Invariant Einstein metrics on SU(N) and complex Stiefel manifolds
aa r X i v : . [ m a t h . DG ] F e b INVARIANT EINSTEIN METRICS ON
SU( N ) ANDCOMPLEX STIEFEL MANIFOLDS
ANDREAS ARVANITOYEORGOS, YUSUKE SAKANE AND MARINA STATHA
Abstract.
We study existence of invariant Einstein metrics on complex Stiefel mani-folds
G/K = SU( ℓ + m + n ) / SU( n ) and the special unitary groups G = SU( ℓ + m + n ).We decompose the Lie algebra g of G and the tangent space p of G/K , by using thegeneralized flag manifolds
G/H = SU( ℓ + m + n ) / S(U( ℓ ) × U( m ) × U( n )). We pa-rametrize scalar products on the 2-dimensional center of the Lie algebra of H , and weconsider G -invariant and left invariant metrics determined by Ad(S(U( ℓ ) × U( m ) × U( n ))-invariant scalar products on g and p respectively. Then we compute their Ricci tensorfor such metrics. We prove existence of Ad(S(U(1) × U(2) × U(2))-invariant Einsteinmetrics on V C = SU(5) / SU(2), Ad(S(U(2) × U(2) × U(2))-invariant Einstein metricson V C = SU(6) / SU(2), and Ad(S(U( m ) × U( m ) × U( n ))-invariant Einstein metrics on V m C m + n = SU(2 m + n ) / SU( n ). We also prove existence of Ad(S(U(1) × U(2) × U(2))-invariant Einstein metrics on the compact Lie group SU(5), which are not naturallyreductive. The Lie group SU(5) is the special unitary group of smallest rank known forthe moment, admitting non naturally reductive Einstein metrics. Finally, we show thatthe compact Lie group SU(4+ n ) admits two non naturally reductive Ad(S(U(2) × U(2) × U( n )))-invariant Einstein metrics for 2 ≤ n ≤
25, and four non naturally reductive Ein-stein metrics for n ≥
26. This extends previous results of K. Mori about non naturallyreductive Einstein metrics on SU(4 + n ) ( n ≥ Introduction
A Riemannian manifold (
M, g ) is called Einstein if it has constant Ricci curvature, i.e.Ric g = λ · g for some λ ∈ R . Besides the detailed exposition on Einstein manifolds in [Be],we refer to [Wa1], [Wa2] for more recent results. General existence results are difficult toobtain and some methods are described in [B¨o], [B¨oWaZi] and [WaZi]. For the case ofhomogeneous spaces the problem of finding all invariant Einstein metrics becomes slightlymore accessible, due to the possibility of making symmetry assumptions, but still it is noteasy. For example, the classification of invariant Einstein metrics for important classes ofhomogeneous spaces, such as the generalized Wallach spaces, was only recently achieved([ChNi]). Also, for other classes of homogeneous spaces, such as the generalized flagmanifolds, a complete classification of invariant Einstein metrics is still open. We referto [Ar2] for more details. For Lie groups the problem of determining all left-invariantEinstein metrics is also quite difficult, even if one makes geometric assumptions, such asthe natural reductivity of the metric. Mathematics Subject Classification.
Primary 53C25; Secondary 53C30, 13P10, 65H10, 68W30.
Keywords : Homogeneous space, Einstein metric, Stiefel manifold, special unitary group, invariantmetric, isotropy representation, Gr¨obner basis.The first author was supported by a Grant from the Empirikion Foundation in Athens and the secondauthor by JSPS KAKENHI Grant Number 16K05130.
In the present paper we study left-invariant Einstein metrics on the compact Lie groupSU( n ) and SU( n )-invariant Einstein metrics on the complex Stiefel manifolds V k C n =SU( n ) / SU( n − k ), of orthonormal k -frames in C n . Two marginal cases are the sphere S n − = SU( n ) / SU( n −
1) = V C n and the compact Lie group SU( n ) = V n − C n . The firstis an irreducible symmetric space, therefore it admits up to scale a unique SU( n )-invariantEinstein metric.Left-invariant Einstein metrics on SU( n ) have not been extensively studied. We recallthat in [D’AZi] J.E. D’Atri and W. Ziller found a large number of left-invariant Einsteinmetrics on the compact Lie groups SU( n ) , SO( n ) and Sp( n ), which are naturally reductiveand they posed the question whether there exist left-invariant Einstein metrics on compactLie groups, which are not naturally reductive. This is not an easy problem in general,especially when the rank of the Lie group is small. For example, the number of left-invariant Einstein metrics on the Lie groups SU(3) and SU(2) × SU(2) is not known(however see recent progress by F. Belgum, V. Cort´es, A.S. Haupt and D. Lindemann in[BeCoHaLi]).In our recent work [ArSaSt2] we proved existence of left-invariant Einstein metrics onSO( n ) ( n ≥ n ) was first considered in the unpublishedwork of K. Mori [Mo], where he proved existence for n ≥
6. He considered SU( n ) fiberedover a generalized flag manifold and used the method of Riemannian submersions (cf.[Be]) to compute the Ricci tensor and to prove existence of left-invariant Einstein metrics.However, he considered a special class of left-invariant metrics on SU( n ). One of our mainresults in this paper is to prove existence of non naturally reductive left-invariant Einsteinmetrics on SU(5). We also extend Mori’s result.The first invariant Einstein metrics on the real Stiefel manifolds V k R n = SO( n ) / SO( n − k ) were obtained by A. Sagle in [Sa]. Later, G. Jensen obtained additional Einstein metricson V k R n as well as on the quaternionic Stiefel manifolds V k H n = Sp( n ) / Sp( n − k ) ([Je]).In the works [ArDzNi1], [ArDzNi2], [ArDzNi3] the first author, V.V. Dzhepko and Yu.G. Nikonorov proved existence of new invariant Einstein metrics on V k R n and V k H n ,by making certain symmetry assumptions. The method was extended by the authorsin [ArSaSt1] and [ArSaSt3] and obtained additional invariant Einstein metrics on thesespaces.Invariant Einstein metrics on complex Stiefel manifolds have not been studied before.Since the isotropy representation of V k C n contains equivalent irreducible subrepresen-tations, the search for invariant Einstein metrics on such homogeneous spaces G/H ,is quite difficult. In fact, a complete description of the set of all G -invariant metrics,and in turn the computation of the Ricci tensor of G/H is complicated. Some otherworks where the authors studied invariant Einstein metrics for such type of homoge-neous spaces, are [Ke] by M. Kerr, and [Ni1], [Ni2] by Yu.G. Nikonorov. Also, in theprevious mentioned works [ArDzNi1], [ArDzNi2], [ArDzNi3], [ArSaSt1] and [ArSaSt3],the Einstein metrics were obtained by using the generalized Wallach spaces
G/H =SO( ℓ + m + n ) / (SO( ℓ ) × SO( m ) × SO( n )) or Sp( ℓ + m + n ) / (Sp( ℓ ) × Sp( m ) × Sp( n )), where instein metrics on SU( N ) and complex Stiefel manifolds 3 the dimension of the center of the Lie algebra of H is at most 1. For the complex Stiefelmanifolds G/K = SU( ℓ + m + n ) / SU( n ) we find SU( ℓ + m + n )-invariant Einstein metricsby using the generalized Wallach space G/H = SU( ℓ + m + n ) / S(U( ℓ ) × U( m ) × U( n ))(a generalized flag manifold). In this case the dimension of the center of the Lie algebraof H is 2, which makes the description of invariant metrics more complicated.In the present work we give a unified treatment for finding left-invariant Einstein metricson the Lie group G = SU( ℓ + m + n ), which are not naturally reductive, as well as SU( ℓ + m + n )-invariant Einstein metrics on the Stiefel manifold G/K = SU( ℓ + m + n ) / SU( n ).Our approach is the following: We consider the generalized flag manifold G/H =SU( ℓ + m + n ) / S(U( ℓ ) × U( m ) × U( n )) whose tangent space decomposes into a directsum of irreducible and inequivalent submodules m = m ⊕ m ⊕ m . We decompose theLie algebra of H into its center h and simple ideals h , h , h . Then the Lie algebra of G decomposes into a direct sum g = h ⊕ h ⊕ h ⊕ h ⊕ m ⊕ m ⊕ m . Also the tangent spaceof the Stiefel manifold G/K decomposes as p = h ⊕ h ⊕ h ⊕ m ⊕ m ⊕ m . Then weparametrize all scalar products in the center h by further decomposing h = h ⊕ h intoone-dimensional ideals, and then consider appropriate Ad(S(U( ℓ ) × U( m ) × U( n ))-invariantscalar products on g and p . These scalar products determine left-invariant metrics on G ,and G -invariant metrics on G/H respectively.Next, we pursue with the computation of the Ricci tensor for such metrics, whichconsists of a non diagonal part at the center h and a diagonal part at h ⊕ h ⊕ h ⊕ m ⊕ m ⊕ m . We introduce the numbers (cid:8) ijk (cid:9) , which generalize the well known numbers (cid:2) ijk (cid:3) introduced by M. Wang and W. Ziller in [WaZi]. As a result, we obtain explicitexpressions for the Ricci tensor in terms of the variables of the metric and ℓ, m, n , so theEinstein equation reduces to an algebraic system of equations r = r = · · · = r = 0with parameters ℓ, m, n . By making a suitable choice of a basis for the center of the Liealgebra of S(U( ℓ ) × U( m ) × U( n )), some of the equations become linear with respect to somevariables (cf. Subsection 5.2). We also take ℓ = 1 , m = 2, and then we use Gr¨obner basesmethods and arguments using the resultant of polynomials, to obtain explicit solutions,or prove existence of positive solutions for such systems.For the case of the complex Stiefel manifold SU( p + n ) / SU( n ) some of the SU( p + n )-invariant Einstein metrics are obtained from solutions of quadratic equations. We callthese Einstein metrics of Jensen’s type , because they are of the form g = B | m + s B | h + t B | su ( p ) , on the total space of fibrations SU( p + n ) / SU( n ) → SU( p + n ) / S(U( p ) × U( n )),where m is the orthogonal complement of s ( u ( p ) + u ( n )) in su ( p + n ), h is the center ofthe Lie algebra of s ( u ( p ) + u ( n )) and B is the negative of the Killing form of g (cf. [Je]).Our results for the special unitary group are the following: Theorem 1.1.
The compact Lie group
SU(5) admits left-invariant Einstein metrics whichare not naturally reductive.
Theorem 1.2.
The compact Lie group
SU(4 + n ) admits at least two non naturally re-ductive left-invariant Einstein metrics for ≤ n ≤ and four non naturally reductiveleft-invariant Einstein metrics for n ≥ . Andreas Arvanitoyeorgos, Yusuke Sakane and Marina Statha
Our results for the complex Stiefel manifold are the following:
Theorem 1.3. The complex Stiefel manifold V C = SU(4) / SU(2) admits two
Ad(S(U(1) × U(1) × U(2)) -invariant Einstein metrics which are of Jensen’s type. The complex Stiefel manifold V C = SU(5) / SU(2) admits four
Ad(S(U(1) × U(2) × U(2)) -invariant Einstein metrics, two of these are of Jensen’s type. The complex Stiefel manifold V C = SU(6) / SU(2) admits eight
Ad(S(U(2) × U(2) × U(2)) -invariant Einstein metrics, two of these are of Jensen’s type.
Theorem 1.4.
The complex Stiefel manifolds V m C m + n ( m ≥ ) admit at least two Ad(S(U( m ) × U( m ) × U( n ))) -invariant Einstein metrics which are not of Jensen’s type,for certain infinite values of m and n . The Ricci tensor for reductive homogeneous spaces
Let G be a compact semisimple Lie group, K a connected closed subgroup of G andlet g and k be the corresponding Lie algebras. The Killing form of g is negative definite,so we can define an Ad( G )-invariant inner product B on g , where B is the negative of theKilling form of g . Let g = k ⊕ m be a reductive decomposition of g with respect to B sothat [ k , m ] ⊂ m and m ∼ = T o ( G/K ).Any G -invariant metric g on G/K is determined by an Ad( K )-invariant scalar product h , i on m . Let { X j } be a h , i -orthonormal basis of m . Then the Ricci tensor r of themetric g is given as follows ([Be, p. 381]): r ( X, Y ) = − X i h [ X, X i ] , [ Y, X i ] i + 12 B ( X, Y ) + 14 X i,j h [ X i , X j ] , X ih [ X i , X j ] , Y i . (1)If the isotropy representation of G/K is decomposed into a sum of non equivalentirreducible summands, then we will also use an alternative expression for the Ricci tensor,which we describe next. Let(2) m = m ⊕ · · · ⊕ m q , be a decomposition into mutually non equivalent irreducible Ad( K )-modules. Then any G -invariant metric on G/K is determined by the scalar product h , i = x B | m + · · · + x q B | m q , (3)for positive real numbers ( x , . . . , x q ) ∈ R q + . Note that G -invariant symmetric covariant2-tensors on G/K are of the same form as the Riemannian metrics (although they are notnecessarily positive definite). In particular, the Ricci tensor r of a G -invariant Riemannianmetric on G/K is of the same form as (3), that is r = z B | m + · · · + z q B | m q , for some real numbers z , . . . , z q .Let { e α } be a B -orthonormal basis adapted to the decomposition of m , i.e. e α ∈ m i forsome i , and α < β if i < j . We put A γαβ = B ([ e α , e β ] , e γ ) so that [ e α , e β ] = P γ A γαβ e γ and set (cid:2) kij (cid:3) = P ( A γαβ ) , where the sum is taken over all indices α, β, γ with e α ∈ m i , e β ∈ instein metrics on SU( N ) and complex Stiefel manifolds 5 m j , e γ ∈ m k (cf. [WaZi]). Then the positive numbers (cid:2) kij (cid:3) are independent of the B -orthonormal bases chosen for m i , m j , m k , and (cid:2) kij (cid:3) = (cid:2) kji (cid:3) = (cid:2) jki (cid:3) . We call these numbers B -structure constants. Let d k = dim m k . Then we have the following: Lemma 2.1. ([PaSa])
The components r , . . . , r q of the Ricci tensor r of the metric h , i of the form (3) on G/K are given by (4) r k = 12 x k + 14 d k X j,i x k x j x i (cid:20) kji (cid:21) − d k X j,i x j x k x i (cid:20) jki (cid:21) ( k = 1 , . . . , q ) , where the sum is taken over i, j = 1 , . . . , q . Since by assumption the submodules m i , m j in the decomposition (2) are mutually nonequivalent for any i = j , it is r ( m i , m j ) = 0 whenever i = j . Thus by Lemma 2.1 it followsthat G -invariant Einstein metrics on M = G/K are exactly the positive real solutions g = ( x , . . . , x q ) ∈ R q + of the polynomial system { r = λ, r = λ, . . . , r q = λ } , where λ ∈ R + is the Einstein constant.3. Invariant metrics on
SU( ℓ + m + n ) and on V ℓ + m C ℓ + m + n = SU( ℓ + m + n ) / SU( n )3.1. Decomposition of tangent spaces.
We will describe decompositions of the tan-gent spaces of the Lie group SU( ℓ + m + n ) and the Stiefel manifold SU( ℓ + m + n ) / SU( n )at corresponding identity elements, which will be convenient for our study. We considerthe homogeneous space G/H = SU( ℓ + m + n ) / S(U( ℓ ) × U( m ) × U( n )), which is a complexgeneralized flag manifold. It is known that the isotropy representation of G/H is a directsum of three non equivalent subrepresentations, hence the tangent space m of G/H at eH decomposes into three non equivalent Ad( H )-submodules m = m ⊕ m ⊕ m , given by m = n A − ¯ A t : A ∈ M ℓ,m C o , m = n B − ¯ B t : B ∈ M ℓ,n C o , m = n C − ¯ C t : C ∈ M m,n C o , where M ℓ,m C denotes the set of all ℓ × m complex matrices. In fact, m is given by k ⊥ in g = su ( ℓ + m + n ) with respect to B . Andreas Arvanitoyeorgos, Yusuke Sakane and Marina Statha
Let h = h ⊕ h ⊕ h ⊕ h be the decomposition of h , the Lie algebra of H , into its2-dimensional center h and simple ideals, given by h = n √− a ℓ I ℓ a m I m
00 0 a n I n : a + a + a = 0 , a , a , a ∈ R o , h = n A : A ∈ su ( ℓ ) o , h = n A
00 0 0 : A ∈ su ( m ) o , h = n A : A ∈ su ( n ) o . Then the Lie algebra g splits into h and three Ad( H )-irreducible modules as(5) g = h ⊕ m = h ⊕ h ⊕ h ⊕ h ⊕ m ⊕ m ⊕ m . This is an orthogonal decomposition with respect to B .Let H = √− b ℓ + m I ℓ b ℓ + m I m
00 0 − b n I n and H = √− b ℓ I ℓ − b m I m
00 0 0 , where b = a + a , b = ( ma − ℓa ) / ( ℓ + m ), and consider the B -orthogonal decomposition h = h ⊕ h , where h = span { H } , h = span { H } . Then decomposition (5) becomes(6) g = h ⊕ h ⊕ h ⊕ h ⊕ h ⊕ m ⊕ m ⊕ m . We also consider the complex Stiefel manifold
G/K = SU( ℓ + m + n ) / SU( n ) and theAd( K )-invariant decomposition of its tangent space p at eK , given by(7) p = h ⊕ h ⊕ h ⊕ h ⊕ m ⊕ m ⊕ m . By a direct computation we obtain the following:
Lemma 3.1.
The submodules in the decompositions (6) , (7) satisfy the following bracketrelations: [ h i , h i ] ⊂ h i , ( i = 1 , ,
3) [ h , h i ] = (0) , ( i = 1 , , , [ h i , h j ] = (0) , ( i = j ) , [ h , m ] ⊂ m , [ h , m ] ⊂ m , [ h , m ] ⊂ m , [ h , m ] ⊂ m , [ h , m ] ⊂ m , [ h , m ] ⊂ m , [ h k , m ij ] = (0) , ( k = 1 , , k = i, j ) [ h , m ij ] ⊂ m ij , [ h , m ij ] ⊂ m ij , [ m , m ] ⊂ m , [ m , m ] ⊂ m , [ m , m ] ⊂ m , [ m , m ] ⊂ h ⊕ h ⊕ h ⊕ h , [ m , m ] ⊂ h ⊕ h ⊕ h ⊕ h , [ m , m ] ⊂ h ⊕ h ⊕ h ⊕ h . instein metrics on SU( N ) and complex Stiefel manifolds 7 Therefore, we see that the only non zero B -structure constants (up to permutation ofindices) are(8) (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (12)1(12) (cid:21) , (cid:20) (13)1(13) (cid:21) , (cid:20) (12)2(12) (cid:21) , (cid:20) (23)2(23) (cid:21) , (cid:20) (13)3(13) (cid:21) , (cid:20) (23)3(23) (cid:21) , (9) (cid:20) (12)4(12) (cid:21) , (cid:20) (23)4(23) (cid:21) , (cid:20) (13)4(13) (cid:21) , (cid:20) (13)5(13) (cid:21) , (cid:20) (23)5(23) (cid:21) , (cid:20) (12)5(12) (cid:21) , (cid:20) (13)(12)(23) (cid:21) . In order to compute the triplets (cid:2) ijk (cid:3) we need the following lemma from [ArDzNi1]adjusted to our case (for a more detailed proof see also [ArSaSt1, Lemma 5.2]).
Lemma 3.2.
Let q be a simple subalgebra of g = su ( N ) . Consider an orthonormal basis { f j } of q with respect to B ( negative of the Killing form of su ( N )) , and denote by B q theKilling form of q . Then, for i = 1 , . . . , dim q , we have dim q X j,k =1 ( B ([ f i , f j ] , f k ) = α qsu ( N ) , where α qsu ( N ) is the constant determined by B q = α qsu ( N ) · B | q . Lemma 3.3.
