Non-naturally reductive Einstein metrics on $\mathrm{SO}(n)$

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by  Huibin Chen

NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON SO(n)
HUIBINCHEN, ZHIQICHEN, AND SHAOQIANGDENG
Abstract. In this article, we prove that every compact simple Lie group SO(n) for n ≥ 10
7 admits at least 2(cid:0)[n−31]−2(cid:1) non-naturally reductiveleft-invariant Einstein metrics.
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1. Introduction
a
J

3 A Riemannian manifold (M,h·,·i) is called Einstein if there exists a constant λ such that
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Ric = λh·,·i, where Ric is the Ricci tensor of the metric h·,·i . For a survey of the results on

]
G
Einstein metrics, we refer to the book [3] and the papers [14, 15]. For the study on Einstein
D
metrics of homogeneous manifolds, see [4, 5, 16].
.
h
t There are a lot of important results on Einstein metrics on Lie groups which are a special
a
m
class of homogeneous manifolds. It is shown in [11] that any Einstein solvmanifold, i.e. a
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simply connected solvable Lie group admitting a left-invariant metric, is standard. According
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v
to a conjecture of Alekseevskii ([3]), this exhausts noncompact Lie groups with left-invariant
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0
Einstein metrics. Also Einstein metrics on solvable Lie groups are unique to isometry and
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scaling [10], which is in sharp contrast to the compact setting.
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.
1 In [9], D’ Atri and Ziller prove every compact simple Lie group except SO(3) admits at
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7 least two left-invariant Einstein metrics which are all naturally reductive. Meanwhile, they
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: pose the question whether there exist non-naturally reductive left-invariant Einstein metrics on
v
i compact simple Lie groups. From then on, there are a lot of studies on non-naturally reductive
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r Einstein metrics on compact simple Lie groups. In [12], Mori obtains non-naturally reductive
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left-invariantEinsteinmetricsonSU(n)forn ≥ 6. Thenin[1],theauthorsprovetheexistenceof
left-invariant Einstein metrics on compact Lie groups SO(n) for n ≥ 11, Sp(n) for n ≥ 3, E ,E
6 7
and E . After that, Chen and Liang [7] give one non-naturally reductive left-invariant Einstein
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metric on F . In [2], Arvanitoyeorgos, Sakane and Statha obtain left-invariant Einstein metrics
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on compact Lie groups SO(n) for n ≥ 7, which are not naturally reductive. Recently, Chrysikos
and Sakane find new non-naturally reductive Einstein metrics on exceptional Lie groups in [8],
especially they give the first non-naturally reductive Einstein metric on G .
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Thatistosay, everycompactsimpleLiegroupexceptsomesmallgroupsadmitsnon-naturally
reductive left-invariant Einstein metrics. But how many non-naturally reductive left-invariant
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2 HUIBINCHEN,ZHIQICHEN,ANDSHAOQIANGDENG
Einstein metrics does every compact simple Lie group possibly admit? It is natural to discuss
the lower bound of the number of non-naturally reductive left-invariant Einstein metrics on
compact simple Lie groups, especially classical Lie groups.
Yan and Deng prove an interesting result in [17]: for any integer n = pl1pl2···pls with p
1 2 s i
prime and p 6= p ,
i j
(1) SO(2n)admitsatleast(l +1)(l +1)···(l +1)−3non-equivalentnon-naturallyreductive
1 2 s
Einstein metrics,
