Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds
aa r X i v : . [ m a t h . DG ] M a r NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMALHOMOGENEOUS EINSTEIN MANIFOLDS
ZAILI YAN AND SHAOQIANG DENG A bstract . It is an important problem in di ff erential geometry to find non-naturally reductive homoge-neous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for somecoset spaces of compact simple Lie groups. A new method to construct invariant non-naturally re-ductive Einstein metrics on normal homogeneous Einstein manifolds is presented. In particular, weshow that on the standard homogeneous Einstein manifolds, except for some special cases, there existplenty of such metrics. A further interesting result of this paper is that on some compact semisimpleLie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics whichare not product metrics. Mathematics Subject Classification (2010) : 53C25, 53C35, 53C30.
Key words : Einstein metrics, Riemannian submersion, naturally reductive metrics, standard ho-mogeneous Einstein manifolds
1. I ntroduction
The study of Einstein metrics has been one of the central problems in Riemannian geometry.Recall that a connected Riemannian manifold ( M , g ) is called Einstein if there exists a constant c such that Ric( g ) = cg , where Ric( g ) is the Ricci tesnor of ( M , g ). In general, the related problems inthis field are rather involved and di ffi cult. For example, till now a su ffi cient and necessary conditionfor a manifold to admit an Einstein metric is still unknown. As another remarkable open problem,it has been a long standing problem whether there is a nonstandard Einstein metric on the 4-sphere S , see for example [20]. This problem particularly reveals the fact that finding new examples ofEinstein metrics is essential in this topic.Although in the homogeneous case many beautiful results have been established, a completeclassification of homogeneous Einstein manifolds still seems to be unreachable. Even if in thecompact case, the classification has only been achieved for spheres, normal homogeneous spacesand naturally reductive metrics; see [9, 21]. See also [5, 6, 7, 22, 19] for some important andinteresting results on the existence (or non-existence) of homogeneous or inhomogeneous Einsteinmetrics on some special manifolds. Meanwhile, in the literature there are some excellent surveys ofthe development of this field, see for example [4, 16, 18].The method of Riemannian submersion is an important tool to construct new examples of Einsteinmetrics, and it has been applied to obtain many interesting existence results; see Chapter 9 of [4]and some results in [1, 2, 10]. Let G / H be a compact connected homogeneous space, and g = h + m a reductive decomposition of g , where g , h denote the Lie algebras of G and H respectively, and m is a subspace of g such that Ad( H )( m ) ⊂ m . Then there is a one-to-one correspondence between the G -invariant Riemannian metric on G / H and the Ad( H )-invariant inner product on m . Recall that aninvariant metric on G / H is called normal if the corresponding inner product on m is the restrictionof a bi-invariant inner product on g . In particular, let B denote the negative Killing form of g , and g B be the standard metric on G / H induced by B | m . Then g B is normal. The coset space G / H iscalled a standard homogeneous Einstein manifold if the standard metric g B is Einstein. In [21], M.Wang and W. Ziller obtained a classification of standard homogeneous Einstein manifolds G / H with G compact simple. Let K / H → G / H → G / K be a Riemannian submersion with totally geodesic Z. Yan is supported by NSFC (no. 11626134, 11401425) and K.C. Wong Magna Fund in Ningbo University. S. Deng is supported by NSFC (no. 11671212, 51535008) of China. AND SHAOQIANG DENG fibres. Assume that the standard metrics on G / H and G / K are Einstein, and there exists a constant c such that B ¯ k = cB | ¯ k , where ¯ K / ¯ H is the corresponding (almost) e ff ective quotient of K / H , and B ¯ k isthe negative Killing form of ¯ k = Lie( ¯ K ). Then besides the standard homogeneous Einstein metric,M. Wang and W. Ziller [21] showed that there exists another (non-naturally reductive) homogeneousEinstein metric on G / H except some special cases; see Table XI of [21] for a complete classificationof the Riemannian submersions K / H → G / H → G / K .This paper is a continuation of our previous work [23]. Inspired by the ideas of Riemanniansubmersion of M. Wang and W. Ziller [21, 22], we consider a family of invariant metrics on G / H depending on two real parameters associated to two Riemannian submersions K / H → G / H → G / K and L / H → G / H → G / L . More precisely, given a basic quadruple ( G , L , K , H ) (see Definition 3.1),where the Lie algebra g has a B -orthogonal decomposition g = l + p = k + u + p = h + n + u + p , m = n + u + p , (1.1)where n , u , p are the subspaces of k , l and g respectively, and k = Lie(K), l = Lie(L), we consider G -invariant metrics of the form h , i = g ( x , y ) = B | n + xB | u + yB | p , x , y ∈ R + , (1.2)on the homogeneous space G / H . Our goal is to find out under what conditions there exist newEinstein metrics, and if so, to classify them. It is clear that the invariant metric g (1 , y ) corresponds tothe Riemannian submersion L / H → G / H → G / L , and the invariant metric g ( x , x ) corresponds to theRiemannian submersion K / H → G / H → G / K .Our first main theorem is the following Theorem 1.1.
Let ( G , L , K , H ) be a basic quadruple with G compact simple. Suppose the standardmetrics on G / L, G / K, G / H are Einstein. If H , { e } , then ( G , L , K , H ) must be one of the quadruplesin Table A; If H = { e } , then ( G , L , K , H ) must be one of the quadruples in Table B. Next we study the Ricci curvature of g ( x , y ) , and obtain a su ffi cient and necessary condition for g ( x , y ) to be Einstein; see Proposition 4.4. Then we prove Theorem 1.2.
Let ( G , L , K , H ) be one of the basic quadruples in Table A and Table B. Then besidesthe three homogeneous Einstein metrics associated to the Riemannian submersions K / H → G / H → G / K and L / H → G / H → G / L, there always exists another Einstein metric on G / H of the formg ( x , y ) with x , , x , y, except for the following three cases: (1) Type A. 4 with n = m + , n = n = , k = m, m ∈ N + , namely, the quadruples (cid:18) sp (8 m (9 m + , (9 m + sp (8 m ) , m + sp (4 m ) , m + sp (2 m ) (cid:19) . (1.3)(2) Type A. 5: (cid:18) e , so (10) ⊕ R , so (8) ⊕ R , R (cid:19) . (1.4)(3) Type B. 3 with n = n = , k = , namely, the quadruple (cid:18) sp (4) , sp (2) , sp (1) , { e } (cid:19) . (1.5)As an application of Theorems 1.1 and 1.2, we obtain some new invariant Einstein metrics onsome flag manifolds G / T , where G = SU( n ) , SO(2n), or E , and T is a maximal compact connectedabelian subgroup of G . Moreover, Table B provides many new invariant Einstein metrics on compactsimple Lie groups which are not naturally reductive. Finally, we prove the following Theorem 1.3.
Let n = p l p l · · · p l s s be a positive integer, where the p i ’s are prime numbers andp i , p j , when i , j. Let H be a compact connected simple Lie group and G = H × H × · · · × H (ntimes). Then G admits at least ( l + l + · · · ( l s + − left invariant non-equivalent non-naturallyreductive Einstein metrics. ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 3
Table A: Standard quadruples with G simple, H , { e } Type A g h l k
Remarks1 su ( n n n k ) s ( n n n u ( k )) s ( n u ( n n k )) s ( n n u ( n k )) k ≥ , n i ≥ so ( n n n k ) n n n so ( k ) n so ( n n k ) n n so ( n k ) k ≥ , n i ≥ so ( n n k ) ⊕ li = h i n so ( n k ) n n so ( k ) k ≥ , n i ≥ sp ( n n n k ) n n n sp ( k ) n sp ( n n k ) n n sp ( n k ) k ≥ , n i ≥ e R so (10) ⊕ R so (8) ⊕ R e su (2) so (12) ⊕ su (2) so (8) ⊕ su (2)7 e R so (16) 2 so (8)8 e R so (16) 8 su (2)9 e R so (8) 8 su (2)10 e su (2) so (16) 2 so (8)11 e su (3) so (16) 2 so (8)Table B: Standard quadruples with G simple, H = { e } Type B g l k
Remarks1 so ( n n k ) n so ( n k ) n n so ( k ) k ≥ , n i ≥ so ( nk ) n so ( k ) ⊕ li = h i k ≥ , n ≥ sp ( n n k ) n sp ( n k ) n n sp ( k ) k ≥ , n i ≥ so (8) so (7) g f so (9) so (8)6 e su (3) 3 so (3)7 e su (8) so (8)8 e so (16) 2 so (8)9 e so (16) so (9)10 e so (16) 8 su (2)11 e so (16) 2 so (5)12 e so (16) 2 su (3)13 e su (9) so (9)14 e su (9) 2 su (3)15 e so (8) 8 su (2)16 e so (8) 2 su (3)17 e su (5) 2 so (5)18 e su (3) 4 so (3) Remark 1.4.