Let N = ℓ + m + n . Then the following expressions are valid: (cid:20) (cid:21) = ℓ ( ℓ − N , (cid:20) (cid:21) = m ( m − N , (cid:20) (cid:21) = n ( n − N , (cid:20) (12)1(12) (cid:21) = m ( ℓ − N , (cid:20) (13)1(13) (cid:21) = n ( ℓ − N , (cid:20) (12)2(12) (cid:21) = ℓ ( m − N , (cid:20) (23)2(23) (cid:21) = n ( m − N , (cid:20) (12)4(12) (cid:21) = 0 , (cid:20) (13)4(13) (cid:21) = ℓℓ + m , (cid:20) (23)4(23) (cid:21) = mℓ + m , (cid:20) (12)5(12) (cid:21) = ℓ + mN , (cid:20) (13)5(13) (cid:21) = mnN ( ℓ + m ) , (cid:20) (23)5(23) (cid:21) = ℓnN ( ℓ + m ) , (cid:20) (23)(12)(13) (cid:21) = ℓmnN , (cid:20) (13)3(13) (cid:21) = ℓ ( n − N , (cid:20) (23)3(23) (cid:21) = m ( n − N .
Proof.
Let g = su ( ℓ + m + n ), q = su ( ℓ ) with corresponding Killing forms B g ( X, Y ) =2( ℓ + m + n ) tr( XY ), B q = 2 ℓ tr( XY ) respectively. Let { f j } be an orthonormal basis of q with respect to − B g (1 ≤ j ≤ dim q ). Then B q = α qg · B g | q with α qg = ℓℓ + m + n . Then wehave (cid:20) (cid:21) = ℓ − X i =1 dim q X j,k =1 B g ([ f i , f j ] , f k ) = (dim q ) α qg = ℓ ( ℓ − N .
The triplets (cid:2) (cid:3) and (cid:2) (cid:3) can be computed in a similar manner, by choosing q = su ( m )and q = su ( n ) respectively.Now let q = su ( ℓ + m ) and { f j } be an orthonormal basis of q with respect to B g . Since su ( ℓ + m ) = su ( ℓ ) ⊕ m ⊕ su ( m ) ⊕ h , we adapt the basis { f j } to this decomposition as fol-lows: { f , . . . , f ℓ − } ∈ su ( ℓ ), { f ℓ , . . . , f ( ℓ − ℓm } ∈ m , { f ℓ +2 ℓm , . . . , f ℓ +2 ℓm + m − } ∈ Andreas Arvanitoyeorgos, Yusuke Sakane and Marina Statha su ( m ), f ( ℓ + m ) − ∈ h . Then ( ℓ + m ) − X j,k =1 B g ([ f i , f j ] , f k ) = α su ( ℓ + m ) g = ℓ + mN . For { f i : i = 1 , . . . , ℓ − } ∈ su ( ℓ ) we have ℓ − X i =1 ( ℓ + m ) − X j,k =1 B g ([ f i , f j ] , f k ) = ℓ + mN ( ℓ − , and for { f i : i = 1 , . . . , ℓ − } ∈ su ( ℓ ), { f j : j = 1 , . . . , m − } ∈ su ( m ) we have that[ f i , f j ] ∈ su ( ℓ ) , f j ∈ su ( ℓ ) m , f j ∈ m , f j ∈ su ( m ) . Therefore, (cid:20) (cid:21) + (cid:20) (cid:21) + 0 + 0 = ℓ + mN ( ℓ − , from which it follows that (cid:20) (cid:21) = mN ( ℓ − . The other B -structure constants can be computed in an analogous way. (cid:3) A parametrization of invariant metrics.
We now consider left-invariant metricson SU( ℓ + m + n ) determined by the Ad( H )-invariant scalar products on g = su ( l + m + n ).Note that in the decomposition (5) the Ad( H )-irreducible modules h , h , h , m , m and m are mutually non equivalent. So any Ad( H )-invariant scalar product h on su ( ℓ + m + n )can be expressed in the form(10) h = β | h + u B | h + u B | h + u B | h + X i 8) with respect to g (resp. g ).Note that the scalar product g is not in general bi-invariant, because g ([ ˜ X ( i ) j , ˜ X ( i ) l ] , U k ) = g ( ˜ X ( i ) j , [ ˜ X ( i ) l , U k ]) , for k = 4 , . Therefore, it is convenient to express the scalar product (11) in terms of B . We havethe following: Proposition 3.4. For every X ( i ) j , X ( i ) l , i = 6 , , the following relations are satisfied: hh [ X ( i ) j , X ( i ) l ] , U ii = √ v (cid:8) aB ([ X ( i ) j , X ( i ) l ] , ˜ H ) + bB ([ X ( i ) j , X ( i ) l ] , ˜ H ) (cid:9) , hh [ X ( i ) j , X ( i ) l ] , U ii = √ v (cid:8) cB ([ X ( i ) j , X ( i ) l ] , ˜ H ) + dB ([ X ( i ) j , X ( i ) l ] , ˜ H ) (cid:9) . Proof. For the first relation we have: hh [ X ( i ) j , X ( i ) l ] , U ii = v h [ X ( i ) j , X ( i ) l ] , U i = √ v h [ X ( i ) j , X ( i ) l ] , V i = √ v (cid:10) B ([ X ( i ) j , X ( i ) l ] , ˜ H ) ˜ H + B ([ X ( i ) j , X ( i ) l ] , ˜ H ) ˜ H , V (cid:11) = √ v (cid:0) h B ([ X ( i ) j , X ( i ) l ] , ˜ H ) ˜ H , V i + h B ([ X ( i ) j , X ( i ) l ] , ˜ H ) ˜ H , V i (cid:1) = √ v (cid:0) h B ([ X ( i ) j , X ( i ) l ] , ˜ H ) ˜ H , p ˜ H + r ˜ H i + h B ([ X ( i ) j , X ( i ) l ] , ˜ H ) ˜ H , p ˜ H + r ˜ H i (cid:1) = √ v (cid:0) B ([ X ( i ) j , X ( i ) l ] , ˜ H ) (cid:0) h ˜ H , p ˜ H i + h ˜ H , r ˜ H i (cid:1) + B ([ X ( i ) j , X ( i ) l ] , ˜ H ) (cid:0) h ˜ H , p ˜ H i + h ˜ H , r ˜ H i (cid:1)(cid:1) . By using the relations h ˜ H , ˜ H i = a + c , h ˜ H , ˜ H i = ab + cd and h ˜ H , ˜ H i = b + d ,the right-hand side in the last equation above can be written as √ v (cid:0) B ([ X ( i ) j , X ( i ) l ] , ˜ H ) (cid:0) p ( a + c ) + r ( ab + cd ) (cid:1) + B ([ X ( i ) j , X ( i ) l ] , ˜ H ) (cid:0) p ( ab + cd ) + r ( b + d ) (cid:1)(cid:1) . On the other hand, by (12) it follows that p = dad − bc , r = − cad − bc , q = − bad − bc , s = aad − bc ,therefore, we finally obtain that hh [ X ( i ) j , X ( i ) l ] , U ii = √ v (cid:0) aB ([ X ( i ) j , X ( i ) l ] , ˜ H ) + bB ([ X ( i ) j , X ( i ) l ] , ˜ H ) (cid:1) . The second relation can be proved by a similar manner. (cid:3) To state the following lemma (which we will use shortly in the next section) we needto choose orthonormal Weyl bases for the modules m , m and m . Let E ij denote instein metrics on SU( N ) and complex Stiefel manifolds 11 the N × N matrix with 1 in the ( i, j )-entry and 0 elsewhere, and define the matrices A ij = E ij − E ij , B ij = √− E ij + E ij ). Lemma 3.5. The following Lie bracket relations are satisfied: (1) If A ij , B ij ∈ n , then [ ˜ H , A ij ] = [ ˜ H , B ij ] = 0 , [ ˜ H , A ij ] = c (cid:0) ℓ + 1 m (cid:1) B ij , [ ˜ H , B ij ] = − c (cid:0) ℓ + 1 m (cid:1) A ij . (2) If A ij , B ij ∈ n , then [ ˜ H , A ij ] = c (cid:0) n + 1 ℓ + m (cid:1) B ij , [ ˜ H , B ij ] = − c (cid:0) n + 1 ℓ + m (cid:1) A ij , [ ˜ H , A ij ] = c ℓ B ij , [ ˜ H , B ij ] = − c ℓ A ij . (3) If A ij , B ij ∈ n , then [ ˜ H , A ij ] = c (cid:0) n + 1 ℓ + m (cid:1) B ij , [ ˜ H , B ij ] = − c (cid:0) n + 1 ℓ + m (cid:1) A ij , [ ˜ H , A ij ] = − c ℓ B ij , [ ˜ H , B ij ] = c ℓ A ij . The Ricci tensor for left-invariant metrics on SU( ℓ + m + n ) andinvariant metrics on SU( ℓ + m + n ) / SU( n )Note that any Ad( H )-invariant symmetric bilinear form of su ( N ) can be expressed as γ | h + w B | h + w B | h + w B | h + X i 3. Then we have the following: Proposition 4.1. Let g denote any of the scalar products (15) , (17) and let Z, W ∈ h .Then it is X i =6 , , n i X j =1 g ([ Z, X ( i ) j ] , [ W, X ( i ) j ]) = B ( Z, W ) . (20) Proof. Let { X ( i ) j : j = 1 , . . . , dim n i } be the orthonormal basis of n i , i = 6 , , g . Then: X i =6 , , X j g ([ Z, X ( i ) j ] , [ W, X ( i ) j ]) = X i =6 , , X j x ( i ) B ([ Z, X ( i ) j ] , [ W, X ( i ) j ])= X i =6 , , X j x ( i ) B ([ Z , √ x ( i ) ˜ X ( i ) j ] , [ W, √ x ( i ) ˜ X ( i ) j ]) = X i =6 , , X j B ([ Z, ˜ X ( i ) j ] , [ W, ˜ X ( i ) j ]) = X i =6 , , X j B (ad( Z ) ˜ X ( i ) j , ad( W ) ˜ X ( i ) j ) = X i =6 , , X j − B ( ˜ X ( i ) j , ad( Z ) ad( W ) ˜ X ( i ) j )= − tr(ad( Z ) ◦ ad( W )) = B ( Z, W ) . (cid:3) In order to compute the Ricci tensor for the invariant metrics (15) we define newnumbers, which we call them Q -structure constants , by (cid:26) kij (cid:27) = X α,β,γ Q ([ ˜ X ( i ) α , ˜ X ( j ) β ] , ˜ X ( k ) γ ) , where Q = h , i| ˜ h + h , i| ˜ h + B | h + B | h + B | h + X B | m ij . Note that it is (cid:26) kij (cid:27) = (cid:26) kji (cid:27) , but (cid:26) kij (cid:27) is not always equal to (cid:26) jik (cid:27) . However, in viewof decomposition (14) we have the following relations:(21) (cid:26) (cid:27) = (cid:20) (cid:21) , (cid:26) (cid:27) = (cid:20) (cid:21) , (cid:26) (cid:27) = (cid:20) (cid:21) , (cid:26) (cid:27) = (cid:20) (cid:21) , (cid:26) jij (cid:27) = (cid:20) jij (cid:21) , for i = 1 , , j = 6 , , . Remark 4.2. For the case of complex Stiefel manifolds we consider the decomposition(16) and metrics (17). Then the term B | h in Q is omitted and { ∗∗∗ } = 0 if there is number3 at any place ∗ .For the center h we need to compute the following numbers:(22) (cid:26) (cid:27) , (cid:26) (cid:27) , (cid:26) (cid:27) , (cid:26) (cid:27) , (cid:26) (cid:27) , (cid:26) (cid:27) . Let ˜ A ij = µA ij , ˜ B ij = µB ij be B -orthonormal vectors of SU( N ), for some real constant µ . Then the sets { ˜ A ij , ˜ B ij : i = ℓ + 1 , . . . , ℓ + m ; j = 1 , . . . , ℓ } , { ˜ A ij , ˜ B ij : i = ℓ + m +1 , . . . , N ; j = 1 , . . . , ℓ } and { ˜ A ij , ˜ B ij : i = ℓ + m +1 , . . . , N ; j = m +1 , . . . , m + ℓ } constituteorthonormal bases for n = m , n = m and n = m respectively. Lemma 4.3. Let N = ℓ + m + n . The numbers (22) are given as follows: (cid:26) (cid:27) = b ( ℓ + m ) N , (cid:26) (cid:27) = d ( ℓ + m ) N (cid:26) (cid:27) = a ℓℓ + m + b mnN ( ℓ + m ) + 2 ab √ ℓmn ( ℓ + m ) √ N (cid:26) (cid:27) = a mℓ + m + b ℓn ( ℓ + m ) − ab √ ℓmn ( ℓ + m ) √ N (23) instein metrics on SU( N ) and complex Stiefel manifolds 13 (cid:26) (cid:27) = c ℓℓ + m + d mn ( ℓ + m ) + 2 cd √ ℓmn ( ℓ + m ) √ N (cid:26) (cid:27) = c mℓ + m + d ℓn ( ℓ + m ) − cd √ ℓmn ( ℓ + m ) √ N . Proof. We will prove the first relation and the others can be calculted similarly. It is (cid:26) (cid:27) = X j,l Q ([ ˜ X (6) j , ˜ X (6) l ] , V ) = X (cid:16) aB ([ ˜ X (6) j , ˜ X (6) l ] , ˜ H ) + bB ([ ˜ X (6) j , ˜ X (6) l ] , ˜ H ) (cid:17) = X (cid:16) a B ([ ˜ X (6) j , ˜ X (6) l ] , ˜ H ) + b B ([ ˜ X (6) j , ˜ X (6) l ] , ˜ H ) +2 abB ([ ˜ X (6) j , ˜ X (6) l ] , ˜ H ) B ([ ˜ X (6) j , ˜ X (6) l ] , ˜ H ) (cid:17) = X (cid:16) a B ( ˜ X (6) j , [ ˜ H , ˜ X (6) l ]) + b B ( ˜ X (6) j , [ ˜ H , ˜ X (6) l ]) +2 abB ( ˜ X (6) j , [ ˜ H , ˜ X (6) l ]) B ( ˜ X (6) j , [ ˜ H , ˜ X (6) l ]) (cid:17) = X b B ( ˜ X (6) j , [ ˜ H , ˜ X (6) l ]) = X ℓ +1 ≤ i,k ≤ ℓ + m ≤ j,l ≤ ℓ b B ( ˜ A ij , [ ˜ H , ˜ A kl ]) + X ℓ +1 ≤ i,k ≤ ℓ + m ≤ j,l ≤ ℓ b B ( ˜ A ij , [ ˜ H , ˜ B kl ]) + X ℓ +1 ≤ i,k ≤ ℓ + m ≤ j,l ≤ ℓ b B ( ˜ B ij , [ ˜ H , ˜ A kl ]) + X ℓ +1 ≤ i,k ≤ ℓ + m ≤ j,l ≤ ℓ b B ( ˜ B ij , [ ˜ H , ˜ B kl ]) = b c ( 1 ℓ + 1 m ) · ℓm. In the second equation above we used (11) and Proposition 3.4, in the forth equationwe used the bi-invariance of the Killing form and in the fifth and seventh equations weused Lemma 3.5. By substituting c from (18) in the last equation we obtain the desiredexpression. (cid:3) Proposition 4.4. The components of the Ricci tensor of the left-invariant metric cor-responding to the scalar product (15) and of the SU( ℓ + m + n ) -invariant metrics corre-sponding to the scalar products (17) for the center h , are given as follows: r = v (cid:18) x (6)2 (cid:26) (cid:27) + 1 x (7)2 (cid:26) (cid:27) + 1 x (8)2 (cid:26) (cid:27)(cid:19) (24) r = v (cid:18) x (6)2 (cid:26) (cid:27) + 1 x (7)2 (cid:26) (cid:27) + 1 x (8)2 (cid:26) (cid:27)(cid:19) (25) r = √ v v ( bdx (6)2 ( ℓ + m )( ℓ + m + n )(26) + 1 x (7)2 ( ℓ + m ) ℓac + √ ℓmn p ( ℓ + m + n ) ( ad + cb ) + bdmn ( ℓ + m + n ) ! + 1 x (8)2 ( ℓ + m ) mac − √ ℓmn p ( ℓ + m + n ) ( ad + cb ) + bdnℓ p ( ℓ + m + n ) ! ) . Proof. We will work with the left-invariant metrics (15) on the Lie group SU( ℓ + m + n ).Let U ∈ ˜ h , where U = 1 √ v V . Then by using equation (1) we have r = r ( U , U ) = − X i =4 , n i X j =1 g ([ U , X ( i ) j ] , [ U , X ( i ) j ]) − g ([ U , U ] , [ U , U ])+ 12 B ( U , U ) + 14 X i,k dim n i X j =1 dim n k X l =1 g ([ X ( i ) j , X ( k ) l ] , U ) g ([ X ( i ) j , X ( k ) l ] , U )= − B ( U , U ) + 12 B ( U , U ) + 14 X i,k dim n i X j =1 dim n k X l =1 g ([ X ( i ) j , X ( k ) l ] , U ) = 14 X i,k dim n i X j =1 dim n k X l =1 g ([ X ( i ) j , X ( k ) l ] , U ) , where the first term in the second equality was obtained by Proposition 4.1. We willsimplify the last term in the above expression for r . It is14 X i,k X j,l g ([ X ( i ) j , X ( k ) l ] , U ) = 14 X i X j,l g ([ X ( i ) j , X ( i ) l ] , U ) + 14 X j,li = k g ([ X ( i ) j , X ( k ) l ] , U ) + 12 X i X l g ([ U , X ( i ) l ] , U ) + 12 X i X l g ([ U , X ( i ) l ] , U ) . By using the Lie bracket relations of Lemma 3.1 it follows that the last three terms in theabove sum are equal to zero. For the first term we have the following:If i = 1 , , X ( i ) j , X ( i ) l ] ⊂ n i and the term vanishes. If i = 6 , , X ( i ) j , X ( i ) l ] ⊂ n ⊕ · · · ⊕ n , so14 X i X j,l g ([ X ( i ) j , X ( i ) l ] , U ) = 14 X i X j,l hh [ X ( i ) j , X ( i ) l ] , U ii = 14 v X i =6 , , X j,l hh [ X ( i ) j , X ( i ) l ] , V ii = X i =6 , , v x ( i )2 X j,l h [ ˜ X ( i ) j , ˜ X ( i ) l ] , V i = v x (6)2 (cid:26) (cid:27) + v x (7)2 (cid:26) (cid:27) + v x (8)2 (cid:26) (cid:27) , instein metrics on SU( N ) and complex Stiefel manifolds 15 from which (24) follows. By similar computations we obtain (25).Now let U ∈ ˜ h and U ∈ ˜ h . Then r = r ( U , U ) = − X i =4 , n i X j =1 g ([ U , X ( i ) j ] , [ U , X ( i ) j ]) − g ([ U , U ] , [ U , U ]) − g ([ U , U ] , [ U , U ]) + 12 B ( U , U )+ 14 X i,k dim n i X j =1 dim n k X l =1 g ([ X ( i ) j , X ( k ) l ] , U ) g ([ X ( i ) j , X ( k ) l ] , U ) = − B ( U , U ) + 12 B ( U , U )+ 14 X i,k dim n i X j =1 dim n k X l =1 g ([ X ( i ) j , X ( k ) l ] , U ) g ([ X ( i ) j , X ( k ) l ] , U ) . We will simplify the last term in the above equation. We have:14 X i,k dim n i X j =1 dim n k X l =1 g ([ X ( i ) j , X ( k ) l ] , U ) g ([ X ( i ) j , X ( k ) l ] , U )= 14 X i dim n i X j,l =1 g ([ X ( i ) j , X ( i ) l ] , U ) g ([ X ( i ) j , X ( i ) l ] , U )+ 14 X i = k dim n i X j =1 dim n k X l =1 g ([ X ( i ) j , X ( k ) l ] , U ) g ([ X ( i ) j , X ( k ) l ] , U )+ 12 X i,l g ([ U , X ( i ) l ] , U ) g ([ U , X ( i ) l ] , U ) + 12 X i,l g ([ U , X ( i ) l ] , U ) g ([ U , X ( i ) l ] , U ) . By using the Lie brackets relations of Lemma 3.1 it is easy to see that the last three termsare equal to zero. For the first term we have:If i = 1 , , X ( i ) j , X ( i ) l ] ∈ n i , hence14 X j,l g ([ X ( i ) j , X ( i ) l ] , U ) g ([ X ( i ) j , X ( i ) l ] , U ) = 0 . If i = 6 , , X ( i ) j , X ( i ) l ] ⊂ n ⊕ · · · ⊕ n , and we introduce the following notations: f ( X, Y ) = aB ( X, [ ˜ H , Y )) + bB ( X, [ ˜ H , Y ])) + cB ( X, [ ˜ H , Y )) + dB ( X, [ ˜ H , Y ])), I = { ( i, j ) : i = ℓ + 1 , . . . , ℓ + m ; j = 1 , . . . , ℓ } , I = { ( i, j ) : i = ℓ + m + 1 , . . . , N ; j = 1 , . . . , ℓ } and I = { ( i, j ) : i = ℓ + m + 1 , . . . , N ; j = m + 1 , . . . , m + ℓ } .Then by using Proposition 3.4 we obtain that14 X i =6 , , n i X j,l =1 g ([ X ( i ) j , X ( i ) l ] , U ) g ([ X ( i ) j , X ( i ) l ] , U )= 14 X i =6 , , n i X j,l =1 hh [ X ( i ) j , X ( i ) l ] , U iihh [ X ( i ) j , X ( i ) l ] , U ii = √ v v X i =6 , , n i X j,l =1 { aB ([ X ( i ) j , X ( i ) l ] , ˜ H ) + bB ([ X ( i ) j , X ( i ) l ] , ˜ H ) }×{ cB ([ X ( i ) j , X ( i ) l ] , ˜ H ) + dB ([ X ( i ) j , X ( i ) l ] , ˜ H ) } = √ v v X i =6 , , n i X j,l =1 { aB ( X ( i ) j , [ ˜ H , X ( i ) l ]) + bB ( X ( i ) j , [ ˜ H , X ( i ) l ]) }×{ cB ( X ( i ) j , [ ˜ H , X ( i ) l ]) + dB ( X ( i ) j , [ ˜ H , X ( i ) l ]) } = √ v v × ( X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ A ij , ˜ A kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ A ij , ˜ B kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ B ij , ˜ A kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ B ij , ˜ B kl )+ X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ A ij , ˜ A kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ A ij , ˜ B kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ B ij , ˜ A kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ B ij , ˜ B kl )+ X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ A ij , ˜ A kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ A ij , ˜ B kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ B ij , ˜ A kl ) + X ( i,j ) ∈ I ( k,l ) ∈ I f ( ˜ B ij , ˜ B kl ) ) = √ v v (cid:26) bdx (6)2 ( ℓ + m )( ℓ + m + n ) + 1 x (7)2 ( ℓ + m ) (cid:18) ℓac + √ ℓmn p ( ℓ + m + n ) ( ad + cb )+ bdmn ( ℓ + m + n ) (cid:19) + 1 x (8)2 ( ℓ + m ) (cid:18) mac − √ ℓmn p ( ℓ + m + n ) ( ad + cb ) + bdnℓ p ( ℓ + m + n ) (cid:19)(cid:27) , where in the third equation we used the bi-invariance of the Killing form and in the fifthequation we used Lemma 3.5. Then equation (27) follows. Similar calculations apply forthe SU( ℓ + m + n )-invariant metrics (17) on the Stiefel manifolds SU( ℓ + m + n ) / SU( n ),where the terms for i = 3 in all sums above are omitted. (cid:3) The Ricci tensor for the diagonal parts of the scalar products (15), (17). We need the following variant of Lemma 2.1. Since the Ricci tensor for the metrics (15)and (17) is Ad( H )-invariant, by using Schur’s lemma and the Q -structure constants, wecan describe the Ricci components of the diagonal parts of these metrics. Lemma 4.5. The components of r , r , r , r , r , r of the Ricci tensor r for the metricscorresponding to the scalar products of the form (15) are given as follows: (27) r k = 12 y k + 14 d k X j,i y k y j y i (cid:26) kji (cid:27) − d k X j,i y j y k y i (cid:26) jki (cid:27) ( k = 1 , , , , , , where the sum is taken over i, j = 1 , . . . , and the variables y i denote correspondingvariables u i , v i , x ( i ) of the metric (15) . By using relations (21) we obtain the following: instein metrics on SU( N ) and complex Stiefel manifolds 17 Proposition 4.6. The components of the Ricci tensor for the diagonal part of the left-invariant metrics corresponding to the scalar products (15) are given as follows: r = 12 u − d u (cid:18)(cid:20) (cid:21) + (cid:20) (cid:21) + (cid:20) (cid:21)(cid:19) + 14 d (cid:18) u (cid:20) (cid:21) + u x (6)2 (cid:20) (cid:21) + u x (7)2 (cid:20) (cid:21)(cid:19) ,r = 12 u − d u (cid:18)(cid:20) (cid:21) + (cid:20) (cid:21) + (cid:20) (cid:21)(cid:19) + 14 d (cid:18) u (cid:20) (cid:21) + u x (6)2 (cid:20) (cid:21) + u x (8)2 (cid:20) (cid:21)(cid:19) ,r = 12 u − d u (cid:18)(cid:20) (cid:21) + (cid:20) (cid:21) + (cid:20) (cid:21)(cid:19) + 14 d (cid:18) u (cid:20) (cid:21) + u x (7)2 (cid:20) (cid:21) + u x (8)2 (cid:20) (cid:21)(cid:19) ,r = 12 x (6) − d x (6)2 (cid:18) u (cid:20) (cid:21) + u (cid:20) (cid:21) + v (cid:26) (cid:27) + v (cid:26) (cid:27)(cid:19) + 12 d (cid:20) (cid:21) (cid:18) x (6) x (7) x (8) − x (7) x (6) x (8) − x (8) x (6) x (7) (cid:19) ,r = 12 x (7) − d x (7)2 (cid:18) u (cid:20) (cid:21) + u (cid:20) (cid:21) + v (cid:26) (cid:27) + v (cid:26) (cid:27)(cid:19) + 12 d (cid:20) (cid:21) (cid:18) x (7) x (6) x (8) − x (6) x (7) x (8) − x (8) x (6) x (7) (cid:19) ,r = 12 x (8) − d x (8)2 (cid:18) u (cid:20) (cid:21) + u (cid:20) (cid:21) + v (cid:26) (cid:27) + v (cid:26) (cid:27)(cid:19) + 12 d (cid:20) (cid:21) (cid:18) x (8) x (6) x (7) − x (6) x (7) x (8) − x (7) x (6) x (8) (cid:19) . For the SU( ℓ + m + n ) -invariant metrics corresponding to the scalar products (17) , thereis no r component and the components r , r simplify by using that (cid:2) ∗∗ (cid:3) = 0 .Proof. It follows from Lemma 4.5 and relations (21). (cid:3) By substituting the values of the numbers (cid:20) ijk (cid:21) , (cid:26) ijk (cid:27) from Lemmas 3.3 and 4.3 re-spectively, to the Ricci components in Propositions 4.4 and 4.6 we finally obtain: Proposition 4.7. The components of the Ricci tensor for the diagonal part of the leftinvariant metric (15) on SU( ℓ + m + n ) are given as follows: r = ℓ N u + u N (cid:18) mx (6)2 + nx (7)2 (cid:19) , r = m N u + u N (cid:18) ℓx (6)2 + nx (8)2 (cid:19) ,r = n N u + u N (cid:18) ℓx (7)2 + mx (8)2 (cid:19) ,r = 12 x (6) + n N (cid:18) x (6) x (7) x (8) − x (7) x (6) x (8) − x (8) x (6) x (7) (cid:19) − ℓmN x (6)2 (cid:0) ( ℓ − mu + ( m − ℓu + ( ℓ + m ) b v + ( ℓ + m ) d v (cid:1) , r = 12 x (7) + m N (cid:18) x (7) x (6) x (8) − x (6) x (7) x (8) − x (8) x (6) x (7) (cid:19) − ℓnN x (7)2 (cid:18) ( ℓ − nu + ( n − ℓu + (cid:18) a ℓNℓ + m + b mnℓ + m + 2 ab √ ℓmn √ Nℓ + m (cid:19) v + (cid:18) c ℓNℓ + m + d mnℓ + m + 2 cd √ ℓmn √ Nℓ + m (cid:19) v (cid:19) ,r = 12 x (8) + ℓ N (cid:18) x (8) x (6) x (7) − x (7) x (6) x (8) − x (6) x (7) x (8) (cid:19) − mnN x (8)2 (cid:18) ( m − nu + ( n − mu + (cid:18) a mNℓ + m + b ℓnℓ + m − ab √ ℓmn √ Nℓ + m (cid:19) v + (cid:18) c mNℓ + m + d ℓnℓ + m − cd √ ℓmn √ Nℓ + m (cid:19) v (cid:19) . Proposition 4.8. The components of the Ricci tensor for the diagonal part of the SU( ℓ + m + n ) -invariant metric (17) on SU( ℓ + m + n ) / SU( n ) are given as follows: r = ℓ N u + u N (cid:18) mx (6)2 + nx (7)2 (cid:19) , r = m N u + u N (cid:18) ℓx (6)2 + nx (8)2 (cid:19) ,r = 12 x (6) + n N (cid:18) x (6) x (7) x (8) − x (7) x (6) x (8) − x (8) x (6) x (7) (cid:19) − ℓmN x (6)2 (cid:0) ( ℓ − mu + ( m − ℓu + ( ℓ + m ) b v + ( ℓ + m ) d v (cid:1) ,r = 12 x (7) + m N (cid:18) x (7) x (6) x (8) − x (6) x (7) x (8) − x (8) x (6) x (7) (cid:19) − ℓnN x (7)2 (cid:18) ( ℓ − nu + (cid:18) a ℓNℓ + m + b mnℓ + m + 2 ab √ ℓmn √ Nℓ + m (cid:19) v + (cid:18) c ℓNℓ + m + d mnℓ + m + 2 cd √ ℓmn √ Nℓ + m (cid:19) v (cid:19) ,r = 12 x (8) + ℓ N (cid:18) x (8) x (6) x (7) − x (7) x (6) x (8) − x (6) x (7) x (8) (cid:19) − mnN x (8)2 (cid:18) ( m − nu + (cid:18) a mNℓ + m + b ℓnℓ + m − ab √ ℓmn √ Nℓ + m (cid:19) v + (cid:18) c mNℓ + m + d ℓnℓ + m − cd √ ℓmn √ Nℓ + m (cid:19) v (cid:19) . Non naturally reductive Einstein metrics on the compact Lie group SU( ℓ + m + n )5.1. Naturally reductive metrics on SU( ℓ + m + n ) . A Riemannian homogeneousspace ( M = G/H, g ) with reductive complement m of h in g is called naturally reductive if h [ X, Y ] m , Z i + h Y, [ X, Z ] m i = 0 for all X, Y, Z ∈ m . Here h , i denotes the inner product on m induced from the Riemannian metric g . Classicalexamples of naturally reductive homogeneous spaces include irreducible symmetric spaces,isotropy irreducible homogeneous manifolds, and Lie groups with bi-invariant metrics. Ingeneral it is not always easy to decide if a given homogeneous Riemannian manifold is instein metrics on SU( N ) and complex Stiefel manifolds 19 naturally reductive, since one has to consider all possible transitive actions of subgroups G of the isometry group of ( M, g ).In [D’AZi] D’Atri and Ziller investigated naturally reductive metrics among left-invariantmetrics on compact Lie groups and gave a complete classification in the case of simpleLie groups. Let G be a compact, connected semisimple Lie group, L a closed subgroup of G and let g be the Lie algebra of G and l the subalgebra corresponding to L .Recall that B is the negative of the Killing form of g , so B is an Ad( G )-invariant innerproduct on g . Let m be an orthogonal complement of l with respect to B . Then we have g = l ⊕ m , Ad( L ) m ⊂ m . Let l = l ⊕ l ⊕ · · · ⊕ l p be a decomposition of l into ideals, where l is the center of l and l i ( i = 1 , . . . , p ) are simple ideals of l . Let A | l be an arbitrary metric on l . Theorem 5.1. ([D’AZi, Theorem 1, p. 9]) Under the notations above a left-invariantmetric on G of the form (28) h , i = x · B | m + A | l + u · B | l + · · · + u p · B | l p , ( x, u , . . . , u p > is naturally reductive with respect to G × L , where G × L acts on G by ( g, l ) y = gyl − .Moreover, if a left-invariant metric h , i on a compact simple Lie group G is naturallyreductive, then there is a closed subgroup L of G and the metric h , i is given by the form (28) . For the Lie group SU( ℓ + m + n ) we consider left-invariant metrics determined byAd(S(U( ℓ ) × U( m ) × U( n )))-invariant scalar products (10). Proposition 5.2. If a left invariant metric of the form (10) on SU( ℓ + m + n ) is naturallyreductive with respect to SU( ℓ + m + n ) × L , for some closed subgroup L of SU( ℓ + m + n ) ,then one of the following holds: (1) the metric (10) is either (i) Ad(S(U( ℓ + m ) × U( n ))) -invariant and x (7) = x (8) , or (ii) Ad(S(U( ℓ ) × U( m + n ))) -invariant and x (6) = x (7) , or (iii) Ad(S(U( ℓ + n ) × U( m ))) -invariant and x (6) = x (8) . (2) x (6) = x (7) = x (8) .Conversely, if one of the conditions (1) , (2) is satisfied, then the metric of the form (10) is naturally reductive with respect to SU( ℓ + m + n ) × L , for some closed subgroup L of SU( ℓ + m + n ) .Proof. Let l be the Lie algebra of L . Then we have that either l ⊂ h = s ( u ( ℓ ) ⊕ u ( m ) ⊕ u ( n ))or l h . First we consider the case of l h . Let k be the subalgebra of g generatedby l and h . Since su ( ℓ + m + n ) splits into h and three Ad( H )-irreducible modules m , m , m as g = h ⊕ m = h ⊕ h ⊕ h ⊕ h ⊕ m ⊕ m ⊕ m , where the Ad( H )-irreducible modules m , m , m are mutually non equivalent, we see that the Lie algebra k contains at least one of m , m , m . Let us assume that k contains m . Note that[ m , m ] ⊂ h ⊕ h ⊕ h and thus k contains su ( ℓ + m ). Thus we see that k containsthe Lie subalgebra s ( u ( ℓ + m ) ⊕ u ( n )). If k = s ( u ( ℓ + m ) ⊕ u ( n )), then we obtain an irreducible decomposition su ( ℓ + m + n ) = k ⊕ n , where n = m ⊕ m . Hence, a naturallyreductive left invariant metric of the form (10) is Ad(S(U( ℓ + m ) × U( n )))-invariant and x (7) = x (8) , so we obtain case (i). Cases (ii) and (iii) are obtained by a similar way.Furthermore, if k = s ( u ( ℓ + m ) ⊕ u ( n )), then k contains m or m . In this case we cansee that k = su ( ℓ + m + n ). Thus the metric is bi-invariant.Now we consider the case l ⊂ h . Since the orthogonal complement l ⊥ of l with respectto B contains the orthogonal complement h ⊥ of h , we see that l ⊥ ⊃ m ⊕ m ⊕ m . Sincethe left invariant metric of the form (10) is naturally reductive with respect to G × L , itfollows that x (6) = x (7) = x (8) by Theorem 5.1. The converse is a direct consequence ofTheorem 5.1. (cid:3) Non naturally reductive Einstein metrics on SU( ℓ + m + n ) . To find nonnaturally reductive Einstein metrics on SU( ℓ + m + n ) we need to solve the system (cf.Propositions 4.4, 4.7) r = 0 , r − r = 0 , r − r = 0 , r − r = 0 ,r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 . (29)We claim that it is possible to choose a basis { V ′ , V ′ } of h so that the matrix of thescalar product (11) with respect to { H , H } is given by(30) t (cid:18) γ (cid:19) (cid:18) v ′ v ′ (cid:19) (cid:18) γ (cid:19) , for some real number γ and v ′ , v ′ > 0. Hence, without loss of generality we may choose a = d = 1 and b = 0.Indeed, by using the QR-decomposition we obtain that (cid:18) a bc d (cid:19) = (cid:18) cos t − sin t sin t cos t (cid:19) (cid:18) x y (cid:19) (cid:18) γ (cid:19) , for some real numbers x, y, γ , with x, y non zero. So the matrix A (cf. (13)) takes theform t (cid:18) γ (cid:19) (cid:18) x y (cid:19) (cid:18) cos t sin t − sin t cos t (cid:19) (cid:18) v v (cid:19) (cid:18) cos t − sin t sin t cos t (cid:19) (cid:18) x y (cid:19) (cid:18) γ (cid:19) . By changing the orthonormal basis { V , V } into some orthonormal basis { V ′ , V ′ } , thematrix A can be expressed as t (cid:18) γ (cid:19) (cid:18) x y (cid:19) (cid:18) v v (cid:19) (cid:18) x y (cid:19) (cid:18) γ (cid:19) , which gives expression (30).We also assume that ℓ = 1 , m = 2. In this case it is h = 0, so for SU(3 + n ) system(29) reduces to r = 0 , r − r = 0 , r − r = 0 ,r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 . (31) instein metrics on SU( N ) and complex Stiefel manifolds 21 Notice that in the above system there is no u variable. By setting x (7) = 1 in the equation r = 0 we obtain that c = p n (3 + n )(3 + n ) (1 − x (8)2 )(2 + x (8)2 ) . We substitute c into the system (31) and interestingly, we observe that the equations r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 , that is, nv x (6)2 x (8)4 + 3 v x (6)2 x (8)4 − v x (8)4 + 4 nv x (6)2 x (8)2 + 12 u x (6)2 x (8)2 − nv x (6)2 x (8)2 − v x (8)2 + 4 nv x (6)2 + 12 v x (6)2 = 0 , nx (6) x (8)4 + 3 u x (8)3 + 9 v x (8)3 − nx (6) x (8)3 − x (6) x (8)3 − nx (6)3 x (8)2 + 6 nx (6) x (8)2 +6 nv x (6)2 x (8) + 6 u x (8) + 18 v x (8) − nx (6) x (8) − x (6) x (8) − nx (6)3 + 4 nx (6) = 0 , − n x (6) x (8)7 − nx (6) x (8)7 − nu x (8)6 − nv x (8)6 + 12 n x (6) x (8)6 + 36 nx (6) x (8)6 +6 n x (6)3 x (8)5 + 6 nx (6)3 x (8)5 − n x (6)2 x (8)5 − nx (6)2 x (8)5 − n x (6) x (8)5 − nx (6) x (8)5 +9 nu x (6)2 x (8)4 + 6 n u x (6)2 x (8)4 − u x (6)2 x (8)4 + 2 nv x (6)2 x (8)4 + 6 u x (6)2 x (8)4 +9 nu x (6)2 x (8)4 − nu x (8)4 − nv x (8)4 + 48 n x (6) x (8)4 + 144 nx (6) x (8)4 + 24 n x (6)3 x (8)3 +24 nx (6)3 x (8)3 − n x (6)2 x (8)3 − nx (6)2 x (8)3 − n x (6) x (8)3 + 36 nu x (6)2 x (8)2 +24 n u x (6)2 x (8)2 − u x (6)2 x (8)2 + 8 nv x (6)2 x (8)2 + 24 v x (6)2 x (8)2 − nu x (8)2 − nv x (8)2 + 48 n x (6) x (8)2 + 144 nx (6) x (8)2 + 24 n x (6)3 x (8) + 24 nx (6)3 x (8) − n x (6)2 x (8) − nx (6)2 x (8) − n x (6) x (8) + 24 nx (6) x (8) + 36 nu x (6)2 + 24 n u x (6)2 − u x (6)2 +8 nv x (6)2 + 24 v x (6)2 = 0 , nx (8)7 − n x (6) x (8)6 − nx (6) x (8)6 + 6 n u x (6) x (8)6 − u x (6) x (8)6 + 2 nu x (6) x (8)6 +6 v x (6) x (8)6 + 6 nx (6)2 x (8)5 + 54 nx (8)5 + 12 n x (6) x (8)5 + 36 nx (6) x (8)5 − n x (6) x (8)4 − nx (6) x (8)4 − nu x (6) x (8)4 + 18 n u x (6) x (8)4 − u x (6) x (8)4 + 6 nv x (6) x (8)4 +18 v x (6) x (8)4 − nv x (6) x (8)4 + 24 nx (6)2 x (8)3 + 48 n x (6) x (8)3 + 144 nx (6) x (8)3 − n x (6) x (8)2 − nx (6) x (8)2 − nu x (6) x (8)2 + 36 nv x (6) x (8)2 + 24 nx (6)2 x (8) − nx (8) + 48 n x (6) x (8) +144 nx (6) x (8) − nu x (6) − n u x (6) + 24 u x (6) − nv x (6) − v x (6) = 0 , are linear with respect to u , u , v and v . By solving the above equations with respectto u , u , v and v we obtain u = − / (3 x (8) (cid:0) − nx (6)4 + nx (8)2 x (6)2 − x (8)2 x (6)2 − nx (6)2 − x (6)2 + x (8)4 + x (8)2 − (cid:1) ) × (cid:0) x (6) (2 nx (8)6 + 3 x (8)6 − nx (8)5 − x (8)5 + 2 n x (6)2 x (8)4 + nx (6)2 x (8)4 − x (6)2 x (8)4 + 4 nx (8)4 +6 nx (6) x (8)4 + 18 x (6) x (8)4 + 9 x (8)4 − n x (6)2 x (8)3 − nx (6)2 x (8)3 − x (6)2 x (8)3 − nx (8)3 − nx (6) x (8)3 − x (6) x (8)3 − x (8)3 − n x (6)4 x (8)2 − nx (6)4 x (8)2 + 4 n x (6)3 x (8)2 +12 nx (6)3 x (8)2 − nx (6)2 x (8)2 − nx (8)2 + 12 nx (6) x (8)2 + 36 x (6) x (8)2 − n x (6)3 x (8) − nx (6)3 x (8) + 4 n x (6)2 x (8) + 16 nx (6)2 x (8) + 12 x (6)2 x (8) + 8 nx (8) − nx (6) x (8) − x (6) x (8) +24 x (8) + 2 n x (6)4 + 6 nx (6)4 − n x (6)2 − nx (6)2 + 12 x (6)2 − n − (cid:1) ,u = − / (cid:0) ( n − n + 1) x (6) x (8) ( nx (6)4 − nx (8)2 x (6)2 + x (8)2 x (6)2 + nx (6)2 + 4 x (6)2 − x (8)4 − x (8)2 + 2) (cid:1) × (cid:0) n x (6)6 + 3 n x (6)6 − n x (8) x (6)5 − n x (8) x (6)5 − n x (6)4 − n x (6)4 − n x (8)2 x (6)4 + n x (8)2 x (6)4 + 4 nx (8)2 x (6)4 + 6 nx (6)4 + 2 n x (8) x (6)4 + 8 n x (8) x (6)4 +6 nx (8) x (6)4 + 2 n x (8)3 x (6)3 + 4 n x (8)3 x (6)3 − nx (8)3 x (6)3 − n x (8)2 x (6)3 − n x (8)2 x (6)3 − nx (8)2 x (6)3 − n x (8) x (6)3 − nx (8) x (6)3 − n x (8)4 x (6)2 − nx (8)4 x (6)2 + x (8)4 x (6)2 +2 n x (8)3 x (6)2 + 8 nx (8)3 x (6)2 + 6 x (8)3 x (6)2 − n x (6)2 + 2 nx (8)2 x (6)2 − x (8)2 x (6)2 − nx (6)2 + 4 n x (8) x (6)2 + 12 nx (8) x (6)2 + 2 n x (8)5 x (6) + 6 nx (8)5 x (6) − n x (8)4 x (6) − nx (8)4 x (6) − x (8)4 x (6) + 4 n x (8)3 x (6) + 14 nx (8)3 x (6) + 6 x (8)3 x (6) − n x (8)2 x (6) − nx (8)2 x (6) − nx (8)6 − x (8)6 − nx (8)4 − x (8)4 + 6 nx (8)2 + 2 x (8)2 (cid:1) ,v = 1 / (cid:0) ( n + 3) x (6) ( x (8)2 + 2)( − nx (6)4 + nx (8)2 x (6)2 − x (8)2 x (6)2 − nx (6)2 − x (6)2 + x (8)4 + x (8)2 − (cid:1) × x (8) ( nx (6)2 + x (8)2 + 2)( − nx (6)4 + nx (8)2 x (6)2 − x (8)2 x (6)2 + nx (6)2 − nx (8) x (6)2 − x (8) x (6)2 + 2 x (6)2 + 2 nx (8)2 x (6) + 6 x (8)2 x (6) − nx (8) x (6) − x (8) x (6) + x (8)4 + x (8)2 − ,v = 1 / (cid:0) x (8) (cid:0) nx (6)4 − nx (8)2 x (6)2 + x (8)2 x (6)2 + nx (6)2 + 4 x (6)2 − x (8)4 − x (8)2 + 2 (cid:1)(cid:1) × x (6) ( x (8)2 + 2)( nx (6)4 − nx (8)2 x (6)2 + x (8)2 x (6)2 − nx (6)2 + 2 nx (8) x (6)2 + 6 x (8) x (6)2 − x (6)2 − nx (8)2 x (6) − x (8)2 x (6) + 2 nx (8) x (6) + 6 x (8) x (6) − x (8)4 − x (8)2 + 2) . We substitute the above expressions for u , u , v and v into the equations r − r = 0 ⇔− u u x (6)2 − u u x (8)2 x (6)2 − n u x (8)2 x (6)2 + 2 u x (8)2 x (6)2 + n u u x (6)2 + u u x (8)2 = 0 ,r − r = 0 ⇔ x (8)2 u + 6 u − nv x (8)2 u − v x (8)2 u − nv u − v u + 3 nx (8)2 = 0and obtain two equations F ( x (6) , x (8) ) = 0 and F ( x (6) , x (8) ) = 0 with parameter n .It is possible to pursue computations for any value of n . However, we will restrictourselves to the case n = 2, not only due to space limitations, but also because thiscorresponds to the Lie group SU(5) (the special unitary group of lowest rank known upto now to admit a non naturally reductive Einstein metric). In this case c = − ( x (8)2 − ) √ ( x (8)2 +2 )and the substitution of u , u , v , v into the equations r − r = 0 and r − r = 0 givesthe equations − x (6) − x (6) F ( x (6) , x (8) ) = 0 and 10( x (8) − x (6) ) F ( x (6) , x (8) ) = 0, where F ( x (6) , x (8) ) = ( − x (8)19 + 273 x (6) x (8)18 + 273 x (8)18 − x (6)2 x (8)17 − x (6) x (8)17 − x (8)17 + 791 tx (6)3 x (8)16 + 2201 x (6)2 x (8)16 + 2201 x (6) x (8)16 + 791 x (8)16 + 42 x (6)4 x (8)15 − x (6)3 x (8)15 + 1092 x (6)2 x (8)15 − x (6) x (8)15 + 42 x (8)15 − x (6)5 x (8)14 +3397 x (6)4 x (8)14 − x (6)3 x (8)14 − x (6)2 x (8)14 + 3397 x (6) x (8)14 − x (8)14 +1330 x (6)6 x (8)13 + 1044 x (6)5 x (8)13 + 17190 x (6)4 x (8)13 + 29176 x (6)3 x (8)13 + 17190 x (6)2 x (8)13 +1044 x (6) x (8)13 + 1330 x (8)13 − x (6)7 x (8)12 − x (6)6 x (8)12 − x (6)5 x (8)12 − x (6)4 x (8)12 − x (6)3 x (8)12 − x (6)2 x (8)12 − x (6) x (8)12 − x (8)12 +336 x (6)8 x (8)11 + 15158 x (6)7 x (8)11 + 41888 x (6)6 x (8)11 + 192250 x (6)5 x (8)11 +144528 x (6)4 x (8)11 + 192250 x (6)3 x (8)11 + 41888 x (6)2 x (8)11 + 15158 x (6) x (8)11 + 336 x (8)11 +1946 x (6)9 x (8)10 − x (6)8 x (8)10 − x (6)7 x (8)10 − x (6)6 x (8)10 − x (6)5 x (8)10 − x (6)4 x (8)10 − x (6)3 x (8)10 − x (6)2 x (8)10 − x (6) x (8)10 + 1946 x (8)10 − x (6)10 x (8)9 − x (6)9 x (8)9 + 15456 x (6)8 x (8)9 +238584 x (6)7 x (8)9 + 373104 x (6)6 x (8)9 + 490608 x (6)5 x (8)9 + 373104 x (6)4 x (8)9 +238584 x (6)3 x (8)9 + 15456 x (6)2 x (8)9 − x (6) x (8)9 − x (8)9 + 3108 x (6)11 x (8)8 instein metrics on SU( N ) and complex Stiefel manifolds 23 +4988 x (6)10 x (8)8 + 42868 x (6)9 x (8)8 − x (6)8 x (8)8 − x (6)7 x (8)8 − x (6)6 x (8)8 − x (6)5 x (8)8 − x (6)4 x (8)8 − x (6)3 x (8)8 + 42868 x (6)2 x (8)8 + 4988 x (6) x (8)8 +3108 x (8)8 − x (6)12 x (8)7 − x (6)11 x (8)7 − x (6)10 x (8)7 − x (6)9 x (8)7 +347248 x (6)8 x (8)7 + 186248 x (6)7 x (8)7 + 747408 x (6)6 x (8)7 + 186248 x (6)5 x (8)7 +347248 x (6)4 x (8)7 − x (6)3 x (8)7 − x (6)2 x (8)7 − x (6) x (8)7 − x (8)7 − x (6)13 x (8)6 + 6280 x (6)12 x (8)6 + 46656 x (6)11 x (8)6 + 78176 x (6)10 x (8)6 − x (6)9 x (8)6 − x (6)8 x (8)6 − x (6)7 x (8)6 − x (6)6 x (8)6 − x (6)5 x (8)6 − x (6)4 x (8)6 + 78176 x (6)3 x (8)6 + 46656 x (6)2 x (8)6 + 6280 x (6) x (8)6 − x (8)6 + 672 x (6)14 x (8)5 + 5824 x (6)13 x (8)5 − x (6)12 x (8)5 − x (6)11 x (8)5 +79968 x (6)10 x (8)5 − x (6)9 x (8)5 + 290720 x (6)8 x (8)5 + 173280 x (6)7 x (8)5 +290720 x (6)6 x (8)5 − x (6)5 x (8)5 + 79968 x (6)4 x (8)5 − x (6)3 x (8)5 − x (6)2 x (8)5 +5824 x (6) x (8)5 + 672 x (8)5 + 224 x (6)15 x (8)4 − x (6)14 x (8)4 + 320 x (6)13 x (8)4 +114880 x (6)12 x (8)4 − x (6)11 x (8)4 + 156576 x (6)10 x (8)4 − x (6)9 x (8)4 − x (6)8 x (8)4 − x (6)7 x (8)4 − x (6)6 x (8)4 + 156576 x (6)5 x (8)4 − x (6)4 x (8)4 +114880 x (6)3 x (8)4 + 320 x (6)2 x (8)4 − x (6) x (8)4 + 224 x (8)4 + 8640 x (6)14 x (8)3 − x (6)13 x (8)3 − x (6)12 x (8)3 + 42560 x (6)11 x (8)3 − x (6)10 x (8)3 +412160 x (6)9 x (8)3 − x (6)8 x (8)3 + 412160 x (6)7 x (8)3 − x (6)6 x (8)3 +42560 x (6)5 x (8)3 − x (6)4 x (8)3 − x (6)3 x (8)3 + 8640 x (6)2 x (8)3 − x (6)15 x (8)2 +7040 x (6)14 x (8)2 + 46720 x (6)13 x (8)2 − x (6)12 x (8)2 + 136960 x (6)11 x (8)2 − x (6)10 x (8)2 + 21760 x (6)9 x (8)2 + 21760 x (6)8 x (8)2 − x (6)7 x (8)2 +136960 x (6)6 x (8)2 − x (6)5 x (8)2 + 46720 x (6)4 x (8)2 + 7040 x (6)3 x (8)2 − x (6)2 x (8)2 − x (6)14 x (8) + 9600 x (6)13 x (8) − x (6)12 x (8) + 38400 x (6)10 x (8) − x (6)9 x (8) +38400 x (6)8 x (8) − x (6)6 x (8) + 9600 x (6)5 x (8) − x (6)4 x (8) + 3200 x (6)15 +3200 x (6)14 + 3200 x (6)13 + 3200 x (6)12 − x (6)11 − x (6)10 − x (6)9 − x (6)8 +3200 x (6)7 + 3200 x (6)6 + 3200 x (6)5 + 3200 x (6)4 ) and F ( x (6) , x (8) ) = ( − x (6)11 x (8)2 − x (6)11 + 144 x (6)10 x (8)3 + 240 x (6)10 x (8) − x (6)9 x (8)4 − x (6)9 x (8)3 − x (6)9 x (8)2 − x (6)9 x (8) − x (6)8 x (8)5 + 720 x (6)8 x (8)4 − x (6)8 x (8)3 + 960 x (6)8 x (8)2 + 480 x (6)8 x (8) + 156 x (6)7 x (8)6 − x (6)7 x (8)5 − x (6)7 x (8)4 − x (6)7 x (8)3 − x (6)7 x (8)2 − x (6)7 x (8) + 160 x (6)7 − x (6)6 x (8)7 + 76 x (6)6 x (8)5 +1280 x (6)6 x (8)4 − x (6)6 x (8)3 + 2080 x (6)6 x (8)2 − x (6)6 x (8) + 49 x (6)5 x (8)8 +324 x (6)5 x (8)7 − x (6)5 x (8)6 + 448 x (6)5 x (8)5 − x (6)5 x (8)4 + 368 x (6)5 x (8)3 − x (6)5 x (8)2 + 480 x (6)5 x (8) + 51 x (6)4 x (8)9 − x (6)4 x (8)8 + 866 x (6)4 x (8)7 − x (6)4 x (8)6 + 1708 x (6)4 x (8)5 − x (6)4 x (8)4 + 240 x (6)4 x (8)3 + 960 x (6)4 x (8)2 − x (6)4 x (8) − x (6)3 x (8)10 + 324 x (6)3 x (8)9 − x (6)3 x (8)8 + 880 x (6)3 x (8)7 − x (6)3 x (8)6 + 448 x (6)3 x (8)5 − x (6)3 x (8)4 + 128 x (6)3 x (8)3 − x (6)3 x (8)2 +320 x (6)3 x (8) − x (6)3 + 50 x (6)2 x (8)11 − x (6)2 x (8)10 + 450 x (6)2 x (8)9 − x (6)2 x (8)8 +1360 x (6)2 x (8)7 − x (6)2 x (8)6 + 1700 x (6)2 x (8)5 − x (6)2 x (8)4 + 720 x (6)2 x (8)3 − x (6)2 x (8) − x (6) x (8)12 + 72 x (6) x (8)11 − x (6) x (8)10 + 284 x (6) x (8)9 − x (6) x (8)8 +204 x (6) x (8)7 + 66 x (6) x (8)6 − x (6) x (8)5 + 396 x (6) x (8)4 − x (6) x (8)3 + 56 x (6) x (8)2 +7 x (8)13 + 28 x (8)11 + 7 x (8)9 − x (8)7 − x (8)5 + 56 x (8)3 ) . Actually, the computer outputs are − x (6) − x (6) F ( x (6) , x (8) ) /A and 10( x (8) − x (6) ) F ( x (6) , x (8) ) /B , where A = 27 x (8)3 (2 x (6)4 − x (6)2 x (8)2 + 6 x (6)2 − x (8)4 − x (8)2 + 2) and B = 3 x (6)2 x (8)2 ( − x (6)4 + x (6)2 x (8)2 − x (6)2 + x (8)4 + x (8)2 − , but we omit A and B since these are non zero.If either x (8) − x (6) = 0 or x (6) = 1 then the solutions obtained correspond to naturallyreductive Einstein metrics.Next, we solve the equations F = 0 , F = 0. We consider the polynomial ring R = Q [ z, x (6) , x (8) ] and the ideal I generated by the polynomials { z x (6) x (8) − , F , F } . Wetake a lexicographic ordering > with z > x (6) > x (8) for a monomial ordering on R . Then,by the aid of computer, we see that a Gr¨obner basis for the ideal I contains a polynomialof x (8) given by( x (8) − (cid:0) x (8)2 + 4 x (8) + 8 (cid:1) (cid:0) x (8)2 + 2 (cid:1) (cid:0) x (8)2 + 5 (cid:1) g ( x (8) ) , where g ( x (8) ) = 806688936348626758637763 x (8)50 − x (8)49 +122600899294485079451070969 x (8)48 − x (8)47 +4399773837901125736682019222 x (8)46 − x (8)45 +77189030664419243971210900356 x (8)44 − x (8)43 +835652804996925618391403544816 x (8)42 − x (8)41 +6267932916334423480011027655836 x (8)40 − x (8)39 +34849018083414433690580776827648 x (8)38 − x (8)37 +149775187669204629807886224706848 x (8)36 − x (8)35 +510957262333041322777745962488048 x (8)34 − x (8)33 +1406785689907755973691204239043056 x (8)32 − x (8)31 +3156573099492698926305125730681312 x (8)30 − x (8)29 +5800052119705927583296215652661568 x (8)28 − x (8)27 +8735998247540128939755053826847872 x (8)26 − x (8)25 +10765377064651986382470072553346112 x (8)24 − x (8)23 +10810407449543829554208165804540672 x (8)22 − x (8)21 +8799062122173126538327519312214016 x (8)20 − x (8)19 instein metrics on SU( N ) and complex Stiefel manifolds 25 +5769126123281919878491583853674496 x (8)18 − x (8)17 +3024464692354751261405789995008000 x (8)16 − x (8)15 +1255570192626380637861179751923712 x (8)14 − x (8)13 +407056027405833286440301394657280 x (8)12 − x (8)11 +100903380382253021546263923916800 x (8)10 − x (8)9 +18483164273291582549151186944000 x (8)8 − x (8)7 +2358038540179746141860003840000 x (8)6 − x (8)5 +187291856473914145872281600000 x (8)4 − x (8)3 +7244299545842737479680000000 x (8)2 − x (8) +59960536211694551040000000 . We consider an ideal J generated by the polynomials { F , F , g } . We take a lexicographicordering > with x (6) > x (8) for a monomial ordering on R . Then, by the aid of computer,we see that a Gr¨obner basis for the ideal J contains the polynomial g of x (8) and apolynomial h of x (6) and x (8) of the form h ( x (6) , x (8) ) = ax (6) + X k =0 b k x (8) k , where a ∈ R and b i ∈ R ( i = 0 , , . . . , g = 0 and h = 0 approximately,we obtain the following results:(1) ( x (6) , x (8) ) ≈ (1 . , . x (6) , x (8) ) ≈ (0 . , . u , u , v , v , we obtain the following: For the solutions(1) we have( u , u , v , v ) ≈ (0 . , . , . , . u , u , v , v ) ≈ (0 . , . , . , . . From the above computations we obtain the following: Theorem 5.3. The compact Lie group SU(5) admits two non naturally reductive Einsteinmetrics which correspond to Ad(S(U(1) × U(2) × U(2))) -invariant inner products of theform (15) . It is possible to show that the compact Lie group SU( n + 3) admits two left-invariantnon naturally reductive Einstein metrics, which correspond to Ad(S(U(1) × U(2) × U( n )))-invariant inner products of the form (15), for 2 ≤ n ≤ 12. Also, we conjecture that for n ≥ 13, SU( n + 3) admits four left-invariant non naturally reductive Einstein metrics.In this case the difficulty is to find, for general n , a Gr¨obner basis for the system ofpolynomials. A generalization of Mori’s result. Now we consider the cases when ℓ = m = 2, n ≥ c = 0, so that x (7) = x (8) = 1. In [Mo] K. Mori