the constructed Einstein solvmanifold is isometric to the complex hyperbolic space. In case n = l, the eigenvalue type of the Einstein solvmanifold is (1 < 2; nm + ml, nl). In case n l, ad H0 has three distinct eigenvalues. 7 Examples of Einstein spaces with higher steps In this section we investigate the solvmanifolds attached to graded Lie algebras of third kind and of fourth kind. Despite of the two-step case, not all of (s,, ) is Einstein. Proposition 7.1 Assume a graded Lie algebra is of third kind and each Z k is parallel to Z. Then the attached solvmanifold (s,, ) is Einstein if and only if dim g 1 = 2 dim g 2. Proof. By assumption, Z 1 + 2Z 2 + 3Z 3 is parallel to Z. Thus, Theorem 5.4 implies that Ric = (1/4) id on RH 0 + g 1 + g 2. Therefore (s,, ) is Einstein if and only if Ric(U 3 ) = (1/4)U 3 for every U 3 g 3. One can calculate this as follows: Ric n (U 3 ) = (1/4)[Z 1, U 3 ] + (1/4)[Z 2, U 3 ], Ric(U 3 ) = Ric n (U 3 ) (1/2)[Z 1 + Z 2 + Z 3, U 3 ] = (1/4)[Z 1, U 3 ] (1/4)[Z 2, U 3 ] (1/2)[Z 3, U 3 ] = (1/4)((dim g 1 + 2 dim g 2 + 6 dim g 3 )/B σ (Z, Z)) [Z, U 3 ] = (1/4)(3 dim g 1 + 6 dim g 2 + 18 dim g 3 )/B σ (Z, Z)) U 3. 16

Thus, Ric(U 3 ) = (1/4)U 3 if and only if 3 dim g 1 + 6 dim g 2 + 18 dim g 3 = B σ (Z, Z) = 2(dim g 1 + 4 dim g 2 + 9 dim g 3 ), that is, dim g 1 = 2 dim g 2. Note that dim g 1 = 2 dim g 2 is not always satisfied. For example, let us consider g := sl(5, R), whose root system is of A 4 -type. Put Z := H 1 + H 2 + H 3, which gives a gradation of third kind. Then, one can calculate that kz k is parallel to Z, but one also has dim g 1 = 4 and dim g 2 = 3. Next we state the similar result for four-step case. Proof is also similar. Proposition 7.2 Assume a graded Lie algebra is of fourth kind and each Z k is parallel to Z. Then the attached solvmanifold (s,, ) is Einstein if and only if dim g 1 = 3 dim g 3 and dim g 1 + 4 dim g 4 = 2 dim g 2. These propositions provide examples of Einstein solvmanifolds whose nilradicals are of three-step or four-step. It is natural to consider the case Z = H i ; in such case each Z k is parallel to Z (Corollary 4.3). The dimension of each g k can be calculated by counting roots and their multiplicities (one can see the table of multiplicities in [20]). Theorem 7.3 The solvmanifolds attached to the following graded Lie algebras are Einstein. The eigenvalue types of the constructed Einstein solvmanifolds are as follows. 17

g Z eigenvalue type g 2(2) G 2 H 1 (1 < 2 < 3 ; 2, 1, 2) g C 2 G 2 H 1 (1 < 2 < 3 ; 4, 2, 4) f 4(4) F 4 H 2 (1 < 2 < 3 ; 12, 6, 2) f C 4 F 4 H 2 (1 < 2 < 3 ; 24, 12, 4) e 6(2) F 4 H 2 (1 < 2 < 3 ; 18, 9, 2) e 7( 5) F 4 H 2 (1 < 2 < 3 ; 30, 15, 2) e 8( 24) F 4 H 2 (1 < 2 < 3 ; 54, 27, 2) e 6(6) E 6 H 4 (1 < 2 < 3 ; 18, 9, 2) e C 6 E 6 H 4 (1 < 2 < 3 ; 36, 18, 4) f 4(4) F 4 H 3 (1 < 2 < 3 < 4 ; 6, 9, 2, 3) f C 4 F 4 H 3 (1 < 2 < 3 < 4 ; 12, 18, 4, 6) e 6(2) F 4 H 3 (1 < 2 < 3 < 4 ; 12, 12, 4, 3) e 7( 5) F 4 H 3 (1 < 2 < 3 < 4 ; 24, 18, 8, 3) e 8( 24) F 4 H 3 (1 < 2 < 3 < 4 ; 48, 30, 16, 3) 8 Rank one reduction of noncompact symmetric spaces In this section we study the mean curvature vectors of noncompact symmetric spaces. For the graded Lie algebra with characteristic element Z := H 1 + + H r, one knows that n is the nilpotent part of the Iwasawa decomposition and the solvmanifold (a + n,, ) is a symmetric space. Proposition 8.1 The mean curvature vector of the above solvmanifold (a + n,, ) is H 0 = h 1 H 1 + + h r H r, where h i = (dim g αi + 4 dim g 2αi )B σ (H αi, H αi ). Proof. We apply Theorem 4.2 (2). In this case i := {β β(z) = β(h i ) > 0} consists of α i and 2α i (if exists). Therefore, h i = (dim g αi )B σ (H αi, H αi ) + (dim g 2αi )B σ (H 2αi, H 2αi ). Proposition is completed by H 2αi = 2H αi. 18

This Proposition implies that, if every root has the same length, then H 0 is parallel to Z. It is remarkable that H 0 may not parallel to Z even for symmetric spaces. But, since symmetric spaces are Einstein, Theorem 5.4 leads that kz k is parallel to Z. Furthermore, as we saw in Propositions 7.1 and 7.2, the Einstein condition requires some equalities of dimensions. It seem to be mysterious that symmetric spaces automatically satisfy these conditions. References [1] D. V. Alekseevskii, Homogeneous Riemannian spaces of negative curvature, Mat. Sb. 25 (1975), 87-109; English translation, Math. USSR-Sb. 96 (1975), 93 117. [2] D. V. Alekseevskii, Classification of quaternionic spaces with transitive solvable group of motions, Math. USSR-Izv. 9 (1975), 297 339. [3] J. Berndt, F. Tricerri, L. Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lect. Notes in Math. 1598 (1995), Springer-Verlag Berlin Heidelberg. [4] A. Besse, Einstein manifolds, Ergeb. Math. 10 (1987), Springer-Verlag, Berlin- Heidelberg. [5] J. Boggino, Generalized Heisenberg groups and solvmanifolds naturally associated, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), 529 547. [6] N. Bourbaki, Groupes et algèbres de Lie, Chapitre IV VI, Éléments de Mathématique, Masson, Paris, 1981. [7] V. Cortés, Alekseevskian spaces, Diff. Geom. Appl. 6 (1996), 129 168. [8] E. Damek, F. Ricci, Harmonic analysis on solvable extensions of H-type groups. J. Geom. Anal. 2 (1992), 213 248. [9] P. Eberlien, J. Heber, Quarter pinched homogeneous spaces of negative curvature, Internat. J. Math. 7 (1996), 441 500. 19

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