In this paper, nG means G × G × · · · × G ( n times).In Section 2, we survey some results on homogeneous Einstein metrics. In particular, we recallsome results of M. Wang and W. Ziller on naturally reductive and non-naturally reductive Einsteinmetrics. In Section 3, we give the definition and classification of standard quadruples. Section 4 isdevoted to the calculation of Ricci curvature of the related coset spaces. The main results of thispaper are proved in Section 5. To make the main proofs of the paper more concise, we collect somerepetitive case by case calculations in Section 5 as two appendixes.2. N aturally reductive and N on - naturally reductive E instein metrics In this section, we recall some results on naturally reductive and non-naturally reductive Einsteinmetrics, for details, see [3, 9].Let ( M , g ) be a connected Riemannian manifold and I ( M , g ) the full group of isometries of M .Given a Lie subgroup G of I ( M , g ), the Riemannian manifold ( M , g ) is said to be G -homogeneousif G acts transitively on M . For a G -homogeneous Riemannian manifold, we fix a point o ∈ M andidentify M with G / H , where H is the isotropy subgroup of G at o . Let g , h be the Lie algebras of ZAILI YAN AND SHAOQIANG DENG G and H respectively. Then g has a reductive decomposition g = h + m (direct sum of subspaces),where m is a subspace of g satisfying Ad( H )( m ) ⊂ m . Then one can identify m with T o M throughthe map X → ddt | t = (exp( tX ) · o ) . In this case, one can pull back the inner product g o on T o M to get an inner product on m , denotedby h , i . Given X ∈ g , we denote by X m the m -component of X . Then a homogeneous Riemannianmetric on M is said to be naturally reductive if there exists a transitive subgroup G and m as abovesuch that h [ Z , X ] m , Y i + h X , [ Z , Y ] m i = , ∀ X , Y , Z ∈ m . In [9], D’Atri and Ziller investigated naturally reductive metrics among the left invariant metricson compact Lie groups, and give a complete description of this type of metrics on simple Lie groups.Now we recall the main results of them.Let G be a compact connected semisimple Lie group, and H a closed subgroup of G . Denote by B the negative of the Killing form of g . Then B is an Ad( G )-invariant inner product on g . Let m bethe orthogonal complement of h with respect to B . Then we have g = h ⊕ m , Ad( H )( m ) ⊂ m . Let h = h ⊕ h ⊕ h ⊕ · · · ⊕ h p be the decomposition of h into ideals, where h is the center of h and h i ( i = , . . . , p ) are simpleideals of h . Let A | h be an arbitrary metric on h . Theorem 2.1 ([9]) . Keep the notation as above. Then a left invariant metric on G of the form h , i = xB | m + A | h + u B | h + · · · + u p B | h p , (2.6) where x , u , . . . , u p are positive real numbers, must be naturally reductive with respect to G × H,where G × H acts on G by ( g , h ) y = gyh − .Moveover, if a left invariant metric h , i on a compact simple Lie group G is naturally reductive,then there exists a closed subgroup H of G such that the metric h , i is given by the form (2.6) . Based on the above theorem, D’Atri and Ziller [9] obtained a large number of naturally reductiveEinstein metrics on compact simple Lie groups.Now we recall some results of Wang and Ziller. Let ( G / H , g B ) be a compact connected homoge-neous space with the reductive decomposition g = h + m . Denote by χ the isotropy representationof h on m . Let C χ, m be the Casimir operator defined by − P i (ad( X i )ad( X i )) | m , where { X i } is a B -orthonormal basis of h . Wang and Ziller obtained a su ffi cient and necessary condition for g B to beEinstein. Theorem 2.2 ([21]) . The standard homogeneous metric g B on G / H is Einstein if and only if thereexists a constant a such that C χ, m = a id , where id denotes the identity transformation. Based on this theorem and some deep results on representation theory, Wang and Ziller give acomplete classification of standard Einstein manifolds G / H for any compact simple Lie group G .Given a subalgebra h of g , one can consider the metric g t = B | h + tB | m , t > , as a left invariantmetric on G . Clearly, g t is naturally reductive. If t =
1, then g t is Einstein since it is bi-invariant. G.Jensen [13] first studied the Einstein metrics of the form g t , where t ,
1. Subsequently D’Atri andZiller proved the following
Theorem 2.3 ([9]) . Suppose h is not an ideal in g . Then there exists a unique t , with g t Einstein if and only if the standard metric on G / H is Einstein and there exists a constant c such thatB h = cB | h . Furthermore, in this case, we have t > and g t must be normal homogeneous withrespect to G × H. In particular, if h is abelian, then t = is the only real number such that g t is anEinstein metric. ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 5
Many non-naturally reductive Einstein metrics can be constructed by Riemannian submersions.We recall the following result, see [21] and page 255 of [4].
Theorem 2.4 ([4]) . Let F → M → B be a Riemannian submersion with totally geodesic fibres.Assume that the metrics on F, M and B are Einstein with Einstein constant r F , r M , r B respectively,and r F > . Furthermore, suppose M is not locally a Riemannian product of F and B. Then themetric g t obtained by scaling the metric on M in the direction of F by a factor t , , t > is Einsteinif and only if r F , r B . Applying this theorem to the homogeneous case, Wang and Ziller obtained a great number ofnon-naturally reductive Einstein metrics on standard homogeneous Einstein manifolds. In fact, in[21], they give a complete classification of the Riemannian submersions K / H → G / H → G / K ,where G is compact simple, such that the standard metrics on G / H , G / K are Einstein, and thereexists a constant c > B ¯ k = cB | ¯ k , where ¯ K / ¯ H is the corresponding e ff ective (almost)quotient of K / H , B and B ¯ k are the negative Killing forms of g = Lie( G ) and ¯ k = Lie( ¯ K ), respectively.Up to now, most known examples of Einstein metrics on compact simple Lie groups are naturallyreductive; see [3, 13, 15, 17]. The problem of finding left invariant Einstein metrics on compact Liegroups which are not naturally reductive is more di ffi cult, and is stressed by J.E. D’Atri and W. Zillerin [9]. In 1994, Mori initiated the study of this problem. Mori showed that there exists non-naturallyreductive Einstein metrics on the Lie group SU( n ) with n ≥ n ) with n ≥
11, Sp( n ) with n ≥
3, and theexceptional groups E , E and E . Recently, some non-naturally reductive Einstein metrics havebeen found on the compact simple Lie groups SU(3), SO(5), G and F ; see [8, 12]. We summarizethe above results as following Theorem 2.5. ([14, 3, 8, 12])
The compact simple Lie groups
SU( n ) ( n ≥ , SO( n ) ( n ≥ , Sp( n ) ( n ≥ , G , F , E , E and E admit non-naturally reductive Einstein metrics. Up to now, it has been an open problem whether there exists a left invariant non-naturally re-ductive Einstein metric on the compact simple Lie groups SU( n ), with n = ,
5, or SO( n ), with n = , , ,
10. 3. C lassification of standard quadruples
Let G be a compact semisimple connected Lie group, and H $ K $ L be three closed propersubgroups of G such that G acts e ff ectively on the coset space G / H . We denote by h , k , l , g the Liealgebras of H , K , L , G , respectively, and B h , B k , B l , B the negative of the Killing forms of h , k , l , g ,respectively. Then g has a B -orthogonal decomposition g = l + p = k + u + p = h + n + u + p , where n , u , p are the subspaces of k , l and g respectively. Denote m = n + u + p . Then it is easily seenthat [ h , n ] ⊂ n , [ h + n , u ] ⊂ u , [ h + n + u , p ] ⊂ p . Let χ h , n , χ h , u , χ h , p , χ k , u , χ k , p , χ l , p be the adjoint representation of h on n , u , p , k on u , p and l on p ,respectively, and C h , n , C h , u , C h , p , C k , u , C k , p , C l , p the corresponding Casimir operators defined by C h , n = − X (ad( h i )ad( h i )) | n , C h , u = − X (ad( h i )ad( h i )) | u , C h , p = − X (ad( h i )ad( h i )) | p , C k , u = − X (ad( k i )ad( k i )) | u , C k , p = − X (ad( k i )ad( k i )) | p , C l , p = − X (ad( l i )ad( l i )) | p , ZAILI YAN AND SHAOQIANG DENG where { h i } , { k i } and { l i } are B-orthonormal basis of h , k , l , respectively.Note that even if G is simple, L / H and K / H need not be e ff ective, so we denote by ¯ L / ¯ H and¯ K / ¯ H the corresponding (almost) e ff ective quotient. Definition 3.1.