proved existence of one Einsteinmetric on SU(4 + n ), which corresponds to Ad(S(U(2) × U(2) × U( n )))-invariant innerproducts of the form (15). We generalize this result as follows: Theorem 5.4. The compact Lie group SU(4 + n ) admits two non naturally reductiveEinstein metrics for ≤ n ≤ and four non naturally reductive Einstein metrics for n ≥ , which correspond to Ad(S(U(2) × U(2) × U( n ))) -invariant inner products of theform (15) .Proof. To find non naturally reductive Einstein metrics on SU(4 + n ) we will use Propo-sitions 4.4 and 4.7. We consider the system of equations r − λ = 0 , r − λ = 0 , r − λ = 0 , r − λ = 0 ,r − λ = 0 , r − λ = 0 , r − λ = 0 , r − λ = 0 . From r − λ = 0 we see that 1 / − λ + v ) = 0 and from r − λ = 0 , r − λ = 0,we obtain that u = u . By substituting these values into r − λ = 0, r − λ = 0 and r − λ = 0, we obtain the equations: − nv x (6)2 + nv x (6)2 − v x (6)2 + 4 v = 0 ,nv x (6)2 − nx (6)3 + 3 u + 4 v x (6)2 + v − x (6) = 0 , − n u − n v + 8 n − nu − nv − nv − nx (6) + 32 n + 4 u − v = 0 . By solving these equations with respect to u , u , v , we obtain u = 13 (cid:0) − nv x (6)2 + nx (6)3 − v + 8 x (6) (cid:1) ,u = − 12 ( n − x (6)2 ( − n v x (6)4 + 2 n v x (6)2 + n x (6)5 − n x (6)2 − nv x (6)2 + 8 nv + 12 nx (6)3 − nx (6)2 + 2 v ) ,v = ( nx (6)2 + 4) v ( n + 4) x (6)2 Now we see that the equations r − λ = 0 and r − λ = 0 become respectively: nu x (6)2 − nu v x (6)2 + 2 u − u v x (6)2 + 2 x (6)2 = 0 , − nu v + n + 4 u − u v = 0 . By substituting the values u , v into r − λ = 0, we obtain19 ( x (6) − v ) (cid:0) − n v x (6)6 + n x (6)7 − n v x (6)4 + 18 n x (6)5 − nv x (6)2 + 96 nx (6)3 − v + 146 x (6) (cid:1) = 0 , (32) instein metrics on SU( N ) and complex Stiefel manifolds 27 and then by substituting the values u , v into r − λ = 0, we obtain n n − ( n + 1) x (6)4 (cid:0) − n v x (6)6 + 2 n v x (6)4 + n v x (6)7 − n v x (6)4 + 2 n x (6)4 +2 n v x (6)8 − n v x (6)6 + 16 n v x (6)2 − n v x (6)9 + 8 n v x (6)7 + 16 n v x (6)6 +16 n v x (6)5 − n v x (6)4 − n v x (6)2 + 2 n x (6)10 − n x (6)7 + 32 n x (6)4 +17 n v x (6)6 − n v x (6)4 + 50 n v x (6)2 + 32 n v − n v x (6)7 + 64 n v x (6)6 +128 n v x (6)5 − n v x (6)4 + 48 n v x (6)3 − n v x (6)2 + 48 n x (6)8 − n x (6)7 − n x (6)5 + 252 n x (6)4 + 32 nv x (6)4 − nv x (6)2 + 136 nv − nv x (6)5 +272 nv x (6)4 + 384 nv x (6)3 − nv x (6)2 + 288 nx (6)6 − nx (6)5 + 512 nx (6)4 − v x (6)2 + 32 v + 48 v x (6)3 − v x (6)2 + 2 x (6)4 (cid:1) = 0 . (33)From equation (32) we obtain that v = x (6) , or v = n x (6)7 + 18 n x (6)5 + 96 nx (6)3 + 146 x (6) n x (6)6 + 15 n x (6)4 + 72 nx (6)2 + 110 . (34)If v = x (6) , then we obtain naturally reductive metrics. We then consider the case (34).By substituting the value v into (33) we obtain an equation for x (6) of degree 16: F ( x (6) , n ) = n (2 n + 5) (cid:0) n + 4 n + 9 (cid:1) x (6)16 − n ( n + 4) (cid:0) n + 8 n + 19 (cid:1) x (6)15 +2 n (cid:0) n + 60 n + 400 n n + 996 n + 763 (cid:1) x (6)14 − n ( n + 4) (cid:0) n + 29 + 607 (cid:1) x (6)13 +4 n (cid:0) n + 650 n + 4333 n + 9854 n + 5310 (cid:1) x (6)12 − n ( n + 4) (cid:0) n + 152 n + 265 (cid:1) x (6)11 +2 n (cid:0) n + 14356 n + 94595 n + 200356 n + 77916 (cid:1) x (6)10 − n ( n + 4) (cid:0) n + 4744 n + 6842 (cid:1) x (6)9 +8 n (cid:0) n + 22698 n + 146413 n + 288990 n + 80098 (cid:1) x (6)8 − n ( n + 4) (cid:0) n + 92200 n + 104939 (cid:1) x (6)7 +8 n (cid:0) n + 84880 n + 529391 n + 958984 n + 179028 (cid:1) x (6)6 − n ( n + 4) (cid:0) n + 21800 n + 17941 (cid:1) x (6)5 +8 n (cid:0) n + 182641 n + 1078736 n + 1715695 n + 186684 (cid:1) x (6)4 − n ( n + 4) (cid:0) n + 200696 n + 98017 (cid:1) x (6)3 +8 (cid:0) n + 203160 n + 1087279 n + 1316352 n + 51424 (cid:1) x (6)2 − n + 4) (cid:0) n + 8 n + 1 (cid:1) x (6) + 170528( n + 4)(4 n + 1) = 0 . For n = 2 we have F ( x (6) , 2) = 288( x (6) − x (6)15 − x (6)14 + 1824 x (6)13 − x (6)12 + 15880 x (6)11 − x (6)10 + 72920 x (6)9 − x (6)8 + 192284 x (6)7 − x (6)6 + 292536 x (6)5 − x (6)4 + 238425 x (6)3 − x (6)2 + 80446 x (6) − . Thus we see that the solutions are given by x (6) = 1 and x (6) ≈ . n ≥ x (6) = 1, F (1 , n ) = − n − n + 2) ( n + 6) (cid:0) n + 14 n + 69 n + 134 n + 76 (cid:1) < . We also see that F (2 , n ) = 32 (cid:0) n + 8704 n + 34368 n + 87488 n + 164144 n + 239040 n +274217 n + 242156 n + 150555 n + 55429 n + 8500 (cid:1) > . We have F (1 / , n ) = 165536 (cid:0) n + 173 n + 4494 n − n − n − n − n − n + 11681267712 n + 39390150656 n + 17762877440 (cid:1) = 165536 (cid:0) n − + 693( n − + 105816( n − + 9342205( n − +525422358( n − + 19518844644( n − + 479005265720( n − +7495082389872( n − + 68109703630368( n − + 279535235098560( n − (cid:1) , and thus we see that F (1 / , n ) > 0, for n ≥ F (292 / (55 n ) , n ) = − n × (cid:0) n + 1131629115268488355792109375000 n +13721578697735127898701000000000 n + 70465121420063004319160003000000 n +66607303696117170858210917600000 n − n − n + 642183745708072078163896886362112 n − n − n − (cid:1) = − n × (cid:0) n − +2533903586202697370684687500000( n − +79701167324216470975283343750000( n − +1430610708561604520873222003000000( n − +16155192972662188907166744001600000( n − +119408859580384221935668378008499200( n − +583172621773938265497841838762557440( n − +1853844985914852789445447754895409152( n − +3622885223042644610041726583690821632( n − +3539894892993878348278554600791605248( n − (cid:1) < , for n ≥ . instein metrics on SU( N ) and complex Stiefel manifolds 29 Thus we obtain that, for 2 ≤ n ≤ 25 there exist two positive solutions and for n ≥ F ( x (6) , n ) = 0.By substituting the value of v into u , u and v , we also have the following: u = 2 x (6) (cid:0) nx (6)2 + 5 (cid:1) n x (6)4 + 10 nx (6)2 + 22 ,u = − n − n + 1) x (6) (cid:0) nx (6)2 + 5 (cid:1) (cid:0) n x (6)4 + 10 nx (6)2 + 22 (cid:1) (35) × (cid:16) n x (6)8 − n x (6)7 + 5 n x (6)8 − n x (6)7 + 44 n x (6)6 − n x (6)5 + 86 n x (6)6 − n x (6)5 + 336 n x (6)4 − n x (6)3 + 480 n x (6)4 − n x (6)3 + 1060 n x (6)2 − n x (6) + 928 nx (6)2 − nx (6) + 1168 n + 292 (cid:17) ,v = (cid:0) nx (6)2 + 4 (cid:1) (cid:0) n x (6)6 + 18 n x (6)4 + 96 nx (6)2 + 146 (cid:1) ( n + 4) x (6) (cid:0) nx (6)2 + 5 (cid:1) (cid:0) n x (6)4 + 10 nx (6)2 + 22 (cid:1) . We claim that the value of u in (35) is positive, whenever x (6) is a solution of F ( x (6) , n ) =0 . Indeed, from equation (35) it follows that G ( x (6) , u ) = 2 n u x (6)7 + 2 n x (6)8 − n x (6)7 + 30 n u x (6)5 + 5 n x (6)8 − n x (6)7 +44 n x (6)6 − n x (6)5 − n u x (6)7 + 144 n u x (6)3 + 86 n x (6)6 − n x (6)5 +336 n x (6)4 − n x (6)3 − n u x (6)5 + 220 n u x (6) + 480 n x (6)4 − n x (6)3 +1060 n x (6)2 − n x (6) − nu x (6)3 + 928 nx (6)2 − nx (6) + 1168 n − u x (6) + 292 = 0 . Now, by taking the resultant of Res x (6) ( F ( x (6) , n ) , G ( x (6) , u ) with respect to x (6) , weobtain the equation of u : Res x (6) ( F, G ) = 79805105467783573929984( n − n ( n + 1) ( n + 3) (2 n + 3) (2 n + 5) × (4 n + 1) (cid:16) n + 4) (cid:0) n + 4 n + 5 (cid:1) (cid:0) n + 4 n + 9 (cid:1) (cid:0) n + 4 n + 11 (cid:1) u − n + 4) × ( n + 4 n + 5)( n + 4 n + 11) (8 n + 96 n + 521 n + 1608 n + 2824 n + 2528 n + 495) u +24 (cid:0) n + 4 n + 11 (cid:1) (160 n + 3821 n + 42144 n + 283768 n + 1292016 n + 4148638 n +9485824 n + 15185192 n + 16188848 n + 10306293 n + 3014816 n + 154272) u − n ( n + 4) (cid:0) n + 4 n + 11 (cid:1) (1120 n + 26481 n + 294364 n + 2029009 n + 9595280 n +32503106 n + 79894120 n + 141057818 n + 172564400 n + 135486477 n + 58005020 n +8732629) u + 4 n (14560 n + 455546 n + 6737593 n + 62487392 n + 405558430 n +1943683970 n + 7067440828 n + 19705190624 n + 42032951806 n + 67552851414 n +79223664915 n + 63957854256 n + 31859786756 n + 7711909438 n + 410101496) u − n ( n + 4)(2912 n + 78078 n + 980205 n + 7626696 n + 40900061 n + 158743428 n +455081712 n + 964514064 n + 1485523623 n + 1597309366 n + 1109259109 n +424815888 n + 59882762) u + 2 n (128128 n + 3871868 n + 54127876 n +464149161 n + 2723268432 n + 11520926166 n + 36006811328 n + 83614509733 n +142792464992 n + 174250392508 n + 143668803716 n + 71910270196 n + 17301717480 n +926030448) u − n ( n + 4)(91520 n + 2334332 n + 27152796 n + 190488357 n +895297632 n + 2951744016 n + 6926787136 n + 11464153999 n + 12921488592 n +9210867932 n + 3562013612 n + 504494076) u + 3 n (137280 n + 3936504 n +51188148 n + 399221022 n + 2077926319 n + 7583085392 n + 19785908796 n +36840346114 n + 47825033196 n + 41093570040 n + 21077483624 n + 5140865464 n +281471440) u − n ( n + 4)(45760 n + 1085656 n + 11454564 n + 70805726 n +283650905 n + 766586712 n + 1404513863 n + 1699119089 n + 1268244972 n +504006728 n + 72600275) u + 4 n (64064 n + 1707706 n + 20248844 n + 140694517 n +634497443 n + 1939967836 n + 4068002539 n + 5765215640 n + 5273901874 n +2822612617 n + 707889047 n + 38868874) u − n ( n + 4)(11648 n + 250068 n +2328840 n + 12317562 n + 40609170 n + 85937948 n + 115337388 n + 92656707 n +38780924 n + 5814614) u + 2 n (29120 n + 703612 n + 7421294 n + 44788306 n +170172755 n + 421324252 n + 678403707 n + 684577731 n + 395658522 n + 105906354 n +6175980) u − n ( n + 3)( n + 4) (cid:0) n + 35252 n + 227370 n + 772912 n + 1475857 n +1549629 n + 797301 n + 140922 (cid:1) u + 6 n ( n + 3) (cid:0) n + 9724 n + 59268 n + 184899 n +310716 n + 268199 n + 98498 n + 7209 (cid:1) u − n ( n + 3) ( n + 4)(2 n + 3) (cid:0) n + 322 n +465 n + 159 (cid:1) u + n ( n + 3) (2 n + 3) (2 n + 5)(4 n + 1) (cid:17) = 0 . By looking at the coefficients of the polynomial, we see that if Res x (6) ( F, G ) has realsolutions, then these are positive. (cid:3) The case of SU(4) and SU(3) . We will show that the compact Lie groups SU(4)and SU(3) admit only naturally reductive Einstein metrics of the form (15). For SU(4)we prove the following: Theorem 5.5. The compact Lie group SU(4) with metrics corresponding to Ad(S(U(1) × U(1) × U(2))) -invariant inner products of the form (15) admits only naturally reductiveEinstein metrics, that is, bi-invariant metric and the metric (15) with x (6) = u = v =5 / , x (7) = x (8) = 1 , v = 73 / .Proof. Let ℓ = m = 1 and n = 2. In this case we have h = h = 0, so we do not have u and u variables. To find Einstein metrics we need to solve the system of equations(36) r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 , r = 0 . We set x (7) = 1. Then from r = 0 we have that c = − ( x (8) − x (8) + 1) / (cid:0) √ x (8)2 + 1) (cid:1) . By substituting c into the first five equations of system (36), we obtain the system f = 2 u x (8)2 + 2 u − u v x (8)2 − u v + 4 x (8)2 = 0 ,f = 4 v x (6)2 x (8)4 + 8 v x (6)2 x (8)2 + 4 v x (6)2 − v x (6)2 x (8)2 − v x (8)4 − v x (8)2 = 0 , instein metrics on SU( N ) and complex Stiefel manifolds 31 f = 4 v x (6)2 x (8) + 4 v x (8)3 + 4 v x (8) − x (6)3 x (8)2 − x (6)3 + 2 x (6) x (8)4 − x (6) x (8)3 + 4 x (6) x (8)2 − x (6) x (8) + 2 x (6) = 0 ,f = 6 u x (6)2 x (8)4 + 12 u x (6)2 x (8)2 + 6 u x (6)2 + 4 v x (6)2 x (8)4 + 8 v x (6)2 x (8)2 +4 v x (6)2 + 8 v x (6)2 x (8)4 − v x (8)6 − v x (8)4 − v x (8)2 + 12 x (6)3 x (8)5 +24 x (6)3 x (8)3 + 12 x (6)3 x (8) − x (6)2 x (8)5 − x (6)2 x (8)3 − x (6)2 x (8) − x (6) x (8)7 +32 x (6) x (8)6 − x (6) x (8)5 + 64 x (6) x (8)4 − x (6) x (8)3 + 32 x (6) x (8)2 − x (6) x (8) = 0 ,f = ( x (8) − u x (6) x (8)5 + 3 u x (6) x (8)4 + 6 u x (6) x (8)3 + 6 u x (6) x (8)2 +3 u x (6) x (8) + 3 u x (6) + 2 v x (6) x (8)5 + 2 v x (6) x (8)4 + 4 v x (6) x (8)3 + 4 v x (6) x (8)2 +2 v x (6) x (8) + 2 v x (6) − v x (6) x (8)3 − v x (6) x (8)2 − x (6) x (8)5 − x (6) x (8)3 − x (6) x (8) + 4 x (8)6 + 4 x (8)5 + 8 x (8)4 + 8 x (8)3 + 4 x (8)2 + 4 x (8) ) = 0 . First we study the case when x (8) = 1. Then c = 0 and equations r − r = 0, r − r = 0, r − r = 0, r − r = 0 reduce to f = u − u v + 1 = 0 ,f = 4 v x (6)2 − v x (6)2 − v = 0 ,f = (cid:0) x (6)2 + 2 (cid:1) ( v − x (6) ) = 0 ,f = 12 u x (6)2 + 8 v x (6)2 + 4 v x (6)2 − v + 24 x (6)3 − x (6)2 + 32 x (6) = 0 . From f = 0 we have v = x (6) , so by substituting this into f , f we obtain: f = − x (6) (cid:0) − v x (6) + x (6)2 + 1 (cid:1) = 0 ,f = 4 x (6) (cid:0) u x (6) + 2 v x (6) + 7 x (6)2 − x (6) + 4 (cid:1) = 0 . By solving f = 0 with respect to v , we have v = ( x + 1) / x (6) . We substitute thisinto f = 0 and f = 0 and thus we obtain:2( u − x (6) )( u x (6) − /x (6) = 0 , x (6) (cid:0) u x (6) + 8 x (6)2 − x (6) + 5 (cid:1) = 0 . From the first equation above we see that u = x (6) or u = 1 /x (6) . We substitute the u = x (6) into the second equation above and we have 11 x − x (6) + 5 = 0 , whosesolutions are x (6) = 5 / 11 and x (6) = 1. Now we substitute u = 1 /x (6) and we obtain8 x − x (6) + 8 = 0 whose solution is x (6) = 1.For x (6) = 1 we have u = v = v = x (8) = 1. This metric corresponds to a bi-invariant metric which is naturally reductive. For x (6) = 5 / 11 from the above we have u = v = 5 / v = 73 / 55, so from Proposition 5.2 we have that this metric is alsonaturally reductive.Now we study the case when x (8) = 1. By solving f = 0 , f = 0 , f = 0, we obtain u = − x (8) x (6) (cid:0) x (6)2 + x (8)2 + 1 (cid:1) (cid:16) x (6)4 − x (6)3 − x (6)2 x (8)2 + 24 x (6)2 x (8) + x (6)2 − x (6) x (8)2 − x (6) − x (8)4 + 8 x (8)3 − x (8)2 + 12 x (8) − (cid:17) , v = x (8) (cid:0) x (6)2 − x (8)2 + 4 x (8) − (cid:1) (cid:0) x (6)2 + x (8)2 + 1 (cid:1) x (6) (cid:0) x (8)2 + 1 (cid:1) (cid:0) x (6)2 + x (8)2 + 1 (cid:1) ,v = − x (6) (cid:0) x (8)2 + 1 (cid:1) (cid:0) − x (6)2 + x (8)2 − x (8) + 1 (cid:1) x (8) (cid:0) x (6)2 + x (8)2 + 1 (cid:1) . By substituting these u , v , v into f , f , we can see that the equations f = 0 and f = 0 reduce to the polynomial equations of x (6) and x (8) : F ( x (6) , x (8) ) = 16 