Let the notation be as above. A quadruple ( G , L , K , H ) is called a basic quadruple ifit satisfies the following conditions:(1) G is compact and acting e ff ectively on G / H ;(2) There exist constants c , c > B ¯ l = c B | ¯ l , B ¯ k = c B | ¯ k .(3) There exist constants h n , h u , h p , k u , k p , l p such that C h , n = h n id, C h , u = h u id, C h , p = h p id, C k , u = k u id, C k , p = k p id, C l , p = l p id, where id denotes the identity transformation.A basic quadruple ( G , L , K , H ) is called standard if the standard metrics on G / H , G / K and G / L areEinstein.We first prove a simple but useful lemma. Lemma 3.2.
Let ( G , L , K , H ) be a basic quadruple. If the standard metrics on G / K and G / H areboth Einstein, then the constants h n , h u , h p , k u , k p , l p are given byh n = h u = h p = G / H X i (1 − α i ) dim H i , (3.7) k u = k p = G / K X i (1 − β i ) dim K i , (3.8) l p = G / L X i (1 − γ i ) dim L i , (3.9) where H i , K i , L i are the simple factors of H, K and L respectively, and B h i = α i B | h i , B k i = β i B | k i ,B l i = γ i B | l i .Moreover, if there exists constants c , c , c ∈ such that B l = c B | l , B k = c B | k , and B h = c B | h ,then we have k u = k p = dim K dim L l p , (3.10) h n = h u = h p = dim H dim L l p , (3.11) c = − dim G − dim L dim L l p , (3.12) c = − dim G − dim K dim L l p . (3.13) Proof.
First, (3.7), (3.8) and (3.9) can be easily calculated by taking the trace of C h , n , C k , u and C l , p .Then (3.10) and (3.11) follows from the facts that k p = dim K dim G / L (1 − c ) , h p = dim H dim G / L (1 − c ) . Finally, (3.12), (3.13) follows from (3.9) and (3.8). (cid:3)
Note that for a general compact simple subgroup H ⊂ G , there always exists a constant c suchthat B h = cB | h . The method of computing c is given in [9]. In particular, if h is a regular subalgebraof g (see [11]), then the constant c is given by c = B h ( α ′ m , α ′ m ) B ( α m , α m ) , (3.14)where α ′ m , α m are the maximal root of h and g , respectively.We must mention that, in this paper, most subalgebras are regular. Note also that the values of B ( α m , α m ) for compact simple Lie groups have been given in Table 3 of [9]. For convenience, wesummarize some of the results as the following table. ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 7
Table C g B ( α m , α m ) dim gsu ( n ) 4 n n − so ( n )( n ≥
4) 4( n − n ( n − sp ( n ) 4( n +
1) 2 n + n g
16 14 f
36 52 e
48 78 e
72 133 e
120 248In the case that H is semisimple, the following results will be useful. Proposition 3.3 ([9]) . Let G / H be a strongly isotropy irreducible space with H not simple. If thereexists a constant c such that B h = cB | h , then the Lie algebra pair ( g , h ) must be one of the followingsix cases: so ( k ) ⊕ so ( k ) ⊂ so (2 k ) , sp ( k ) ⊕ sp ( k ) ⊂ sp (2 k ) , su ( k ) ⊕ su ( k ) ⊂ su ( k ) , su (3) ⊕ su (3) ⊕ su (3) ⊂ e , sp (3) ⊕ g ⊂ e , sp (1) ⊕ so (4) ⊂ sp (4) . Theorem 3.4 ([21]) . Let G be a compact connected simple Lie group and H a semi-simple subgroupsuch that G / H is standard homogeneous Einstein but not strongly isotropy irreducible. Then thereexists a constant c such that B h = cB | h except for the following two cases: sp (1) ⊕ sp (5) ⊕ so (6) ⊂ so (26) , so (8) ⊕ su (2) ⊂ e . Proof of Theorem 1.1
Let ( G , L , K , H ) be a basic quadruple with G compact simple, such thatthe standard metrics on G / L , G / K , G / H are Einstein. Then K / H → G / H → G / K is one of thefibrations listed in Table XI of [21]. Combining Table IA and Table XI of [21], we can find out allthe subgroups L of G which contains K such that the standard metric on G / L is Einstein. ApplyingProposition 3.3 and Theorem 3.4, we can determine all the ones such that there exists a constant c with B ¯ l = cB | ¯ l among the above subgroups . The result is listed in Table A.On the other hand, if H = { e } , then L / K → G / K → G / L is also a fibration of Einstein metricslisted in Table XI of [21]. According to Definition 3.1, we only need to find out all the subgroups L such that K ⊂ L and there exists a constant c > B l = c B | l . Combining this with Proposition3.3 and Theorem 3.4, we get Table B. This completes the proof of Theorem 1.1.There are two types of the basic quadruples which need some more interpretation, namely, Type A. 3 so ( n n k ) ⊃ n so ( n k ) ⊃ n n so ( k ) ⊃ ⊕ li = h i , k ≥ , n i ≥ , and Type B. 2 so ( nk ) ⊃ n so ( k ) ⊃ ⊕ li = h i ⊃ { e } , k ≥ , n ≥ . These two types of basic quadruples are constructed through the following observation.Let G i / H i , i = , . . . , l ( l ≥
2) be a family of irreducible symmetric spaces such that either H i issimple or G i / H i is one of the types SO(2 k ) / SO( k ) × SO( k ) and Sp(2 k ) / Sp( k ) × Sp( k ). Then G / H = G / H × · · · × G s / H s is also a symmetric space. Let π be the isotropy representation of G / H . Thenit has been shown in [21] that SO(dim G / H ) /π ( H ) is a standard homogeneous Einstein manifold ifand only if dim G i dim H i is independent of i . In particular, if the standard metric on SO(dim G / H ) /π ( H ) isEinstein, then by Theorem 3.4, there exists a constant c such that B h = cB | h . Now in the above twotypes, ⊕ li = h i ⊂ n so ( k ) ⊂ so ( nk ) can be expressed as l M i h i = n − M s = t s + M i = t s + h i ⊂ n so ( k ) , = t < t < · · · < t n = l , (3.15) ZAILI YAN AND SHAOQIANG DENG where ⊕ t s + i = t s + h i ⊂ so ( k ) and SO( nk ) / ⊕ li H i , SO( k ) / ⊕ t s + i = t s + H i ( s = , , . . . , n −
1) are standardhomogeneous Einstein manifolds. Moreover, it is easy to check thatdim H i ≤ dim SO(dim G i / H i ) , ∀ ≤ i ≤ l . (3.16)Then it follows that dim ⊕ li = h i <
12 dim n so ( k ) . (3.17)This assertion will be useful in the following sections.4. R icci curvature of the invariant metrics As in Section 1, given a basic quadruple ( G , L , K , H ), we consider G -invariant metrics of the form h , i = g ( x , y ) = B | n + xB | u + yB | p , x , y ∈ R + , on the homogeneous space G / H . In this section, we mainly study the condition for g ( x , y ) to beEinstein.First, we have Lemma 4.1.
Let ( G , L , K , { e } ) be a basic quadruple with G simple, then the left invariant metricg ( x , y ) on G is naturally reductive with respect to G × N for some closed subgroup N of G, if and onlyif at least one of the following holds: (1) x = y .(2) x = k is an ideal in l . Proof.
It follows from Theorem 2.1 and the fact that k and l are subalgebras of g . (cid:3) The following result is obvious, so we omit the proof.
Proposition 4.2.
Let ( G , L , K , H ) and ( G ′ , L ′ , K ′ , H ′ ) be two basic quadruples, and g ( x , y ) , g ′ ( x ′ , y ′ ) be two invariant metrics on G / H and G ′ / H ′ defined as above, respectively. Then ( G / H , g ( x , y ) ) isisometric to ( G ′ / H ′ , g ′ ( x ′ , y ′ ) ) if and only if there exists an isomorphism ϕ : G → G ′ , such that ϕ ( L ) = L ′ , ϕ ( K ) = K ′ , ϕ ( H ) = H ′ , and x = x ′ , y = y ′ . Now we compute the Ricci curvature of g ( x , y ) . It is well known that the sectional curvature andRicci curvature of a homogeneous Riemannian manifold can be explicitly expressed using the innerproduct on the tangent space and the Lie algebraic structure. In the literature, there are severalversions of the formulas. Here we will use the formula of the Ricci curvature of an invariant metricon a homogeneous compact Riemannian manifold given by [4] (see (7.38) of [4]):Ric( X , Y ) = B ( X , Y ) − X i h [ X , X i ] m , [ Y , X i ] m i + X i , j h [ X i , X j ] m , X ih [ X i , X j ] m , Y i , (4.18)where { X i } is an orthonormal basis of m with respect to the restriction of the inner product h , i to m .Now we have Lemma 4.3.
Let ( G , L , K , H ) be a basic quadruple. Then the Ricci curvature of ( G / H , g ( x , y ) ) is givenas follows: (1) Ric( n , u ) = Ric( n , p ) = Ric( u , p ) = n , n ) = (cid:2) c + h n + x ( c − c ) + y (1 − c ) (cid:3) B ( n , n ),(3) Ric( u , u ) = (cid:2) k u + c − x ( k u − h u ) + x y (1 − c ) (cid:3) B ( u , u ),(4) Ric( p , p ) = (cid:2) + l p − y ( k p − h p ) − x y ( l p − k p ) (cid:3) B ( p , p ),where n ∈ n , u ∈ u , p ∈ p . ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 9
Proof.
The formulas will be proved through a direct computation. Let { h i } ⊂ h , { n i } ⊂ n , { u i } ⊂ u , { p i } ⊂ p be a B-orthonormal basis of g . Then { n i } ∪ { u i √ x } ∪ { p i √ y } is an orthonormal basis of m withrespect to g ( x , y ) . Given n ∈ n , u ∈ u , and p ∈ p , by (4.18), one hasRic( u , p ) = B ( u , p ) − X i h [ u , n i ] m , [ p , n i ] m i + X i , j h u , [ p i √ y , p j √ y ] m ih u , [ p i √ y , p j √ y ] m i− X i h [ u , u i √ x ] m , [ p , u i √ x ] m i − X i h [ u , p i √ y ] m , [ p , p i √ y ] m i = − y X i B ([ u , p i √ y ] , [ p , p i √ y ]) + xy X i , j B ( u , [ p i √ y , p j √ y ]) B ( p , [ p i √ y , p j √ y ]) = , Similarly, Ric( n , p ) =
0. On the other hand, we haveRic( n , u ) = B ( n , u ) − X i h [ n , n i ] m , [ u , n i ] m i + X i , j h n , [ p i √ y , p j √ y ] m ih u , [ p i √ y , p j √ y ] m i− X i h [ n , u i √ x ] m , [ u , u i √ x ] m i − X i h [ n , p i √ y ] m , [ u , p i √ y ] m i + X i , j h n , [ n i , n j ] m ih u , [ n i , n j ] m i + X i , j h n , [ u i √ x , u j √ x ] m ih u , [ u i √ x , u j √ x ] m i = − X i B ([ n , u i ] , [ u , u i ]) − X i B ([ n , p i ] , [ u , p i ]) + x X i , j B ( n , [ u i , u j ]) B ( u , [ u i , u j ]) + x y X i , j B ( n , [ p i , p j ]) B ( u , [ p i , p j ]) = x X i , j B ([ n , u i ] , u j ) B ([ u , u i ] , u j ) + x y X i , j B ([ n , p i ] , p j ) B ([ u , p i ] , p j ) = x X i B ([ n , u i ] , [ u , u i ]) + x y X i B ([ n , p i ] , [ u , p i ]) = x B ¯ l ( n , u ) + x y [ B ( n , u ) − B ¯ l ( n , u )] = , which proves the first assertion.Now, a direct calculation shows thatRic( n , n ) = B ( n , n ) − X i h [ n , n i ] m , [ n , n i ] m i − X i h [ n , u i √ x ] m , [ n , u i √ x ] m i− X i h [ n , p i √ y ] m , [ n , p i √ y ] m i + X i , j h n , [ n i , n j ] m i + X i , j h n , [ u i √ x , u j √ x ] m i + X i , j h n , [ p i √ y , p j √ y ] m i = X i , j B ([ n , n i ] , h j ) + X i , j B ( n , [ n i , n j ]) + x X i B ([ n , u i ] , [ n , u i ]) + y X i B ([ n , p i ] , [ n , p i ]) . AND SHAOQIANG DENG Since X i , j B ( n , [ n i , n j ]) = X i B ([ n , n i ] , [ n , n i ]) − X i , j B ( n j , [ n , h i ]) = B ¯ k ( n , n ) − B ( C h , n ( n ) , n ) , we have Ric( n , n ) = X i B ([ n , h i ] , [ n , h i ]) +
14 [ B ¯ k ( n , n ) − B ( C h , n ( n ) , n )] + x [ B ¯ l ( n , n ) − B ¯ k ( n , n )] + y [ B ( n , n ) − B ¯ l ( n , n )] = (cid:2) c + h n + x ( c − c ) + y (1 − c ) (cid:3) B ( n , n ) . (4.19)Furthermore, using a similar argument, we getRic( u , u ) = B ( u , u ) − X i h [ u , n i ] m , [ u , n i ] m i − X i h [ u , u i √ x ] m , [ u , u i √ x ] m i− X i h [ u , p i √ y ] m , [ u , p i √ y ] m i + X i , j h u , [ n i , u j √ x ] m i + X i , j h u , [ u i √ x , u j √ x ] m i + X i , j h u , [ p i √ y , p j √ y ] m i = B ( u , u ) − x X i B ([ u , n i ] , [ u , n i ]) − x X i h [ u , u i ] m , [ u , u i ] m i− X i B ([ u , p i ] , [ u , p i ]) + x X i , j B ( u , [ n i , u j ]) + X i , j B ( u , [ u i , u j ]) + x y X i , j B ( u , [ p i , p j ]) . Next, since X i , j B ( u , [ u i , u j ]) = X i , j B ([ u , u i ] , u j ) = X i B ([ u , u i ] , [ u , u i ]) − X i , j B ([ u , u i ] , n j ) − X i , j B ([ u , u i ] , h j ) = B ¯ l ( u , u ) − X i B ([ u , n i ] , [ u , n i ]) − X i B ([ u , h i ] , [ u , h i ]) = ( c − k u ) B ( u , u ) , ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 11 we have X i h [ u , u i ] m , [ u , u i ] m i = X i , j (cid:2) h [ u , u i ] m , u j √ x i + h [ u , u i ] m , n j i (cid:3) = X i , j (cid:2) xB ([ u , u i ] , u j ) + B ([ u , u i ] , n j ) (cid:3) = X i , j (cid:2) xB ( u , [ u i , u j ]) (cid:3) + X i B ([ u , n i ] , [ u , n i ]) = ( c − k u ) xB ( u , u ) + ( k u − h u ) B ( u , u ) . Therefore we haveRic( u , u ) = B ( u , u ) − x (cid:2) ( c − k u ) xB ( u , u ) + ( k u − h u ) B ( u , u ) (cid:3) −
12 (1 − c ) B ( u , u ) +
14 ( c − k u ) B ( u , u ) + x y (1 − c ) B ( u , u ) = (cid:2) k u + c − x ( k u − h u ) + x y (1 − c ) (cid:3) B ( u , u ) , (4.20)and Ric( p , p ) = B ( p , p ) − X i h [ p , n i ] m , [ p , n i ] m i− X i h [ p , u i √ x ] m , [ p , u i √ x ] m i − X i h [ p , p i √ y ] m , [ p , p i √ y ] m i + X i , j h p , [ n i , p j √ y ] m i + X i , j h p , [ u i √ x , p j √ y ] m i + X i , j h p , [ p i √ y , p j √ y ] m i = B ( p , p ) − y X i B ([ p , n i ] , [ p , n i ]) − y x X i B ([ p , u i ] , [ p , u i ]) − y X i h [ p , p i ] m , [ p , p i ] m i + y X i B ([ p , n i ] , [ p , n i ]) + y x X i B ([ p , u i ] , [ p , u i ]) + X i , j B ( p , [ p i , p j ]) = B ( p , p ) − y X i h [ p , p i ] m , [ p , p i ] m i +
14 (1 − l p ) B ( p , p ) . Finally, since X i h [ p , p i ] m , [ p , p i ] m i = X i , j (cid:2) h [ p , p i ] m , p j √ y i + h [ p , p i ] m , u j √ x i + h [ p , p i ] m , n j i (cid:3) = X i , j (cid:2) yB ([ p , p i ] , p j ) + x B ([ p , p i ] , u j √ x ) + B ([ p , p i ] , n j ) (cid:3) = (1 − l p ) yB ( p , p ) + x X i B ([ p , u i ] , [ p , u i ]) + X i B ([ p , n i ] , [ p , n i ]) = (1 − l p ) yB ( p , p ) + x ( l p − k p ) B ( p , p ) + ( k p − h p ) B ( p , p ) , AND SHAOQIANG DENG we have Ric( p , p ) = B ( p , p ) − y (cid:2) (1 − l p ) y + x ( l p − k p ) + ( k p − h p ) (cid:3) B ( p , p ) +
14 (1 − l p ) B ( p , p ) = (cid:2) + l p − y ( k p − h p ) − x y ( l p − k p ) (cid:3) B ( p , p ) . (4.21)This completes the proof of the lemma. (cid:3) Proposition 4.4.