x (6)8 x (8)2 + 28 x (6)8 − x (6)7 x (8)2 − x (6)7 − x (6)6 x (8)4 +96 x (6)6 x (8)3 + 40 x (6)6 x (8)2 + 180 x (6)6 x (8) + 68 x (6)6 − x (6)5 x (8)4 − x (6)5 x (8)3 − x (6)5 x (8)2 − x (6)5 x (8) − x (6)5 − x (6)4 x (8)6 − x (6)4 x (8)5 + 229 x (6)4 x (8)4 +38 x (6)4 x (8)3 + 510 x (6)4 x (8)2 + 90 x (6)4 x (8) + 125 x (6)4 + 64 x (6)3 x (8)6 − x (6)3 x (8)5 +152 x (6)3 x (8)4 − x (6)3 x (8)3 + 112 x (6)3 x (8)2 − x (6)3 x (8) + 24 x (6)3 + 8 x (6)2 x (8)8 − x (6)2 x (8)7 + 182 x (6)2 x (8)6 − x (6)2 x (8)5 + 576 x (6)2 x (8)4 − x (6)2 x (8)3 +494 x (6)2 x (8)2 − x (6)2 x (8) + 68 x (6)2 + 32 x (6) x (8)8 − x (6) x (8)7 + 116 x (6) x (8)6 − x (6) x (8)5 + 156 x (6) x (8)4 − x (6) x (8)3 + 92 x (6) x (8)2 − x (6) x (8) + 20 x (6) +4 x (8)10 − x (8)9 + 33 x (8)8 − x (8)7 + 116 x (8)6 − x (8)5 + 166 x (8)4 − x (8)3 +80 x (8)2 − x (8) + 1 = 0 ,F ( x (6) , x (8) ) = − x (6)4 x (8) − x (6)4 + 4 x (6)3 x (8) + x (6)2 x (8)3 − x (6)2 x (8)2 − x (6)2 x (8) + x (6)2 + 4 x (6) x (8)3 + 4 x (6) x (8) + x (8)5 − x (8)4 − x (8) + 1 = 0 . By taking the resultant Res x (8) ( F , F ) of F and F with respect to x (8) , we obtainRes x (8) ( F , F ) = 2304( x (6) − x (6)4 (cid:0) x (6)2 + 2 (cid:1) (cid:0) x (6)2 + 2 x (6) + 1 (cid:1)(cid:0) x (6)16 − x (6)15 + 33986766 x (6)14 − x (6)13 +106943061 x (6)12 − x (6)11 + 193524028 x (6)10 − x (6)9 +135191836 x (6)8 − x (6)7 + 64876480 x (6)6 − x (6)5 +15920560 x (6)4 − x (6)3 + 1785280 x (6)2 − x (6) + 72000 (cid:1) . Now we can see that the factor of degree 16 in the polynomial Res x (8) ( F , F ) of x (6) does not have real roots by using computer manipulation, thus there are no solutions forthe system of equations for x (8) = 1. If x (6) = 1 then Proposition 5.2 (1, (ii)) implies thatthe metric is naturally reductive and this concludes the proof. (cid:3) We now proceed with SU(3) and we prove the following: Theorem 5.6. The compact Lie group SU(3) admits only naturally reductive Einsteinmetrics which correspond to Ad(S(U(1) × U(1) × U(1))) -invariant inner products of theform (15) Proof. Let ℓ = m = n = 1. In this case we have h = h = h = 0, so we do not have u , u and u variable. To find Einstein metrics we need to solve the system(37) r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 . instein metrics on SU( N ) and complex Stiefel manifolds 33 We set x (7) = 1. Then from r = 0 we have that c = − ( x (8)2 − / (cid:0) √ x (8)2 + 1) (cid:1) . Bysubstituting c into the system (37), this reduces to the system g = 3 v x (6)2 x (8)4 + 6 v x (6)2 x (8)2 + 3 v x (6)2 − v x (6)2 x (8)2 − v x (8)4 − v x (8)2 = 0 ,g = 2 v x (6)2 x (8) + 4 v x (8)3 + 4 v x (8) − x (6)3 x (8)2 − x (6)3 + x (6) x (8)4 − x (6) x (8)3 +2 x (6) x (8)2 − x (6) x (8) + x (6) = 0 ,g = 3 v x (6)2 x (8)4 + 6 v x (6)2 x (8)2 + 3 v x (6)2 + 4 v x (6)2 x (8)4 − v x (8)6 − v x (8)4 − v x (8)2 + 4 x (6)3 x (8)5 + 8 x (6)3 x (8)3 + 4 x (6)3 x (8) − x (6)2 x (8)5 − x (6)2 x (8)3 − x (6)2 x (8) − x (6) x (8)7 + 12 x (6) x (8)6 − x (6) x (8)5 + 24 x (6) x (8)4 − x (6) x (8)3 +12 x (6) x (8)2 = 0 ,g = ( x (8) − v x (6) x (8)5 + 3 v x (6) x (8)4 + 6 v x (6) x (8)3 + 6 v x (6) x (8)2 + 3 v x (6) x (8) +3 v x (6) − v x (6) x (8)3 − v x (6) x (8)2 − x (6) x (8)5 − x (6) x (8)3 − x (6) x (8) + 4 x (8)6 +4 x (8)5 + 8 x (8)4 + 8 x (8)3 + 4 x (8)2 + 4 x (8) ) = 0 . First we study the case where x (8) = 1. Then the equations r − r = 0 , r − r =0 , r − r = 0 reduce to the system g = − v x (6)2 + v x (6)2 + 2 v = 0 ,g = (cid:0) x (6)2 + 4 (cid:1) ( v − x (6) ) = 0 ,g = 3 v x (6)2 + v x (6)2 − v + 4 x (6)3 − x (6)2 + 8 x (6) = 0 . From g = 0 we have v = x (6) , so we substitute into g , g and we take g = − v x (6) + x (6)2 + 2 = 0 , g = 3 v x (6) + 5 x (6)2 − x (6) + 4 = 0 . We solve g with respect to v and we have v = ( x (6)2 + 2) / (3 x (6) ) . We substituteinto g and we take 6 x − x (6) + 6 = 0 , whose solution is x (6) = 1. From the abovecalculations we have that v = v = 1. So the only Einstein metric is the bi-invariantmetric which is naturally reductive.Now we study the case when x (8) = 1. By solving g = 0 , g = 0, we obtain v = − x (8) x (6) (cid:0) x (8)2 + 1 (cid:1) (cid:0) x (6)2 + 2 x (8)2 + 2 (cid:1) (cid:0) x (6)4 x (8)2 + 2 x (6)4 − x (6)3 x (8)2 − x (6)3 +12 x (6)2 x (8)3 + 3 x (6)2 x (8)2 + 6 x (6)2 x (8) + 3 x (6)2 − x (6) x (8)4 − x (6) x (8)2 − x (6) − x (8)6 +6 x (8)5 − x (8)4 + 12 x (8)3 − x (8)2 + 6 x (8) + 1 (cid:1) ,v = − x (6) (cid:0) x (8)2 + 1 (cid:1) (cid:0) − x (6)2 + x (8)2 − x (8) + 1 (cid:1) x (8) (cid:0) x (6)2 + 2 x (8)2 + 2 (cid:1) . By substituting these v , v into g , g , we can see that the equations g = 0 and g = 0reduce to the polynomial equations of x (6) and x (8) : G ( x (6) , x (8) ) = ( x (6) − x (8) )( x (6)3 + x (6)2 x (8) − x (6)2 + x (6) x (8)2 + 2 x (6) x (8) + x (6) + x (8)3 − x (8)2 + x (8) − 4) = 0 , G ( x (6) , x (8) ) = − x (6)4 x (8) − x (6)4 + 2 x (6)3 x (8) − x (6)2 x (8)2 − x (6)2 x (8) + 4 x (6) x (8)3 +4 x (6) x (8) + x (8)5 − x (8)4 − x (8) + 1 = 0 . By taking the resultant Res x (8) ( G , G ) of G and G with respect to x (8) , we obtainRes x (8) ( G , G ) = − x (6) − ( x (6)2 + 4) ( x (6)2 + x (6) + 1)(4 x (6)2 + 1) . Thus the polynomial Res x (8) ( G , G ) of x (6) does not have real roots, for x (6) = 1. If x (6) = 1 then Proposition 5.2 (1, (ii)) implies that the metric is naturally reductive andthis concludes the proof. (cid:3) Invariant Einstein metrics on certain Stiefel manifolds V ℓ + m C ℓ + m + n A complete description for the set of all SU( ℓ + m + n )-invariant metrics on the Stiefelmanifolds V ℓ + m C ℓ + m + n ∼ = SU( ℓ + m + n ) / SU( n ) is not easy. This is because the isotropyrepresentation χ of U( ℓ + m + n ) / U( n ) ∼ = SU( ℓ + m + n ) / SU( n ) contains some equivalentsubrepresentations. In fact, it is given as χ = 1 ⊕ · · · ⊕ | {z } ( ℓ + m ) -times ⊕ ( ( µ n ⊕ ¯ µ n ) ⊕ · · · ⊕ ( µ n ⊕ ¯ µ n ) ) | {z } ( ℓ + m ) -times , where µ n : U( n ) → Aut( C n ) is the standard representation of U( n ) and Ad U( n ) ⊗ C = µ n ⊗ ¯ µ n is its complexified adjoint representation. In this section we search for Ad(S(U( ℓ ) × U( m ) × U( n )))-invariant Einstein metrics of the form (17), which correspond to a subsetof all SU( ℓ + m + n )-invariant metrics on V ℓ + m C ℓ + m + n .The metric (17) is Einstein if and only if the system r = 0 , r − r = 0 , r − r = 0 , r − r = 0 ,r − r = 0 , r − r = 0 , r − r = 0(38)has positive solutions (cf. Propositions 4.4, 4.8). As for the case of the special unitarygroup, in the above system we may assume that a = d = 1, b = 0.6.1. The Stiefel manifold V C ∼ = SU(4) / SU(2) . In this case we have ℓ = m = 1, n = 2and in system (38) the second and third equations are absent. From the equation r = 0we obtain that c = 1 − x (8)2 √ x (8)2 ) . Next, we observe that in the equations r − r = 0 , r − r = 0 the variables v , v are linearexpressions of x (6) , x (8) . We substitute v , v and the above value of c in the equations r − r = 0 , r − r = 0 and we obtain the solutions x (8) = 0 , − x (8) = 1, which implies that c = 0. We set x (7) = 1 in the last four equations of system(38) and this reduces to the system2 v x (6)2 − v x (6)2 − v = 0 , (cid:0) x (6)2 + 2 (cid:1) ( v − x (6) ) = 0 , v x (6)2 + v x (6)2 − v + 6 x (6)3 − x (6)2 + 8 x (6) = 0 . instein metrics on SU( N ) and complex Stiefel manifolds 35 From the second equation above we have v = x (6) . We substitute into the first and thirdequations above and we obtain two polynomials of x (6) and v : g ( x (6) , u ) = 2 v x (6)2 − x (6)3 − x (6) , g ( x (6) , v ) = 2 v x (6)2 + 7 x (6)3 − x (6)2 + 4 x (6) . Thus we have a polynomial of x (6) given by 8 x (6)2 − x (6) + 5 = 0 whose solutionsare x (6) = (4 ± √ / 4. We substitute these values into g = 0 and we see that, for x (6) = (4 − √ / , v = (52 + 3 √ / x (6) = (4 + √ / , v = (52 − √ / . Therefore, we obtain two Einstein metrics for V C which are of Jensen’s type:(1) ( v , v , x (6) , x (7) , x (8) ) = ((52 + 3 √ / , (4 − √ / , (4 − √ / , , v , v , x (6) , x (7) , x (8) ) = ((52 − √ / , (4 + √ / , (4 + √ / , , The Stiefel manifold V C ∼ = SU(5) / SU(2) . In this case we have ℓ = 1, m = n = 2and in system (38) the second equation is absent. To find Einstein metrics we solve thesystem r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 . We set x (7) = 1 and this reduces to the system: f = 6 u x (6)2 + 3 u x (8)2 − u v x (6)2 x (8)2 − u v x (6)2 + 6 x (6)2 x (8)2 = 0 ,f = 5 v x (6)2 x (8)4 + 20 v x (6)2 x (8)2 + 20 u x (6)2 − v x (6)2 x (8)2 − v x (8)4 − v x (8)2 = 0 ,f = 3 u x (8)3 + 6 u x (8) + 12 v x (6)2 x (8) + 9 u x (8)3 + 18 v x (8) − x (6)3 x (8)2 − x (6)3 +4 x (6) x (8)4 − x (6) x (8)3 + 12 x (6) x (8)2 − x (6) x (8) + 8 x (6) = 0 ,f = 9 u x (6)2 x (8)4 + 36 u x (6)2 x (8)2 + 36 u x (6)2 − u x (8)6 − u x (8)4 − u x (8)2 +5 u x (6)2 x (8)4 + 20 v x (6)2 x (8)2 + 20 v x (6)2 + 9 v x (6)2 x (8)4 − v x (8)6 − v x (8)4 − v x (8)2 + 18 x (6)3 x (8)5 + 72 x (6)3 x (8)3 + 72 x (6)3 x (8) − x (6)2 x (8)5 − x (6)2 x (8)3 − x (6)2 x (8) − x (6) x (8)7 + 60 x (6) x (8)6 − x (6) x (8)5 + 240 x (6) x (8)4 − x (6) x (8)3 +240 x (6) x (8)2 − x (6) x (8) = 0 ,f = − u x (6) x (8)4 − u x (6) x (8)2 − u x (6) + 5 v x (6) x (8)6 + 15 v x (6) x (8)4 − u x (6) − u x (6) x (8)4 + 36 v x (6) x (8)2 + 6 x (6)2 x (8)5 + 24 x (6)2 x (8)3 + 24 x (6)2 x (8) − x (6) x (8)6 + 60 x (6) x (8)5 − x (6) x (8)4 + 240 x (6) x (8)3 − x (6) x (8)2 + 240 x (6) x (8) +18 x (8)7 + 54 x (8)5 − x (8) = 0 . Also, from r = 0 we obtain that c = − (cid:0) x (8)2 − (cid:1) √ (cid:0) x (8)2 + 2 (cid:1) . We observe that the equations f , f and f are linear with respect to u , u and v andby solving we obtain the following: u = 1 / (cid:0) x (8) ( − x (6)4 + x (6)2 x (8)2 − x (6)2 + x (8)4 + x (8)2 − (cid:1) × (cid:0) x (6) (14 x (6)4 x (8)2 − x (6)4 − x (6)3 x (8)2 + 40 x (6)3 x (8) − x (6)2 x (8)4 + 50 x (6)2 x (8)3 + 2 x (6)2 x (8)2 − x (6)2 x (8) − x (6) x (8)4 + 30 x (6) x (8)3 − x (6) x (8)2 + 60 x (6) x (8) − x (8)6 +20 x (8)5 − x (8)4 + 20 x (8)3 + 4 x (8)2 − x (8) + 20) (cid:1) v = − / (cid:0) x (6) x (8) ( x (8)2 + 2)(2 x (6)4 − x (6)2 x (8)2 + 6 x (6)2 − x (8)4 − x (8)2 + 2) (cid:1) × (cid:0) x (6)6 x (8)2 + 20 x (6)6 − x (6)5 x (8)3 − x (6)5 x (8) + 2 x (6)4 x (8)4 + 20 x (6)4 x (8)3 +6 x (6)4 x (8)2 + 60 x (6)4 x (8) + 10 x (6)3 x (8)5 − x (6)3 x (8)4 − x (6)3 x (8)3 − x (6)3 x (8)2 − x (6)3 x (8) − x (6)2 x (8)6 + 10 x (6)2 x (8)5 − x (6)2 x (8)4 + 40 x (6)2 x (8)3 − x (6)2 x (8)2 +40 x (6)2 x (8) − x (6)2 + 10 x (6) x (8)7 − x (6) x (8)6 + 40 x (6) x (8)5 − x (6) x (8)4 +40 x (6) x (8)3 − x (6) x (8)2 − x (8)8 − x (8)6 + 12 x (8)2 ) (cid:1) v = − / (3 x (8) (cid:0) − x (6)4 + x (6)2 x (8)2 − x (6)2 + x (8)4 + x (8)2 − (cid:1) ) × (cid:0) x (6)5 x (8)2 + 4 x (6)5 − x (6)3 x (8)4 + 10 x (6)3 x (8)3 − x (6)3 x (8)2 + 20 x (6)3 x (8) − x (6)3 − x (6)2 x (8)4 + 10 x (6)2 x (8)3 − x (6)2 x (8)2 + 20 x (6)2 x (8) − x (6) x (8)6 − x (6) x (8)4 + 4 x (6) (cid:1) . We substitute the above expressions into f = 0, f = 0 and we obtain − x (6)2 g ( x (6) , x (8) ) = 0 and − x (6) ( x (8)2 + 2) g ( x (6) , x (8) ) = 0 , where g and g are given as follows: g ( x (6) , x (8) ) = 280 x (6)10 x (8)4 + 1840 x (6)10 x (8)2 − x (6)10 − x (6)9 x (8)5 + 320 x (6)9 x (8)4 − x (6)9 x (8)3 − x (6)9 x (8)2 + 12800 x (6)9 x (8) + 28 x (6)8 x (8)6 + 6080 x (6)8 x (8)5 − x (6)8 x (8)4 + 29120 x (6)8 x (8)3 − x (6)8 x (8)2 − x (6)8 x (8) + 1680 x (6)7 x (8)7 − x (6)7 x (8)6 + 2000 x (6)7 x (8)5 − x (6)7 x (8)4 − x (6)7 x (8)3 + 33600 x (6)7 x (8)2 +19200 x (6)7 x (8) − x (6)6 x (8)8 + 1640 x (6)6 x (8)7 − x (6)6 x (8)6 + 33600 x (6)6 x (8)5 − x (6)6 x (8)4 + 33120 x (6)6 x (8)3 − x (6)6 x (8)2 − x (6)6 x (8) + 6400 x (6)6 +1260 x (6)5 x (8)9 + 240 x (6)5 x (8)8 + 8400 x (6)5 x (8)7 − x (6)5 x (8)6 − x (6)5 x (8)5 +12320 x (6)5 x (8)4 − x (6)5 x (8)3 + 96000 x (6)5 x (8)2 − x (6)5 x (8) − x (6)4 x (8)10 − x (6)4 x (8)9 + 4326 x (6)4 x (8)8 − x (6)4 x (8)7 + 46164 x (6)4 x (8)6 − x (6)4 x (8)5 +14296 x (6)4 x (8)4 − x (6)4 x (8)3 − x (6)4 x (8)2 + 19200 x (6)4 x (8) − x (6)3 x (8)11 +5620 x (6)3 x (8)10 − x (6)3 x (8)9 + 9160 x (6)3 x (8)8 − x (6)3 x (8)7 − x (6)3 x (8)6 − x (6)3 x (8)5 + 18720 x (6)3 x (8)4 − x (6)3 x (8)3 + 43200 x (6)3 x (8)2 − x (6)3 x (8) +224 x (6)2 x (8)12 − x (6)2 x (8)11 + 1938 x (6)2 x (8)10 − x (6)2 x (8)9 + 22938 x (6)2 x (8)8 − x (6)2 x (8)7 + 53204 x (6)2 x (8)6 − x (6)2 x (8)5 + 20216 x (6)2 x (8)4 − x (6)2 x (8)3 − x (6)2 x (8)2 + 12800 x (6)2 x (8) − x (6)2 − x (6) x (8)13 + 1740 x (6) x (8)12 − x (6) x (8)11 + 8340 x (6) x (8)10 − x (6) x (8)9 + 10560 x (6) x (8)8 − x (6) x (8)7 − x (6) x (8)6 + 6240 x (6) x (8)5 − x (6) x (8)4 + 16320 x (6) x (8)3 − x (6) x (8)2 instein metrics on SU( N ) and complex Stiefel manifolds 37 +77 x (8)14 − x (8)13 + 28 x (8)12 − x (8)11 − x (8)10 − x (8)9 + 590 x (8)8 +1280 x (8)7 + 32 x (8)6 + 960 x (8)5 − x (8)4 − x (8)3 + 1040 x (8)2 and g ( x (6) , x (8) ) = − x (6) ( x (8)2 + 2) ( − x (6)6 + 40 x (6)5 x (8) − x (6)4 x (8)2 − x (6)4 x (8) − x (6)3 x (8)3 + 60 x (6)3 x (8)2 + 60 x (6)3 x (8) + 17 x (6)2 x (8)4 − x (6)2 x (8)3 − x (6)2 x (8)2 − x (6)2 x (8) + 20 x (6)2 − x (6) x (8)5 + 30 x (6) x (8)4 − x (6) x (8)3 + 40 x (6) x (8)2 + 7 x (8)6 +7 x (8)4 − x (8)2 ) . We consider a polynomial ring R = Q [ z, x (6) , x (8) ] and an ideal I generated by { g , g ,z x (6) x (8) − } to find non zero solutions for the equations g = 0 , g = 0. We take alexicographic order > with z > x (6) > x (8) for a monomial ordering on R . Then by theaid of computer we see that a Gr¨obner basis for the ideal I contains the polynomials( x (8) − (5 x (8)2 + 2) h ( x (8) ) , where h ( x (8) ) is given by h ( x (8) ) = 6525496468915200 x (8)26 − x (8)25 + 281982445913589120 x (8)24 − x (8)23 + 3741072893659661028 x (8)22 − x (8)21 +25787611597344221492 x (8)20 − x (8)19 + 112460183799287026516 x (8)18 − x (8)17 + 336020119285435562648 x (8)16 − x (8)15 +701457087993906599199 x (8)14 − x (8)13 + 1029897386253507589663 x (8)12 − x (8)11 + 1029965936819584117520 x (8)10 − x (8)9 +678539858188720076700 x (8)8 − x (8)7 + 284744896720187390000 x (8)6 − x (8)5 + 71039304710981500000 x (8)4 − x (8)3 +8664161487275000000 x (8)2 − x (8) + 216106141875000000 , and(39) ( x (8) − ( x (6) − w ( x (8) )) = 0 , where w ( x (8) ) is a polynomial with rational coefficients. We solve the equation h ( x (8) ) = 0numerically and we obtain two positive solutions, which are given approximately as x (8) ≈ . , x (8) ≈ . . By substituting the values of x (8) into (39) we obtain two positive solutions of the systemof equations g = 0 , g = 0, approximately as( x (6) , x (8) ) ≈ (0 . , . x (6) , x (8) ) ≈ (1 . , . u , u and u and obtain two Einsteinmetrics on V C ∼ = SU(5) / SU(2) which are given as follows:(1) ( u , v , v , x (6) , x (8) , x (7) ) ≈ (0 . , . , . , . , . , u , v , v , x (6) , x (8) , x (7) ) ≈ (0 . , . , . , . , . , In the case where x (8) = 1 then c = 0 and from g = 0 we obtain that9 (cid:0) − x (6)6 + 40 x (6)5 − x (6)4 + 100 x (6)3 − x (6)2 (cid:1) = 0 . The solutions of the above equation are x (6) = (10 − √ / x (6) = (10 + √ / 10. Wesubstitute into u , v and v and we obtain two more Einstein metrics of Jensen’s type asfollows:(3) ( u , v , v , x (6) , x (7) , x (8) )= (1 − p / , √ / , − p / , (10 − √ / , , , (4) ( u , v , v , x (6) , x (7) , x (8) )= (1 + p / , − √ / , p / , (10 + √ / , , The Stiefel manifold V C ∼ = SU(6) / SU(2) . In this case we have ℓ = m = n = 2.To find Einstein metrics we solve the system r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 , r − r = 0 . We set x (7) = 1 and the above system reduces to f = u u x (6)2 x (8)2 + u u x (8)2 − u u x (6)2 − u u x (8)2 − u x (6)2 x (8)2 + u x (6)2 x (8)2 = 0 ,f = 2 u x (6)2 + 2 u x (8)2 − u v x (6)2 x (8)2 − u v x (6)2 + 2 x (6)2 x (8)2 = 0 ,f = 3 v x (6)2 x (8)4 + 6 v x (6)2 x (8)2 + 3 v x (6)2 − v x (6)2 x (8)2 − v x (8)4 − v x (8)2 = 0 ,f = 3 u x (8)3 + 3 u x (8) + 3 u x (8)3 + 3 u x (8) + 8 v x (6)2 x (8) + 10 u x (8)3 + 10 v x (8) − x (6)3 x (8)2 − x (6)3 + 4 x (6) x (8)4 − x (6) x (8)3 + 8 x (6) x (8)2 − x (6) x (8) + 4 x (6) = 0 ,f = − u x (8)6 − u x (8)4 − u x (8)2 + 6 u x (6)2 x (8)4 + 12 u x (6)2 x (8)2 + 6 u x (6)2 − u x (8)6 − u x (8)4 − u x (8)2 + 3 v x (6)2 x (8)4 + 6 v x (6)2 x (8)2 + 3 v x (6)2 +4 v x (6)2 x (8)4 − v x (8)6 − v x (8)4 − v x (8)2 + 16 x (6)3 x (8)5 + 32 x (6)3 x (8)3 +16 x (6)3 x (8) − x (6)2 x (8)5 − x (6)2 x (8)3 − x (6)2 x (8) − x (6) x (8)7 + 48 x (6) x (8)6 − x (6) x (8)5 + 96 x (6) x (8)4 − x (6) x (8)3 + 48 x (6) x (8)2 = 0 ,f = 6 u x (6) x (8)6 + 12 u x (6) x (8)4 + 6 u x (6) x (8)2 − u x (6) x (8)4 − u x (6) x (8)2 − u x (6) + 3 v x (6) x (8)6 + 3 v x (6) x (8)4 − v x (6) x (8)2 − v x (6) − v x (6) x (8)4 +4 v x (6) x (8)2 − x (6) x (8)6 + 48 x (6) x (8)5 − x (6) x (8)4 + 96 x (6) x (8)3 − x (6) x (8)2 +48 x (6) x (8) + 16 x (8)7 + 16 x (8)5 − x (8)3 − x (8) = 0 . Also, from r = 0 we obtain that c = − x (8)2 − √ (cid:0) x (8)2 + 1 (cid:1) . instein metrics on SU( N ) and complex Stiefel manifolds 39 We observe that the polynomials f , f , f and f are linear with respect to u , u , v and v and by solving we obtain the following: u = − x (6) − x (8) ) (cid:0) x (6)5 − x (6)4 x (8) − x (6)3 x (8)2 + 10 x (6)3 x (8) − x (6)2 x (8)3 − x (6)2 x (8) − x (6) x (8)4 + 10 x (6) x (8)3 − x (6) x (8)2 + 8 x (6) x (8) − x (6) + 3 x (8)5 − x (8) (cid:1) / (cid:0) x (6) x (8) (cid:0) x (6)4 − x (8)4 − x (8)2 − (cid:1) (cid:1) ,u = − x (6) − x (8) (3 x (6)5 − x (6)4 + 10 x (6)3 x (8) − x (6)3 − x (6)2 x (8)2 − x (6)2 − x (6) x (8)4 + 8 x (6) x (8)3 − x (6) x (8)2 + 10 x (6) x (8) − x (6) − x (8)4 + 3) / (cid:0) x (6) (cid:0) x (6)4 − x (8)4 − x (8)2 − (cid:1) (cid:1) ,v = 4 x (8) (cid:0) x (6)2 + x (8)2 + 1 (cid:1) (cid:0) x (6)4 + 6 x (6)2 x (8) − x (6) x (8)2 − x (6) x (8) − x (8)4 +2 x (8)2 − (cid:1) / (cid:0) x (6) (cid:0) x (8)2 + 1 (cid:1) (cid:0) x (6)4 − x (8)4 − x (8)2 − (cid:1) (cid:1) ,v = x (6) (cid:0) x (8)2 + 1 (cid:1) (cid:0) x (6)4 + 6 x (6)2 x (8) − x (6) x (8)2 − x (6) x (8) − x (8)4 + 2 x (8)2 − (cid:1) / (cid:0) x (8) (cid:0) x (6)4 − x (8)4 − x (8)2 − (cid:1) (cid:1) . We substitute the above expressions into f and f and we obtain2 x (8) ( x (8) − g ( x (6) , x (8) ) and 2 x (8)2 g ( x (6) , x (8) ) , where g and g given as follows: g ( x (6) , x (8) ) = 288 x (6)19 − x (8) x (6)18 − x (6)18 + 288 x (8)2 x (6)17 + 3712 x (8) x (6)17 +288 x (6)17 − x (8)2 x (6)16 − x (8) x (6)16 − x (8)4 x (6)15 + 3944 x (8)3 x (6)15 +1888 x (8)2 x (6)15 + 3944 x (8) x (6)15 − x (6)15 + 2340 x (8)5 x (6)14 − x (8)4 x (6)14 +4532 x (8)3 x (6)14 + 4532 x (8)2 x (6)14 − x (8) x (6)14 + 2340 x (6)14 − x (8)6 x (6)13 − x (8)5 x (6)13 − x (8)4 x (6)13 − x (8)3 x (6)13 − x (8)2 x (6)13 − x (8) x (6)13 − x (6)13 + 8752 x (8)6 x (6)12 + 16840 x (8)5 x (6)12 + 56608 x (8)4 x (6)12 + 56608 x (8)3 x (6)12 +16840 x (8)2 x (6)12 + 8752 x (8) x (6)12 + 864 x (8)8 x (6)11 − x (8)7 x (6)11 − x (8)6 x (6)11 − x (8)5 x (6)11 − x (8)4 x (6)11 − x (8)3 x (6)11 − x (8)2 x (6)11 − x (8) x (6)11 +864 x (6)11 − x (8)9 x (6)10 + 4508 x (8)8 x (6)10 − x (8)7 x (6)10 + 82480 x (8)6 x (6)10 +117717 x (8)5 x (6)10 + 117717 x (8)4 x (6)10 + 82480 x (8)3 x (6)10 − x (8)2 x (6)10 + 4508 x (8) x (6)10 − x (6)10 + 864 x (8)10 x (6)9 + 4288 x (8)9 x (6)9 + 896 x (8)8 x (6)9 − x (8)7 x (6)9 − x (8)6 x (6)9 − x (8)5 x (6)9 − x (8)4 x (6)9 − x (8)3 x (6)9 + 896 x (8)2 x (6)9 +4288 x (8) x (6)9 + 864 x (6)9 − x (8)10 x (6)8 − x (8)9 x (6)8 + 4616 x (8)8 x (6)8 +91112 x (8)7 x (6)8 + 120582 x (8)6 x (6)8 + 120582 x (8)5 x (6)8 + 91112 x (8)4 x (6)8 + 4616 x (8)3 x (6)8 − x (8)2 x (6)8 − x (8) x (6)8 − x (8)12 x (6)7 + 3944 x (8)11 x (6)7 + 12064 x (8)10 x (6)7 +29328 x (8)9 x (6)7 − x (8)8 x (6)7 − x (8)7 x (6)7 − x (8)6 x (6)7 − x (8)5 x (6)7 − x (8)4 x (6)7 + 29328 x (8)3 x (6)7 + 12064 x (8)2 x (6)7 + 3944 x (8) x (6)7 − x (6)7 +780 x (8)13 x (6)6 − x (8)12 x (6)6 − x (8)11 x (6)6 − x (8)10 x (6)6 + 34975 x (8)9 x (6)6 +5303 x (8)8 x (6)6 + 79136 x (8)7 x (6)6 + 79136 x (8)6 x (6)6 + 5303 x (8)5 x (6)6 + 34975 x (8)4 x (6)6 − x (8)3 x (6)6 − x (8)2 x (6)6 − x (8) x (6)6 + 780 x (6)6 − x (8)14 x (6)5 − x (8)13 x (6)5 + 3920 x (8)12 x (6)5 + 20576 x (8)11 x (6)5 + 7120 x (8)10 x (6)5 − x (8)9 x (6)5 +912 x (8)8 x (6)5 − x (8)7 x (6)5 + 912 x (8)6 x (6)5 − x (8)5 x (6)5 + 7120 x (8)4 x (6)5 +20576 x (8)3 x (6)5 + 3920 x (8)2 x (6)5 − x (8) x (6)5 − x (6)5 + 288 x (8)14 x (6)4 − x (8)13 x (6)4 − x (8)12 x (6)4 − x (8)11 x (6)4 − x (8)10 x (6)4 + 11730 x (8)9 x (6)4 +19986 x (8)8 x (6)4 + 19986 x (8)7 x (6)4 + 11730 x (8)6 x (6)4 − x (8)5 x (6)4 − x (8)4 x (6)4 − x (8)3 x (6)4 − x (8)2 x (6)4 + 288 x (8) x (6)4 − x (8)14 x (6)3 + 5904 x (8)13 x (6)3 +6832 x (8)12 x (6)3 − x (8)11 x (6)3 + 24160 x (8)10 x (6)3 − x (8)9 x (6)3 + 33056 x (8)8 x (6)3 − x (8)7 x (6)3 + 24160 x (8)6 x (6)3 − x (8)5 x (6)3 + 6832 x (8)4 x (6)3 + 5904 x (8)3 x (6)3 − x (8)2 x (6)3 + 216 x (8)15 x (6)2 − x (8)14 x (6)2 − x (8)13 x (6)2 + 5832 x (8)12 x (6)2 − x (8)11 x (6)2 + 13320 x (8)10 x (6)2 − x (8)9 x (6)2 − x (8)8 x (6)2 + 13320 x (8)7 x (6)2 − x (8)6 x (6)2 + 5832 x (8)5 x (6)2 − x (8)4 x (6)2 − x (8)3 x (6)2 + 216 x (8)2 x (6)2 +1728 x (8)14 x (6) − x (8)13 x (6) + 1728 x (8)12 x (6) − x (8)10 x (6) + 864 x (8)9 x (6) − x (8)8 x (6) + 1728 x (8)6 x (6) − x (8)5 x (6) + 1728 x (8)4 x (6) − x (8)15 − x (8)14 − x (8)13 − x (8)12 + 432 x (8)11 + 432 x (8)10 + 432 x (8)9 + 432 x (8)8 − x (8)7 − x (8)6 − x (8)5 − x (8)4 and g ( x (6) , x (8) ) = 48 x (6)14 − x (6)13 + 48 x (6)12 x (8)2 + 352 x (6)12 x (8) + 356 x (6)12 − x (6)11 x (8)2 − x (6)11 x (8) − x (6)11 − x (6)10 x (8)4 + 576 x (6)10 x (8)3 +1836 x (6)10 x (8)2 + 1536 x (6)10 x (8) − x (6)10 − x (6)9 x (8)4 − x (6)9 x (8)3 − x (6)9 x (8)2 − x (6)9 x (8) + 448 x (6)9 − x (6)8 x (8)6 − x (6)8 x (8)5 + 2936 x (6)8 x (8)4 +3520 x (6)8 x (8)3 + 1440 x (6)8 x (8)2 + 832 x (6)8 x (8) − x (6)8 + 576 x (6)7 x (8)6 − x (6)7 x (8)5 − x (6)7 x (8)4 − x (6)7 x (8)3 − x (6)7 x (8)2 + 832 x (6)7 x (8) + 448 x (6)7 + 48 x (6)6 x (8)8 − x (6)6 x (8)7 + 1704 x (6)6 x (8)6 + 2880 x (6)6 x (8)5 + 2096 x (6)6 x (8)4 + 944 x (6)6 x (8)3 +408 x (6)6 x (8)2 − x (6)6 x (8) − x (6)6 + 352 x (6)5 x (8)8 − x (6)5 x (8)7 − x (6)5 x (8)6 +192 x (6)5 x (8)5 − x (6)5 x (8)4 + 2320 x (6)5 x (8)3 − x (6)5 x (8)2 + 1536 x (6)5 x (8) − x (6)5 +48 x (6)4 x (8)10 − x (6)4 x (8)9 + 164 x (6)4 x (8)8 + 1088 x (6)4 x (8)7 + 876 x (6)4 x (8)6 − x (6)4 x (8)5 + 1633 x (6)4 x (8)4 − x (6)4 x (8)3 + 1036 x (6)4 x (8)2 − x (6)4 x (8) +356 x (6)4 + 224 x (6)3 x (8)9 − x (6)3 x (8)8 + 384 x (6)3 x (8)7 − x (6)3 x (8)6 + 1120 x (6)3 x (8)5 − x (6)3 x (8)4 + 1376 x (6)3 x (8)3 − x (6)3 x (8)2 + 352 x (6)3 x (8) − x (6)3 − x (6)2 x (8)10 +192 x (6)2 x (8)9 + 304 x (6)2 x (8)8 − x (6)2 x (8)7 + 1064 x (6)2 x (8)6 − x (6)2 x (8)5 +944 x (6)2 x (8)4 − x (6)2 x (8)3 + 316 x (6)2 x (8)2 + 48 x (6)2 − x (6) x (8)9 + 240 x (6) x (8)8 − x (6) x (8)7 + 192 x (6) x (8)6 + 192 x (6) x (8)5 − x (6) x (8)4 + 240 x (6) x (8)3 − x (6) x (8)2 +36 x (8)10 − x (8)6 + 36 x (8)2 . instein metrics on SU( N ) and complex Stiefel manifolds 41 We consider a polynomial ring R = Q [ z, x (6) , x (8) ] and an ideal I generated by { g , g ,zx (6) x (8) − } to find non zero solutions for the equations g = 0 , g = 0. We take alexicographic order > with z > x (6) > x (8) for a monomial ordering on R . Then by theaid of computer we see that a Gr¨obner basis for the ideal I contains the polynomial(2 x (8)2 + 5) (cid:0) x (8)6 − x (8)5 + 896 x (8)4 − x (8)3 + 896 x (8)2 − x (8) + 256 (cid:1) h ( x (8) )where h ( x (8) ) = x (8) 58 − x (8) 57 +5191100419195183352367435153408000000 x (8) 56 − x (8) 55 +135192627680445877856223796819845120000 x (8) 54 − x (8) 53 +1714176845340411043085057653911857971200 x (8) 52 − x (8) 51 +13463173946418384084581239673092042268160 x (8) 50 − x (8) 49 +73074478080162709767298450382256432673024 x (8) 48 − x (8) 47 +290859738683958218715335374171387713117587 x (8) 46 − x (8) 45 +878838272222037101767064678576069389361339 x (8) 44 − x (8) 43 +2054687728647824826465309756000191412577616 x (8) 42 − x (8) 41 +3739378101475752750734169569969105594053280 x (8) 40 − x (8) 39 +5241820414109727354087818276473527894123404 x (8) 38 − x (8) 37 +5424363186128466881300155939542628484559548 x (8) 36 − x (8) 35 +3591716347523905215063410201185994361815744 x (8) 34 − x (8) 33 +454714233323193018091497888909946789089488 x (8) 32 + 995461754392201718690335133719733779341968 x (8) 31 − x (8) 30 + 2356145206000070543512745518613265350730260 x (8) 29 − x (8) 28 + 995461754392201718690335133719733779341968 x (8) 27 +454714233323193018091497888909946789089488 x (8) 26 − x (8) 25 +3591716347523905215063410201185994361815744 x (8) 24 − x (8) 23 +5424363186128466881300155939542628484559548 x (8) 22 − x (8) 21 +5241820414109727354087818276473527894123404 x (8) 20 − x (8) 19 +3739378101475752750734169569969105594053280 x (8) 18 − x (8) 17 +2054687728647824826465309756000191412577616 x (8) 16 − x (8) 15 +878838272222037101767064678576069389361339 x (8) 14 − x (8) 13 +290859738683958218715335374171387713117587 x (8) 12 − x (8) 11 +73074478080162709767298450382256432673024 x (8) 10 − x (8) 9 +13463173946418384084581239673092042268160 x (8) 8 − x (8) 7 +1714176845340411043085057653911857971200 x (8) 6 − x (8) 5 +135192627680445877856223796819845120000 x (8) 4 − x (8) 3 +5191100419195183352367435153408000000 x (8) 2 − x (8) +40644078463495519177723084800000000 . We solve the equation h ( x (8) ) = 0 numerically and we obtain four positive solutionswhich are given approximately as x (8) ≈ . , x (8) ≈ . , x (8) ≈ . , x (8) ≈ . . Also the Gr¨obner basis of the ideal I contains the polynomial(2 x (8)2 + 5) (cid:0) x (8)6 − x (8)5 + 896 x (8)4 − x (8)3 + 896 x (8)2 − x (8) + 256 (cid:1) ( x (6) − w ( x (8) ))where w ( x (8) ) is a polynomial of x (8) with integer coefficients. We substitute the abovevalues of x (8) in the above equation and we obtain the solutions( x (6) , x (8) ) ≈ (1 . , . , ( x (6) , x (8) ) ≈ (0 . , . , ( x (6) , x (8) ) ≈ (0 . , . , ( x (6) , x (8) ) ≈ (1 . , . . We substitute the above solutions into u , u , u and u and we find three Einstein metricson V C ∼ = SU(6) / SU(2) which are given as follows:(1) ( u , u , v , v , x (6) , x (8) , x (7) ) ≈ (0 . , . , . , . , . , . , u , u , v , v , x (6) , x (8) , x (7) ) ≈ (0 . . , . , . , . , . , u , u , v , v , x (6) , x (8) , x (7) ) ≈ (0 . , . , . , . , . , . , u , u , v , v , x (6) , x (8) , x (7) ) ≈ (1 . , . , . , . , . , . , x (8) = 1 then c = 0, and from g ( x (6) , x (8) ) = 0 we obtain(2 x (8) − x (8) − x (8)8 − x (8)7 + 116 x (8)6 − x (8)5 + 384 x (8)4 − x (8)3 + 508 x (8)2 − x (8) + 219) = 0 . The solutions of the above equation are x (6) = 1 / , x (6) = 3 / , x (6) ≈ . , x (6) ≈ . . Then by substituting the above solutions into u , u , v and v we find two Einstein metricsof Jensen’s type and two more Einstein metrics on V C ∼ = SU(6) / SU(2), which are givenas follows:(1) ( u , u , v , v , x (6) , x (8) , x (7) ) = (1 / , / , / , / , / , , u , u , v , v , x (6) , x (8) , x (7) ) = (3 / , / , / , / , / , , u , u , v , v , x (6) , x (8) , x (7) ) ≈ (0 . , . , . , . , . , , u , u , v , v , x (6) , x (8) , x (7) ) ≈ (0 . , . , . , . , . , , Theorem 6.1. The complex Stiefel manifold V C = SU(4) / SU(2) admits two Ad(S(U(1) × U(1) × U(2)) -invariant