Let ( G , L , K , H ) be a basic quadruple. Then the following two assertions hold: (1) If h n , h u , then the invariant metric g ( x , y ) on G / H is Einstein if and only if ( x , y ) satisfies thefollowing equations: − c + l p ) x ( x − ∆ ( x ) = (cid:16)(cid:2)
12 ( k p − h p ) + − c + x l p − k p ) (cid:3) ∆ ( x ) + − c x − (cid:2) ( 14 c + h n ) x + c − c (cid:3)(cid:17) , (4.22) y = x s (1 − c )( x − ∆ ( x ) , (4.23) where ∆ ( x ) = ( 14 c + h n ) x − ( 12 k u + c ) x +
12 ( k u − h u ) + c − c . (4.24)(2) If h n = h u , then invariant metric g (1 , y ) on G / H is Einstein if and only if y satisfies thefollowing equation: ( c + h n ) y − ( 14 + l p ) y +
12 ( 12 + l p − c − h p ) = . (4.25) Moreover, in this case, the invariant metric g ( x , y ) ( x , on G / H is Einstein if and only if ( x , y ) satisfies the conditions: − c + l p ) x δ ( x ) = (cid:16)(cid:2)
12 ( k p − h p ) + − c + x l p − k p ) (cid:3) δ ( x ) + − c (cid:2) ( 14 c + h n ) x + c − c (cid:3)(cid:17) , (4.26) and y = x s − c δ ( x ) , (4.27) where δ ( x ) = ( 14 c + h n ) x −
12 ( k u − h u ) − c − c . (4.28) Proof.
By Lemma 4.3, the invariant metric g ( x , y ) on G / H is Einstein with Ricci constant λ if andonly if ( x , y ) satisfies the following equations:14 c + h n + x ( c − c ) + y (1 − c ) = λ, (4.29)12 k u + c − x ( k u − h u ) + x y (1 − c ) = λ x , (4.30)14 + l p − y ( k p − h p ) − x y ( l p − k p ) = λ y . (4.31)Now assume ( x = , y ) is a solution of equations (4.29), (4.30) and (4.31). Then one has14 c + h n +
14 ( c − c ) + y (1 − c ) = λ = k u + c −
12 ( k u − h u ) + y (1 − c ) , ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 13 hence h n = h u . (4.32)Moreover, plugging (4.29) into (4.31), we have14 + l p − y ( k p − h p ) − y ( l p − k p ) = [ 12 h n + c + y (1 − c )] y . (4.33)Thus ( c + h n ) y − ( 14 + l p ) y +
12 ( 12 + l p − c − h p ) = . (4.34)Now assume ( x , y ) ( x ,
1) is a solution of equations (4.29), (4.30) and (4.31). Then plugging(4.29) into (4.30), we get12 k u + c − x ( k u − h u ) + x y (1 − c ) = (cid:2) c + h n + x ( c − c ) + y (1 − c ) (cid:3) x . Therefore we have y ∆ ( x ) = − c x ( x − , (4.35)and y = x s (1 − c )( x − ∆ ( x ) , (4.36)where ∆ ( x ) = ( 14 c + h n ) x − ( 12 k u + c ) x +
12 ( k u − h u ) + c − c . (4.37)Now plugging (4.29) into (4.31), we have14 + l p − y ( k p − h p ) − x y ( l p − k p ) = (cid:2) c + h n + x ( c − c ) + y (1 − c ) (cid:3) y . (4.38)Then we have( 14 + l p ) y −
12 ( k p − h p ) − x l p − k p ) = (cid:2) c + h n + x ( c − c ) + y (1 − c ) (cid:3) y , and ( 14 + l p ) y ∆ ( x ) = (cid:2) ( 14 c + h n + c − c x ) y + − c +
12 ( k p − h p ) + x l p − k p ) (cid:3) ∆ ( x ) . (4.39)Now substituting (4.35) into (4.39), we obtain1 − c + l p ) x ( x − ∆ ( x ) = (cid:16)(cid:2)
12 ( k p − h p ) + − c + x l p − k p ) (cid:3) ∆ ( x ) + − c x − (cid:2) ( 14 c + h n ) x + c − c (cid:3)(cid:17) . (4.40)Notice that if h n = h u , then ∆ ( x ) = ( x − δ ( x ), where δ ( x ) = ( 14 c + h n ) x −
12 ( k u − h u ) − c − c . (4.41)Thus, in this case, equation (4.22) can be divided by ( x − , which leads to the following equation:1 − c + l p ) x δ ( x ) = (cid:16)(cid:2)
12 ( k p − h p ) + − c + x l p − k p ) (cid:3) δ ( x ) + − c (cid:2) ( 14 c + h n ) x + c − c (cid:3)(cid:17) . (4.42)Conversely, if x = z , ∆ ( z ) , (cid:3) AND SHAOQIANG DENG Notice that the equation (4.22) is an equation of order six in one variable, hence it might admit noreal solutions. Moreover, if the isotropy representation of H on T eH ( G / H ) decomposes into exactlythree non-equivalent irreducible summands, then the G -invariant metrics must be of the form g ( x , y ) up to scaling. These facts may provide us with a method to obtain new homogeneous spaces whichadmit no G -invariant Einstein metrics. However, we will not deal with this problem here.5. E instein metrics on normal homogeneous E instein manifolds To prove the main theorem of this paper, we need the following result.
Proposition 5.1.
Keep the notation as above. Let ( G , L , K , H ) be a standard quadruple listed inTable A and Table B, and denote ω = + l p − k p − c , ω = k p + c − c − h p . Then we have ω ≥ , ω ≥ except for the following cases: : (a) Type A. 4. n = n = . sp (4 n k ) ⊃ sp (2 n k ) ⊃ sp ( n k ) ⊃ n sp ( k ) , k ≥ , n ≥ .ω = − n k + , ω = n k − k − n k + . Or n = n = . sp (4 n k ) ⊃ n sp (4 k ) ⊃ n sp (2 k ) ⊃ n sp ( k ) , k ≥ , n ≥ .ω = n k − k − n k + , ω = − n k + . : (b) Type A. 5. e ⊃ so (10) ⊕ R ⊃ so (8) ⊕ R ⊃ R .ω = − , ω = . : (c) Type A. 6. e ⊃ so (12) ⊕ su (2) ⊃ so (8) ⊕ su (2) ⊃ su (2) .ω = − , ω = − . : (d) Type B. 3. n = n = . sp (4 k ) ⊃ sp (2 k ) ⊃ sp ( k ) ⊃ { e } , k ≥ .ω = − k + , ω = k k + . : (e) Type B. 4. so (8) ⊃ so (7) ⊃ g ⊃ { e } .ω = − , ω = . : (f) Type B. 5. f ⊃ so (9) ⊃ so (8) ⊃ { e } .ω = − , ω = . Proof.
Let ( G , L , K , H ) be one of the standard quadruples listed in Table A and Table B which is notof Type A 1, A 5, or A 6. Then there exist constants c , c , c such that B l = c B | l , B k = c B | k , and B h = c B | h . By Lemma 3.2, one has l p = dim L dim G / L (1 − c ) , k p = dim K dim L l p , h p = dim H dim L l p . (5.43) ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 15
Therefore we have ω = + l p − k p − c = + l p − dim K dim L l p −
12 (1 − dim G − dim L dim L l p ) = − + dim G − K L l p , (5.44)and ω = k p + c − c − h p = K dim L l p + (1 − dim G − dim L dim L l p ) − H dim L l p − − dim G − dim K dim L l p ) = dim G + dim L − H dim L l p − . (5.45)In particular, if G / L is also a symmetric space, then l p = , and hence we have ω =
14 dim L (dim G − dim L − K ) , (5.46) ω =
12 dim L (dim G − dim L − H ) . (5.47)Moreover, it is obvious that, if H = { e } , then ω > G / L = E / SO(16) is symmetric, and we have l p = . Then by (5.46) and (5.47), we have ω =
14 dim L (dim G − dim L − K ) ≥ − − × > , and ω =
12 dim L (dim G − dim L − H ) ≥ − − × > , where we have used the facts that dim G = dim e = L = dim so (16) = K ≤ dim 2 so (8) =
56, and dim H ≤ dim 8 su (2) = (cid:3) Now we can prove the main theorem of this section.
Theorem 5.2.
Let ( G , L , K , H ) be a basic quadruple, such that k u = k p , h u = h p . Then G / H admitsat least one invariant Einstein metric of the form g ( x , y ) , with x , , x , y, if one of the followingconditions holds:(1) h n < h u ; (2) h n = h u , and G is simple. That is, ( G , L , K , H ) is one of the standard quadruples listed inTable A and Table B which is not the following ones: AND SHAOQIANG DENG (I) Type A. 4. n = m + , n = n = , k = m, m ∈ N + . sp (8 m (9 m + ⊃ (9 m + sp (8 m ) ⊃ m + sp (4 m ) ⊃ m + sp (2 m ) . (II) Type A. 5. e ⊃ so (10) ⊕ R ⊃ so (8) ⊕ R ⊃ R . (III) Type B. 3. n = n = , k = . sp (4) ⊃ sp (2) ⊃ sp (1) ⊃ { e } . Proof.