Einstein metrics of the form (17) , which are of Jensen’stype. The complex Stiefel manifold V C = SU(5) / SU(2) admits four Ad(S(U(1) × U(2) × U(2)) -invariant Einstein metrics of the form (17) , two of which are of Jensen’s type. instein metrics on SU( N ) and complex Stiefel manifolds 43 The complex Stiefel manifold V C = SU(6) / SU(2) admits eight Ad(S(U(2) × U(2) × U(2)) -invariant Einstein metrics of the form (17) , two of which are of Jensen’s type. The Stiefel manifolds V m C m + n . In the next theorem we prove existence of Ein-stein metrics, which are not of Jensen’s type, on large families of complex Stiefel manifolds. Theorem 6.2. The complex Stiefel manifolds V m C m + n admit at least two Ad(S(U( m ) × U( m ) × U( n ))) -invariant Einstein metrics, which are not of Jensen’s type, for the followingvalues of m and n : m ≥ n ≥ m/ m = 6 , n ≥ m = 4 , n ≥ m = 2 , n ≥ .Proof. We consider the metric on V m C m + n with a = 1 , b = 0 , c = 0 and d = 1 (diagonalmetric). Then for ℓ = m we take from r = 0 that r = ( x (8) − x (7) )( x (8) + x (7) ) x (8)2 x (7)2 r n m + n = 0 . We set x (8) = x (7) = 1. Then system (38) reduces to the system( ⋆ ) f = ( u − u )( mu u − mx (6)2 + nu u x (6)2 ) = 0 ,f = mu − mu v x (6)2 + mx (6)2 + nu x (6)2 − nu v x (6)2 = 0 ,f = 2 mv x (6)2 − mv + nv x (6)2 − nv x (6)2 = 0 f = m u + m u + 2 m v − m x (6) + mnv x (6)2 − mnx (6)3 − u − u + 2 v = 0 ,f = − m nu + 2 m nu x (6)2 − m nu + 2 m nx (6)3 − m nx (6)2 +8 m nx (6) + 2 mn x (6)3 − mn x (6)2 + 2 mv x (6)2 + 2 nu − nu x (6)2 +2 nu + nv x (6)2 + nv x (6)2 − nv = 0 ,f = ( m − m + 1)( u − u ) = 0 . From f = 0 and f = 0 we have that u = u and by substituting into the system ( ⋆ )we obtain f = mu − mu v x (6)2 + mx (6)2 + nu x (6)2 − nu v x (6)2 = 0 ,f = 2 mv x (6)2 − mv + nv x (6)2 − nv x (6)2 = 0 ,f = 2 m u + 2 m v − m x (6) + mnv x (6)2 − mnx (6)3 − u + 2 v = 0 ,f = 2 m nu x (6)2 − m nu + 2 m nx (6)3 − m nx (6)2 + 8 m nx (6) + 2 mn x (6)3 − mn x (6)2 + 2 mv x (6)2 − nu x (6)2 + 4 nu + nv x (6)2 + nv x (6)2 − nv = 0 . We observe that the equations f = 0 , f = 0 and f = 0 are linear with respect to u , v , v . By solving the system of equations { f = 0 , f = 0 , f = 0 } we obtain that u = H ( x (6) ) ≡ − x (6) (2 m nx (6)2 − m nx (6) + 8 m n + m n x (6)4 − m n x (6)3 (40) +6 m n x (6)2 − m n x (6) + 4 m + mn x (6)4 − mn x (6)3 + 7 mnx (6)2 − mnx (6) + n x (6)4 − n x (6) ) / (cid:0) ( m − mnx (6)2 − mn + n x (6)4 − n x (6)2 − (cid:1) v = H ( x (6) ) ≡ (2 m + nx (6)2 )(6 mnx (6) − mn + n x (6)3 − n )(2 m + n )(2 mnx (6)2 − mn + n x (6)4 − n x (6)2 − v = H ( x (6) ) ≡ x (6)2 (6 mnx (6) − mn + n x (6)3 − n )2 mnx (6)2 − mn + n x (6)4 − n x (6)2 − . (42)From equations (41) and (42), we see that the value of v is positive if and only if thevalue of v is positive.Now we substitute the values u and v into f and we see that f = mx A ( x (6) ) B n,m ( x (6) )Γ( x (6) ) = 0 , where A ( x (6) ) = 2 mnx (6)2 − mnx (6) + 2 mn + n x (6)2 − n x (6) + 1 ,B n,m ( x (6) ) = 16 m nx (6)2 − m nx (6) + 40 m n + 20 m n x (6)4 − m n x (6)3 +68 m n x (6)2 − m n x (6) + 20 m + 8 m n x (6)6 − m n x (6)5 + 42 m n x (6)4 − m n x (6)3 + 56 m nx (6)2 − m nx (6) − m n + m n x (6)8 − m n x (6)7 +11 m n x (6)6 − m n x (6)5 + 41 m n x (6)4 − m n x (6)3 − m n x (6)2 − m n x (6) − m + mn x (6)8 − mn x (6)7 + 11 mn x (6)6 − mn x (6)5 − mn x (6)3 − mnx (6)2 +16 mnx (6) + 8 mn + n x (6)8 − n x (6)5 − n x (6)4 + 4 n x (6)2 + 8 n x (6) + 4 , Γ( x (6) ) = ( m − ( m + 1) (2 mnx (6)2 − mn + n x (6)4 − n x (6)2 − . Case of A ( x (6) ) = 0We consider the polynomial ring R = Q [ x (6) , u , v , v ] and the ideal I generated by thepolynomials { f , f , f , f , A ( x (6) ) , z x (6) u v v − } . We take a lexicographic ordering > with z > v > v > u > x (6) for a monomial ordering on R . Then, by the aid ofcomputer, we see that a Gr¨obner basis for the ideal I contains the polynomial A ( x (6) ) and F ( x (6) , u ) = ( m − m ( m + 1)(2 mn + 1)(32 m n + 144 m − m n + 336 m n − m − m n + 384 m n − mn + 168 mn − mn − n + 24 n + 1)( u − x (6) ) . Now we consider the lexicographic order > with z > v > u > v > x (6) . Then a Gr¨obnerbasis for the ideal I contains the polynomial A ( x (6) ) and the polynomial G ( x (6) , v ) = m (2 mn + 1)(32 m n + 144 m − m n + 336 m n − m n +384 m n − m − mn + 168 mn − mn − n + 24 n + 1)( v − x (6) ) . instein metrics on SU( N ) and complex Stiefel manifolds 45 Therefore from the equations F ( x (6) , u ) = 0 and G ( x (6) , v ) = 0 we have x (6) = u = v .We substitute into f i , i = 2 , , , g = − x (6)2 (2 mv x (6) − m + nv x (6) − nx (6)2 ) = 0 ,A ( x (6) ) = 2 mnx (6)2 − mnx (6) + 2 mn + n x (6)2 − n x (6) + 1 = 0 ,g = x (6) (2 mv x (6) − m + nv x (6) − nx (6)2 ) = 0 ,g = x (6) (4 m nx (6)2 − m nx (6) + 4 m n + 2 mn x (6)2 − mn x (6) +2 mv x (6) + nv x (6) − nx (6)2 ) = 0 . From A ( x (6) ) = 0 we have the solutions for x (6) as follows: x = −√ mn − mn + n − n + 2 mn + n mn + n x = √ mn − mn + n − n + 2 mn + n mn + n . We substitute into g = 0 and we find v = n (cid:16) − m + 2 p ( n − n ) (2 m + n ) − mn − n + 1 (cid:17) (2 m + n ) (cid:16)p ( n − n ) (2 m + n ) − mn − n (cid:17) v = n (cid:16) m + 2 p ( n − n ) (2 m + n ) + 4 mn + 2 n − (cid:17) (2 m + n ) (cid:16)p ( n − n ) (2 m + n ) + 2 mn + n (cid:17) . These metrics correspond to Jensen’s type metrics. Case of B n,m ( x (6) ) = 0For this case we observe that the value of B n,m ( x (6) ) at x (6) = 0 is written as B n,m (0) = 4 (cid:0) m − m + 1 (cid:1) (2 mn + 1) > . We also observe that the value of B n,m ( x (6) ) at x (6) = 3 / B n,m (3 / 2) = 4 m n + 9 m n m − m n m n − m n m n − m − mn 256 + 675 mn − mn − n 256 + 3 n . We see that B n,m (3 / 2) = (cid:18) m − m (cid:19) (cid:16) n − m (cid:17) + (cid:18) − m − (cid:19) (cid:16) n − m (cid:17) + ( − m + 512408 m − m )4096 (cid:16) n − m (cid:17) − m − m − × (cid:16) n − m (cid:17) + − m + 544050 m − m + 327688192 − m (cid:16) n − m (cid:17) . We also see that − m + 544050 m − m + 32768= − m − − m − − m − − B n,m (3 / < n ≥ m/ m ≥ m can be examined separately as follows.For m = 7 we see that B n, (3 / 2) = − (cid:0) n − + 605070( n − + 8694540( n − +53940288( n − + 126839488( n − 4) + 49348608 (cid:1) , and thus B n, (3 / < n ≥ 4. (In fact, we can show that B , ( x (6) ) > x (6) .)Similarly, we obtain the following table: m = 6 , B n,m (3 / < n ≥ B ,m ( x (6) ) > x (6) m = 4 , B n,m (3 / < n ≥ B ,m ( x (6) ) > x (6) m = 2 , B n,m (4 / < n ≥ B n,m ( x (6) ) at x (6) = 2 is B n,m (2) = 4(2 m n + m (8 n + 5) + 4 m n (2 n + 9) + m (84 n − 2) + m (80 n − n )+32 n − n + 1)and we observe that for all positive integers n, m we have B n,m (2) > x (6) = α , β ,where m ≥ n ≥ m/ < α < / 2, 3 / < β < m = 6 , n ≥ < α < / 2, 3 / < β < m = 4 , n ≥ < α < / 3, 4 / < β < m = 2 , n ≥ < α < / 3, 4 / < β < u , v and v , so we mustprove that these are positive. We take the resultant of the polynomials B n,m ( x (6) ) andthe numerator of the rational function u − H ( x (6) ) and we obtain the polynomial q ( u ) = (cid:16) m − n ( m + mn + 1)(2 mn + 1)( m + 2 mn + 2) × (cid:0) m n + 144 m − m n + 336 m n − m n + 384 m n − m − mn + 168 mn − mn − n + 24 n + 1 (cid:1)(cid:17) h ( u ) , where h ( u ) = ( m − ( m + 1) n (3 m + 2 n ) u − m − m + 1)(7 m − n (2 m + n ) × (3 m + 2 n ) u + n (3 m + 2 n ) (4 (cid:0) m − m + 24 (cid:1) n + m (cid:0) m − m + 404 (cid:1) n +(1219 m − m + 396 m + 16) n + m (cid:0) m − m − m + 24 (cid:1) ) u − n (2 m + n )(3 m + 2 n ) (cid:0) m − m − n + 2 m (1495 m − m + 316) n +(3235 m − m + 492 m + 96) n + m (cid:0) m − m − m + 144 (cid:1) (cid:1) u instein metrics on SU( N ) and complex Stiefel manifolds 47 + n (cid:0) (cid:0) m − (cid:1) n + 64 m (cid:0) m − m + 47 (cid:1) n + 8(10011 m − m + 1138 m +96) n + 4 m (cid:0) m − m + 1176 m + 1116 (cid:1) n + 2(50219 m − m − m + 4176 m + 48) n + m (32406 m − m − m + 5192 m + 288) n + m (3060 m − m − m + 172 m + 216) (cid:1) u − n (2 m + n ) (cid:0) m (37 m − m +5) n + 4 (cid:0) m − m + 26 m + 16 (cid:1) n + 4 m (1033 m − m − m + 94) n +(2131 m + 1265 m − m + 356 m + 48) n + m (351 m + 467 m − m − m +72) (cid:1) u + 4 n (cid:0) m (cid:0) m − m + 3 (cid:1) n + 4 m (cid:0) m − m − m + 60 (cid:1) n +4(1915 m + 750 m − m + 186 m + 24) n + m (4827 m + 6834 m − m − m +444) n + (1093 m + 2998 m + 713 m − m + 472 m + 16) n + m (36 m + 230 m +93 m − m + 19 m + 24) (cid:1) u − n (2 m + n )( m + 2 mn + 2)(9 m + 46 m + 21 m − m + 8 + 2 (cid:0) m − m + 7 (cid:1) m n + (cid:0) m + 88 m − m + 22 (cid:1) mn ) u + 4(5 m − m + 1)( m + mn + 1)(2 mn + 1)( m + 2 mn + 2) . We observe that the coefficients of the polynomial h ( u ) are positive for even degreeterms and negative for odd degree terms. Thus if the equation h ( u ) = 0 has realsolutions, then these are all positive. By the same way we take the resultant for thepolynomials B n,m ( x (6) ) and the numerator of the rational function v − H ( x (6) ) and weobtain the polynomial q ( v ) = − n (cid:0) − m n − m + 48 m (2 n − n + 8 m (22 n − n + 3)+6 m (16 n − n + n ) + 16 n − n − (cid:1) h ( v ) , where h ( v ) = (2 m + n ) (3 m + 2 n ) ( m + mn + 1)(2 mn + 1)( m + 2 mn + 2) v − n (2 m + n ) (3 m + 2 n ) ( m + 2 mn + 2) (cid:0) m + 68 m n + 128 m n + 41 m + 64 m n +172 m n + 132 m n + 54 m + 84 mn + 16 (cid:1) v + n (2 m + n ) (3 m + 2 n ) (cid:0) m +3060 m n + 22010 m n + 544 m + 64352 m n + 14245 m n + 92232 m n + 75004 m n +619 m + 68384 m n + 151044 m n + 20896 m n + 25088 m n + 139312 m n +75512 m n + 224 m + 3584 m n + 59328 m n + 94496 m n + 11436 m n + 9408 m n +48432 m n + 24064 m n + 192 m + 8640 m n + 15776 m n + 1968 m n + 3200 mn +1776 mn + 96 m + 384 n + 64 n (cid:1) v − n (2 m + n ) (3 m + 2 n ) (cid:0) m + 37908 m n +177352 m n + 9199 m + 401744 m n + 108038 m n + 485856 m n + 404452 m n +6988 m + 319744 m n + 667416 m n + 96400 m n + 107520 m n + 541664 m n +269168 m n + 2948 m + 14336 m n + 211520 m n + 293408 m n + 34312 m n +31360 m n + 138304 m n + 59424 m n + 2064 m + 23040 m n + 35264 m n +6080 m n + 6400 mn + 3968 mn + 576 m + 512 n + 384 n (cid:1) v + n (2 m + n ) × (cid:0) m + 520608 m n + 4322720 m n + 14040 m + 16938600 m n + 967712 m n +37781312 m n + 7795248 m n + 15289 m + 51970688 m n + 26578272 m n +524192 m n + 45604736 m n + 48503328 m n + 4250984 m n + 11480 m +25566080 m n + 52079872 m n + 12309456 m n + 241944 m n + 8849920 m n +33943424 m n + 17616272 m n + 1308512 m n + 1544 m + 1720320 m n +13190400 m n + 13936000 m n + 2594464 m n + 133056 m n + 143360 m n +2805760 m n + 6201088 m n + 2454848 m n + 368864 m n + 1264 m + 250880 m n +1450240 m n + 1192960 m n + 379328 m n + 21536 m n + 138240 m n + 283648 m n +172608 m n + 33664 m n + 864 m + 25600 mn + 31232 mn + 17856 mn + 1152 mn +1024 n + 3072 n + 384 n (cid:1) v − n (2 m + n ) (cid:0) m + 111360 m n + 594784 m n +6356 m + 1664096 m n + 125340 m n + 2765152 m n + 642880 m n + 5036 m +2863040 m n + 1579252 m n + 48298 m n + 1856640 m n + 2159232 m n +219000 m n + 2819 m + 729216 m n + 1718224 m n + 454480 m n + 26488 m n +157696 m n + 786208 m n + 472096 m n + 78116 m n + 408 m + 14336 m n +190464 m n + 254976 m n + 92712 m n + 9592 m n + 18816 m n + 67712 m n +49408 m n + 17184 m n + 388 m + 6912 m n + 10880 m n + 10272 m n + 1248 m n +640 mn + 1920 mn + 872 mn + 144 m + 128 n + 96 n (cid:1) v + 16 n (2 m + n ) (cid:0) m +96128 m n + 848320 m n + 2328 m + 3499648 m n + 56304 m n + 8416192 m n +500432 m n + 544 m + 12975200 m n + 1956172 m n + 38600 m n + 13339312 m n +4349536 m n + 180104 m n + 620 m + 9219520 m n + 5940704 m n + 543914 m n +14624 m n + 4215392 m n + 5109920 m n + 976656 m n + 109640 m n + 431 m +1217920 m n + 2760400 m n + 1006392 m n + 272120 m n + 1808 m n + 200704 m n +904704 m n + 597264 m n + 311304 m n + 19228 m n − m + 14336 m n +163840 m n + 199680 m n + 180544 m n + 36856 m n + 2872 m n + 12544 m n +34368 m n + 51584 m n + 27072 m n + 5776 m n + 5 m + 2304 m n + 5760 m n +8544 m n + 3616 m n + 400 m n + 960 mn + 720 mn + 404 mn + 24 m + 96 n + 16 n (cid:1) v − n (2 m + n ) (cid:0) m + 8080 m n + 44736 m n + 472 m + 130952 m n +2136 m n + 236456 m n + 7056 m n + 84 m + 280320 m n + 20276 m n +1968 m n + 221432 m n + 40056 m n + 7344 m n + 56 m + 114488 m n + 47760 m n +14298 m n + 350 m n + 36864 m n + 33170 m n + 15930 m n + 2028 m n + 38 m +6656 m n + 13056 m n + 9600 m n + 4107 m n + 111 m n + 512 m n + 2688 m n +2880 m n + 3312 m n + 384 m n − m + 224 m n + 336 m n + 1152 m n +312 m n + 108 m n + 144 m n + 72 m n + 120 m n + 2 m + 30 mn + 5 mn + 2 (cid:1) v +64 n (4 m + 32 m n + 32 m n + 4 m + 8 mn + 1)(64 m n + 768 m n + 3840 m n +4 m (cid:0) m − m + 12 (cid:1) n + 96 m (cid:0) m − m + 4 (cid:1) n + 96 m (197 m − m +12) n + 4 m (cid:0) m − m + 417 m − m + 3 (cid:1) n + 48 m (cid:0) m − m + 1 (cid:1) × (15 m − m + 1) n + 48 m (cid:0) m − m + 11 m − m + 1 (cid:1) n + (cid:0) m + 2 m − (cid:1) × (cid:0) m − m + 1 (cid:1) ) . instein metrics on SU( N ) and complex Stiefel manifolds 49 We observe that the coefficients of the polynomial h ( v ) are positive for even degreeterms and negative for odd degree terms. 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(ALM), 22, Int. Press,Somerville, MA, 2012.[WaZi] M. Wang and W. Ziller, Existence and non-existence of homogeneous Einstein metrics , Invent.Math. 84 (1986), no. 1, 177–194. University of Patras, Department of Mathematics, GR-26500 Rion, Greece E-mail address : [email protected] Osaka University, Department of Pure and Applied Mathematics, Graduate School ofInformation Science and Technology, Suita, Osaka 565-0871, Japan E-mail address : [email protected] University of Patras, Department of Mathematics, GR-26500 Rion, Greece E-mail address ::