Keep the notation as above. Suppose ∆ ( x ) − − c x − = M ( x − α )( x − β ) , (5.48)where M = c + h n >
0, and α, β ∈ C . Then it follows easily from (4.23) that x = y if and only if x = α , or x = β .Now plugging (5.48) into the right side of (4.22), one has (cid:2)
12 ( k p − h p ) + − c + x l p − k p ) (cid:3) ∆ ( x ) + − c x − (cid:2) ( 14 c + h n ) x + c − c (cid:3) = (cid:2)
12 ( k p − h p ) + − c + x l p − k p ) (cid:3) M ( x − α )( x − β ) + − c x − (cid:2)
12 ( k p − h p ) + − c + x l p − k p ) + ( 14 c + h n ) x + c − c (cid:3) = (cid:2)
12 ( k p − h p ) + − c + x l p − k p ) (cid:3) M ( x − α )( x − β ) + − c x − (cid:2) M ( x − α )( x − β ) + ( 14 + l p ) x (cid:3) = M ( x − α )( x − β ) η ( x ) + − c + l p )( x − x , (5.49)where η ( x ) = x + l p − k p − c +
12 ( k p − h p ) . (5.50)Then equation (4.22) can be simplified as:1 − c + l p ) x ( x − ∆ ( x ) = (cid:2) M ( x − α )( x − β ) η ( x ) + − c + l p )( x − x (cid:3) . This implies that M ( x − α )( x − β ) (cid:2) M ( x − α )( x − β ) η ( x ) + − c + l p ) x ( x − η ( x ) −
12 ( 14 + l p ) x ) (cid:3) = . (5.51)Thus to prove the theorem, it is su ffi cient to show that the equation (of x ) f ( x ) = M ( x − α )( x − β ) η ( x ) + − c + l p ) x ( x − η ( x ) −
12 ( 14 + l p ) x ] = , (5.52)admits a real positive solution x , α or β . Now we prove this assertion case by case. Case 1 h n < h u . In this case, ∆ (1) = M (1 − α )(1 − β ) = ( h n − h u ) <
0, so we can assume 0 < α < < β withoutlosing generality. Notice that f (0) = M αβ ( k p − h p ) > , f (1) =
14 ( 12 + l p − h p − c ∆ (1) < , and lim x → + ∞ f ( x ) = + ∞ . Thus there exist real numbers z , z ∈ R such that 0 < z < < z , and f ( z ) = f ( z ) =
0. Noticealso that the equation η ( x ) − ( + l p ) x = f ( α ) and f ( β ) can not be ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 17 equal to zero at the same time. Hence we have either z , α or z , β , which proves the theorem inthis case. Case 2 h n = h u , and G is simple.In this case, the standard metrics on G / L , G / K and G / H are Einstein, and these spaces have beenclassified in Section 3, which are listed in Table A and Table B.Clearly, x = y = x = α, β is equal to 1. Without losing generality, we assume α =
1. Then by (5.48), we have β = M [ ( k u − h u ) + − c ] = k p − h p + − c c + h p . Then can easily deduce thefact β > f ( x ) = M ( x − β ) η ( x ) + − c + l p ) x [ η ( x ) −
12 ( 14 + l p ) x ] = M ( x − β ) η ( x ) + − c + l p ) x ( ω x + k p − h p ) , (5.53)where ω = + l p − k p − c .Clearly, f ( x ) = ¯ f ( x )( x − f ( x ) = x , , β except for the three cases (I), (II) and (III).First, by the facts that ¯ f (0) = − M β ( k p − h p ) < , and lim x → + ∞ ¯ f ( x ) = + ∞ , there exists a unique positive number z ∈ R such that ¯ f ( z ) = ω ≥
0, then ¯ f ( β ) = − c β ( + l p )( ω β + k p − h p ) >
0, and so z < β .By Proposition 4.4, x = z is a solution of equation (4.26). Then we have z > δ , where δ = k u − h u ) + c − c c + h n is a solution of the equation δ ( x ) =
0. Now δ ≥ ω ≥
0, where ω = k p + c − c − h p . In summarizing, we have the following facts: ( z < β if ω ≥ , z > ω ≥ . (5.54)Notice that (5.54) is also valid when G is only semisimple.Now by Proposition 5.1, for any standard quadruple ( G , L , K , H ) listed in Table A and Table B,there exists an invariant Einstein metric on G / H of the form g ( x , y ) with x , x , y , except for thecases (a)-(f) therein. We will deal with the cases of (a)-(f) listed in Proposition 5.1 in Appendix B.Now the proof of the theorem is completed. (cid:3) It is clear that Theorem 1.2 is the second case of this Theorem.In particular, from Table A, we obtain some new invariant Einstein metrics on some flag manifolds G / T , where G = SU( n ) , SO(2n), or E , and T is a maximal compact connected abelian subgroup of G . These Einstein metrics on flag manifolds are clearly neither Kahlerian [4] nor naturally reductive.We should also mention that, by the above result, the standard quadruple (cid:18) Sp(8 m (9 m + , (9 m + m ) , m + m ) , m + m ) (cid:19) , m ∈ N + , doesn’t correspond to any new invariant Einstein metric on the homogeneous spaceSp(8 m (9 m + / m + m ) . However, we do find at least two new invariant Einstein metrics on the space associated to thestandard quadruples (cid:18)
Sp(8 m (9 m + , m (9 m + , m + m ) , m + m ) (cid:19) AND SHAOQIANG DENG and (cid:18) Sp(8 m (9 m + , m (9 m + , m (9 m + , m + m ) (cid:19) . Finally, we give some new examples of homogeneous Einstein manifolds G / H with G semisim-ple. Theorem 5.3.
Let G = n n n H, L = n n H, K = n H with H compact simple, where n , n , n ∈ N ,and n i ≥ . Let H be embedded into G by the map h ( h , h , · · · , h ) . Then (1) ( G , L , K , H ) and ( G , L , K , { e } ) are both standard quadruples, and the standard metrics onG / L, G / K, G / H are Einstein. (2) G / H admits an invariant non-naturally reductive Einstein metric of the form g ( x , y ) withx , , x , y, associated to the quadruple ( G , L , K , H ) .Proof. The first assertion follows from Proposition 5.5 of [21]. For the basic quadruple ( G , L , K , H ),one has c = n , c = n n , l p = dim L dim G / L (1 − c ) = n , k p = n n , h p = n n n . Then we have ω = + l p − k p − c = − n n ≥ ,ω = k p + c − c − h p = n − n n n ≥ . Now the second assertion follows from (5.54) of Theorem 5.2. (cid:3)
Now we can prove
Theorem 5.4.
Let H be a compact simple Lie group, and G = H × H ×· · ·× H (n times, n ≥ ), wheren = p l p l · · · p l s s , with p i prime, and p i , p j , i , j. Then G admits at least ( l + l + · · · ( l s + − non-equivalent non-naturally reductive Einstein metrics.Proof. Given an integer pair ( p , q ), denote n = pq , p , q ≥
2, and let L = pH , K = H . Then( G , L , K , { e } ) is a basic quadruple, and by Theorem 5.3, the standard metrics on G / L , G / K are Ein-stein. For the basic quadruple ( G , L , K , { e } ), we have c = q , c = pq , l p = dim L dim G / L (1 − c ) = q , k p = pq . Then ω = + l p − k p − c = − pq ≥ ,ω = k p + c − c − h p = q ≥ . Thus by (5.54) of Theorem 5.2, G admits a left invariant Einstein metric of the form g ( x , y ) with x , , x , y , associated to ( G , L , K , { e } ), which is not naturally reductive. This completes the proofof the theorem. (cid:3) To the best knowledge of the authors, the Einstein metrics on compact semisimple Lie groupsdescribed on the above theorem are the first known examples of non-naturally reductive Einsteinmetrics which are not a product of Einstein metrics.
ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 19 A ppendix A. T he related quantities in the P roof of P roposition Type A. 1 : su ( n n n k ) ⊃ s ( n u ( n n k )) ⊃ s ( n n u ( n k )) ⊃ s ( n n n u ( k )), k ≥ , n i ≥ c = n , c = n n . Then by Lemma 3.2, we have l p = G / L [ n dim su ( n n k )(1 − c ) + ( n − = n , and similarly k p = n n , h p = n n n . Therefore we have ω = + l p − k p − c = − n n ≥ ,ω = k p + c − c − h p = n − n n n ≥ . Type A. 2 : so ( n n n k ) ⊃ n so ( n n k ) ⊃ n n so ( n k ) ⊃ n n n so ( k ), k ≥ , n i ≥ c = n n k − n n n k − , c = n k − n n n k − . Then l p = dim L dim G / L (1 − c ) = n n n k ( n n k − n n n k ( n n n k − − n n n k ( n n k −
1) (1 − n n k − n n n k − = n n k − n n n k − , and similarly k p = n k − n n n k − , h p = k − n n n k − . Therefore we have ω = + l p − k p − c = n n n k − n k + n n n k − > ,ω = k p + c − c − h p = n n k − k + n n n k − > . Type A. 3 : so ( n n k ) ⊃ n so ( n k ) ⊃ n n so ( k ) ⊃ ⊕ li = h i , k ≥ , n i ≥ c = n k − n n k − , c = k − n n k − , l p = n k − n n k − . Therefore ω = n n k − k + n n k − > . AND SHAOQIANG DENG Moreover, by (5.45), we have ω = dim G + dim L − H dim L l p − > dim G + dim L − K dim L l p − = n n k ( n n k − + n n k ( n k − − n n k ( k − n n k ( n k − × n k − n n k − − = n k − k + n n k − > , since dim H < dim K . Type A. 4 : sp ( n n n k ) ⊃ n sp ( n n k ) ⊃ n n sp ( n k ) ⊃ n n n sp ( k ), k ≥ , n i ≥ c = n n k + n n n k + , c = n k + n n n k + . Thus l p = dim L dim G / L (1 − c ) = n n n k (2 n n k + n n n k (2 n n n k + − n n n k (2 n n k +
1) (1 − n n k + n n n k + = n n k + n n n k + , and similarly k p = n k + n n n k + , h p = k + n n n k + . Therefore ω = + l p − k p − c = n n n k − n k − n n n k + ,ω = k p + c − c − h p = n n k − k − n n n k + . It follows that ω < n = n = ω <
0, if and only if n = n = Type A. 5 : e ⊃ so (10) ⊕ R ⊃ so (8) ⊕ R ⊃ R .Note that so (10) and so (8) are regular subalgebras of e , hence we have c = , c = . Since G / L and ¯ L / ¯ K are both symmetric, we have l p = , k p = , and h p = . Thus ω = + l p − k p − c = + × − − × = − ,ω = k p + c − c − h p = × + − × − × = . Type A. 6 : e ⊃ so (12) ⊕ su (2) ⊃ so (8) ⊕ su (2) ⊃ su (2).Note that so (12), so (8) and 7 su (2) are regular subalgebras of e , hence we have c = , c = ,and B su (2) = B | su (2) .Since G / L and ¯ L / ¯ K are both symmetric, we have l p = , k p = , and h p = . Therefore ω = + l p − k p − c = + × − − × = − ,ω = k p + c − c − h p = × + − × − × = − . Type A. 9 : e ⊃ so (8) ⊃ su (2) ⊃ R . ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 21
Note that 2 so (8) and 8 su (8) are regular subalgebras of e , hence we have c = , l p = . It follows that ω = − + dim G − K L l p = − + − × × × = , and ω = dim G + dim L − H dim L l p − = + − × × − = . Now we deal with the cases of Type B. Notice that for any standard quadruple ( G , L , K , H ) listedin Table B with H = { e } , one has h n = h u = h p = Type B. 1 : so ( n n k ) ⊃ n so ( n k ) ⊃ n n so ( k ) , k ≥ , n i ≥ c = n k − n n k − , c = k − n n k − , l p = n k − n n k − , k p = k − n n k − . It follows that ω = + l p − k p − c = n n k − k + n n k − > , and ω = k p + c − c − h p = n kn n k − . Type B. 2 : so ( nk ) ⊃ n so ( k ) ⊃ ⊕ li = h i , k ≥ , n ≥ c = k − nk − , l p = k − nk − . since dim K < dim L , we have ω = − + dim G − K L l p > − + dim G − dim L L l p = − + nk ( nk − − nk ( k − nk ( k − × k − nk − = nk − k + nk − > . AND SHAOQIANG DENG On the other hand, we have ω = dim G + dim L − H dim L l p − = nk ( nk − + nk ( k − nk ( k − × k − nk − − = knk − . Type B. 3 : sp ( n n k ) ⊃ n sp ( n k ) ⊃ n n sp ( k ), k ≥ , n i ≥ c = n k + n n k + , c = k + n n k + , l p = n k + n n k + , k p = k + n n k + . It follows that ω = + l p − k p − c = + × n k + n n k + − k + n n k + − × n k + n n k + = n n k − k − n n k + , and ω = k p + c − c − h p = × k + n n k + + n k + n n k + − × k + n n k + = n kn n k + > . It is easily seen that ω < n = n = Type B. 4 : so (8) ⊃ so (7) ⊃ g .Since SO(8) / SO(7) is symmetric, we have l p = . By (5.46) and (5.47), we have ω =
14 dim L (dim G − dim L − K ) = ×
21 (28 − − × = − , and ω =
12 dim L (dim G − dim L ) = ×
21 (28 − = . Type B. 5 : f ⊃ so (9) ⊃ so (8). ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 23
Since F / SO(9) is symmetric, we have l p = . By (5.46) and (5.47), we have ω =
14 dim L (dim G − dim L − K ) = ×
36 (52 − − × = − , and ω =
12 dim L (dim G − dim L ) = ×
36 (52 − = . Type B. 6 : e ⊃ su (3) ⊃ so (3).Note that 3 su (3) is a regular subalgebra of e , hence we have c = , c = , l p = × − × −
14 ) = , k p = × × × = . Therefore ω = + l p − k p − c = + × − − × = ,ω = k p + c − c − h p = × + − × = . Type B. 7 : e ⊃ su (8) ⊃ so (8).Since E / SU(8) is symmetric, we have l p = . By (5.46) and (5.47), we get ω =
14 dim L (dim G − dim L − K ) = ×
63 (133 − − × = , and ω =
12 dim L (dim G − dim L ) = ×
63 (133 − = . Type B. 13 : e ⊃ su (9) ⊃ so (9), and Type B. 14 : e ⊃ su (9) ⊃ su (3). AND SHAOQIANG DENG Clearly, G / L = E / SU(9), su (9) is a regular subalgebra of e , hence we have c = , and l p = − × (1 − ) = . Since dim 2 su (3) < dim so (9) =
36, we have ω = − + dim G − K L l p ≥ − + − × × × = , and ω = dim G + dim L − H dim L l p − = + × − = . Type B. 15 : e ⊃ so (8) ⊃ su (2), and Type B. 16 : e ⊃ so (8) ⊃ su (3).Clearly, G / L = E / SO(8) × SO(8), 2 so (8) is a regular subalgebra of e , hence we have c = ,and l p = − (1 − ) = . Since dim 2 su (3) < dim 8 su (2) =
24, we have ω = − + dim G − K L l p ≥ − + − × × × = , and ω = dim G + dim L − H dim L l p − = + × − = . Type B. 17 : e ⊃ su (5) ⊃ so (5).Note that 2 su (5) is a regular subalgebra of e , SU(5) / SO(5) is symmetric, hence we have c = , c = , l p = × − × = , k p = × × = . Therefore ω = + l p − k p − c = + × − − × = ,ω = k p + c − c − h p = × + − × = . Type B. 18 : e ⊃ su (3) ⊃ so (3). ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 25
Note that 4 su (3) is a regular subalgebra of e , SU(3) / SO(3) is symmetric, hence we have c = , c = , l p = − × −
110 ) = , k p = × × × = . Therefore ω = + l p − k p − c = + × − − × = ,ω = k p + c − c − h p = × + − × = . A ppendix B. T he values ¯ f (1) and ¯ f ( β ) in the proof of T heorem f (1) and ¯ f ( β ) in the proof of Theorem 5.2. First recall theformula (5.53) ¯ f ( x ) = M ( x − β ) η ( x ) + − c + l p ) x ( ω x + k p − h p ) , where M = c + h n > ω = + l p − k p − c . z is the unique positive number such that ¯ f ( z ) = f (1) and ¯ f ( β ) of the cases (a)-(f) listed in Proposition 5.1. This willbe completed case by case below. Case (a)
Type A. 4 with n = n =
2, namely, sp (4 n k ) ⊃ sp (2 n k ) ⊃ sp ( n k ) ⊃ n sp ( k ) , k ≥ , n ≥ . In this case, it is easily seen that c = n k + n k + , c = n k + n k + , l p = , k p = n k + n k + , h p = k + n k + , and ω = − n k + , ω = n k − k − n k + . Therefore we have β = k p − h p + − c c + h p = n k − kn k + k + , and ¯ f ( β ) = − c β ( ω β + k p − h p ) = − c β [ − n k + × n k − kn k + k + + n k − k n k +
1) ] = − c β × k ( n − n k + k + − k (5 n − n k + n k + k + > . AND SHAOQIANG DENG It is clear that ω < n =
2. On the other hand, if n =
2, then we have¯ f (1) = M (1 − β ) η (1) + − c ω + k p − h p ) = ( 14 × k + k + + × k + k +
1) )(1 − k k + − k + k + − × k + k + +
18 (1 − k + k + (cid:2) − k + + k + k + − k + k + (cid:3) = × k + k + × − k k + × ( 5 k k + + × k k + × k − k + = k (2 k − − k )8(8 k + < . Thus 1 < z < β .In the case n = n =
2, we have sp (4 n k ) ⊃ n sp (4 k ) ⊃ n sp (2 k ) ⊃ n sp ( k ) , k ≥ , n ≥ . It follows that c = k + n k + , c = k + n k + , l p = k + n k + , k p = k + n k + , h p = k + n k + , and ω = n k − k − n k + , ω = − n k + . Then we have β = k p − h p + − c c + h p = n k k + . Notice that the inequality ω < n =
2, and we have studied this case in theabove. Therefore in the following we assume that z < β . Now¯ f (1) = M (1 − β ) η (1) + − c ω + k p − h p )( 12 + l p ) = ( 14 × k + n k + + × k + n k +
1) )(1 − n k k + × × ( 12 + k + n k + − k + n k + − × k + n k + +
18 (1 − k + n k + n k − k − n k + + k + n k + − k + n k +
1) )( 12 + k + n k +
1) ) = × k + n k + × k + − n k k + × ( 2 n k + k n k + + × n k − k n k + × n k − k − n k + × n k + k + n k + = k (2 n k − k − n k + [2( n − n k + k + − k (2 n + ] = k (2 n k − k − n k + [2( n − − k ] . Thus ¯ f (1) = n − − k = . Since k , n ∈ N + , n ≥
2, it is clear that ¯ f (1) = n = m + k = m , where m ∈ N + . Case (b)
Type A. 5: e ⊃ so (10) ⊕ R ⊃ so (8) ⊕ R ⊃ R . ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 27
In this case, we have c = , c = , l p = , k p = , h p = , and ω = − , ω = . Then we have β = k p − h p + − c c + h p = , and ¯ f ( β ) = − c β ( ω β + k p − h p ) = × ×
32 ( − × + −
112 ) = . So x = β = is the only real solution of ¯ f ( x ) = Case (c)
Type A. 6: e ⊃ so (12) ⊕ su (2) ⊃ so (8) ⊕ su (2) ⊃ su (2) . In this case, we have c = , c = , B su (2) = B | su (2) , l p = , k p = , h p = , and ω = − , ω = − . Then we have β = k p − h p + − c c + h p = , and ¯ f ( β ) = − c β ( ω β + k p − h p ) = × ×
43 ( − × + −
16 ) = . Moreover,¯ f (1) = M (1 − β ) η (1) + − c ω + k p − h p ) = ( 14 × + ×
16 )(1 −
43 ) 14 ( 12 + − − ×
59 ) +
18 (1 −
59 )( − + −
16 ) = − . Thus 1 < z < . Case (d)
Type B. 3 with n = n = sp (4 k ) ⊃ sp (2 k ) ⊃ sp ( k ) ⊃ { e } , k ≥ . AND SHAOQIANG DENG In this case, we have c = k + k + , c = k + k + , l p = , k p = k + k + , and ω = − k + , ω = k k + > . Then we have β = k p − h p + − c c + h p = k + + k + − k − k + = k + k + , and ¯ f ( β ) = − c β ( ω β + k p − h p ) = k k + × × k + k + − k + × k + k + + k + k +
1) ) = k (5 k + k − k + ( k + ≥ . Thus ¯ f ( β ) = k = Case (e)
Type B. 4: so (8) ⊃ so (7) ⊃ g ⊃ { e } . In this case, we have c = , c = , l p = , k p = , and ω = − , ω = . Then we have β = k p − h p + − c c + h p = , and ¯ f ( β ) = − c β ( ω β + k p − h p ) = × ×
32 ( − × +
13 ) = − . So z > . Case (f)
Type B. 5: f ⊃ so (9) ⊃ so (8) ⊃ { e } . In this case, we have c = , c = , l p = , k p = , and ω = − , ω = . ON-NATURALLY REDUCTIVE EINSTEIN METRICS ON NORMAL HOMOGENEOUS EINSTEIN MANIFOLDS 29
Then we have β = k p − h p + − c c + h p = , and ¯ f ( β ) = − c β ( ω β + k p − h p ) = × ×
53 ( − × +
718 ) = − . So z > β = . R eferences [1] F. Ara´ujo, Some Einstein homogeneous Riemannian fibrations, Di ff . Geom. Appl., 28 (2010), 241–263.[2] F. Ara´ujo, Einstein homogeneous bisymmetric fibrations, Geom. Ded., 154 (2011), 133–160.[3] A. Arvanitoyeorgos, K. Mori, Y. Sakana, Einstein metrics on compact Lie groups which are not naturally reductive,Geom. Dedicata, 160 (2012), 261–285.[4] A. L. Besse, Einstein Manifolds, Springer, Berlin, 1987.[5] C. B¨ohm, Homogeneous Einstein metrics and simplicial complexes, J. Di ff . Geom, 67 (2004), 79–165.[6] C. B¨ohm, M.M. Kerr, Low-dimensional homogeneous Einstein manifolds, Trans. Amer. Math. Soc., 358 (2006), 1455–1468.[7] C. B¨ohm, M. Wang, W. Ziller, A variational approach for homogeneous Einstein metrics, Geom. Functional Analysis,14 (2004), 681–733.[8] Z. Chen, K. Liang, Non-naturally reductive Einstein metrics on the compact simple Lie group F , Ann. Glob. Anal.Geom., 46 (2014), 103–115.[9] J.E. D’Atri, W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc.,215, 1979.[10] W. Dickinson, M.M. Kerr, The geometry of compact homogeneous spaces with two isotropy summands, Ann. Glob.Anal. Geom., 34 (2008), 329–350.[11] E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Translations Amer. Math. Soc., ser. 2, 6 (1957),111–244.[12] G.W. Gibbons, H. L¨u, C.N. Pope, Einstein metrics on group manifolds and cosets, J. Geom. Phys., 61 (2011), 947–960.[13] G.R. Jensen, Einstein metrics on principal fibre bundles, J. Di ff er. Geom., 8 (1973), 599–614.[14] K. Mori, Left invariant Einstein metrics on SU(n) that are not naturally reductive, Master thesis (in Japanese), OsakaUniversity 1994, English translation Osaka University RPM 96-10 (prepreint series) (1996).[15] A. H. Mujtaba, Homogeneous Einstein metrics on SU(n), J. Geom. Phys., 62 (2012), 976–980.[16] Yu.G. Nikonorov, E.D. Rodionov, V.V. Slavskii, Geometry of homogeneous Riemannian manifolds, J. Math. Sci. 146(6) (2007), 6313–6390.[17] C.N. Pope, Homogeneous Einstein metrics on SO(n), arXiv:1001.2776(2010).[18] M. Wang, Einstein metrics from symmetry and bundle constructions in surveys in di ff erential geometry, VI: essay onEinstein manifolds. International Press(1999).[19] J.A. Wolf, The geometry and structure of isotropy irreducible homogeneous spaces, Acta Math. 120 (1968), 59–148.[20] S.T. Yau, H. Ma, C.J. Tsai, M.T. Wang, E.T. Zhao, Open Problems in Di ff erential Geometry, in: Open Problems andSurveys of Contemporary Mathematics (eds: L.Z. Ji, Y.S. Poon, S.T. Yau), 397-477, High Education Press, Beijing,2013.[21] M. Wang, W. Ziller, On normal homogeneous Einstein manifolds, Ann. Sci. Ecole. Norm. Super., 4 e serie 18 (1985),563–633.[22] M. Wang, W. Ziller, Existence and non-existence of homogeneous Einstein metrics, Invent. Math. 84 (1986), 177–194.[23] Z. Yan, S. Deng, Einstein metrics on compact simple Lie groups attached to standard triples, Trans. Amer. Math. Soc.,to appear.[24] W. Ziller, Homogeneous Einstein metrics on spheres and projetive spaces, Math. Ann. 259 (1982), 351–358.(Zaili Yan) D epartment of M athematics , N ingbo U niversity , N ingbo , Z hejiang P rovince , 315211, P eople ’ s R epublic of C hina E-mail address : [email protected] (Shaoqiang Deng) S chool of M athematical S ciences and LPMC, N ankai U niversity , T ianjin eople ’ s R epublicof C hina E-mail address ::