(2) and Sp(2n) admits at least (l +1)(l +1)···(l +1)−1 non-equivalent non-naturally
1 2 s
reductive Einstein metrics.
That is, they give lower bounds of the number for SO(2n) and Sp(2n), which depend on the
integer n. But we point out that the above results hold only when n is not a prime. A much
better estimate for Sp(n) is given in [6] that every Sp(n) admits at least 2[n−1] non-naturally
3
reductive left-invariant Einstein metrics for n ≥ 4.
In this article, we obtain the following lower bound of the number of non-naturally reductive
left-invariant Einstein metrics on SO(n).
Theorem 1.1. For every integer n ≥ 10, SO(n) admits at least 2 [n−1]−2 non-naturally
3
(cid:0) (cid:1)
reductive left-invariant Einstein metrics.
Thepaperis organized as follows. In section 2, we recall the study on non-naturally reductive
left-invariant metrics on SO(n) in [2], in particular the Ricci tensor of a class of left-invariant
metrics h·,·i on SO(n),andsufficientandnecessary conditions forh·,·i tobenaturally reductive.
Furthermore, they prove that SO(n) admits non-naturally reductive Einstein metrics which are
Ad(SO(n−6)×SO(3)×SO(3))-invariant. In section 3, based on the Ricci tensor formulae in [2]
and the technique of Gro¨bner basis, we prove that SO(2k+l) admits at least two non-naturally
reductive Einstein metrics which are Ad(SO(k)×SO(k)×SO(l))-invariant when l > k ≥ 3. It
implies Theorem 1.1.
2. The study on SO(n) in [2]
LetGbeacompact semisimpleLiegroupandlet K beaconnected closed subgroupofGwith
Liealgebras gandk. SincetheKillingformB ofgisnegative definite,−B isanAd(G)-invariant
inner product on g. Let g = k⊕m be the reductive decomposition of g such that [k,m] ⊂ m.
Then m can be identified with the tangent space of G/K at the origin. Assume that m admits
NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON SO(n) 3
a decomposition into mutually non-equivalent irreducible Ad(K)-modules:
m = m ⊕···⊕m (2.1)
1 q
Then any G-invariant metric on G/K has the following form
h·,·i = x (−B)| +···+x (−B)| , (2.2)
1 m1 q mq
where x ,··· ,x ∈ R+. The G-invariant metric h·,·i on G/K is called naturally reductive if
1 q
h[X,Y] ,Zi+hY,[X,Z] i= 0, ∀X,Y,Z ∈m.
m m
NotethatG-invariantsymmetriccovariant2-tensorsonG/K areofthesameformasRiemannian
metrics. In particular, the Ricci tensor r of a G-invariant Riemannian metric on G/K is of the
same form as (2.2), that is,
r = y (−B)| +···+y (−B)| , (2.3)
1 m1 q mq
where y ,··· ,y ∈ R. Let d = dimm and let {ei }di be a (−B)-orthonormal basis of m .
1 q i i α α=1 i
Denote Aγ = −B([ei ,ej],ek), i.e. [ei ,ej]= Aγ ek, and define
α,β α β γ α β γ α,β γ
P
(ijk) := (Aγ )2,
α,β
X
where the sum is taken over all indices α,β,γ with ei ∈ m ,ej ∈ m ,ek ∈ m . Then (ijk) is
α i β j γ k
independentof thechoice forthe−B-orthonormalbasisof m ,m ,m , and(ijk) = (jik) = (jki).
i j k
Then we have the following result.
Lemma 2.1 ([13]). The components r ,··· ,r of the Ricci tensor r associated to h·,·i of the
1 p+q
form (2.2) on G/K
1 1 x 1 x
k j
r = + (kji)− (jki), (k = 0,1,··· ,p+q).
k
2x 4d x x 2d x x
k k Xj,i j i k Xj,i k i
Here, the sums are taken over all i= 1,··· ,p+q.
For G= SO(k +k +k ), we consider the closed subgroup K = SO(k )×SO(k )×SO(k ),
1 2 3 1 2 3
where the embedding of K in G is diagonal, and we have the fibration
SO(k )×SO(k )×SO(k ) → SO(k +k +k )→ SO(k +k +k )/(SO(k )×SO(k )×SO(k )).
1 2 3 1 2 3 1 2 3 1 2 3
Denote the tangent space of SO(k +k +k ), SO(k +k +k )/(SO(k )×SO(k )×SO(k ))
1 2 3 1 2 3 1 2 3
and SO(k )×SO(k )×SO(k ) at the origin by so(k +k +k ), m and so(k )⊕so(k )⊕so(k ),
1 2 3 1 2 3 1 2 3
respectively. Then we have
so(k +k +k ) = so(k )⊕so(k )⊕so(k )⊕m.
1 2 3 1 2 3
4 HUIBINCHEN,ZHIQICHEN,ANDSHAOQIANGDENG
By setting m = so(k ),m = so(k ) and m = so(k ), we have the following decomposition:
1 1 2 2 3 3
so(k +k +k ) = m ⊕m ⊕m ⊕m ⊕m ⊕m ,
1 2 3 1 2 3 12 13 23
where m ,m and m are irreducible submodules of m. Note that there is a diffeomorphism
12 13 23
G/{e} ∼= (G×SO(k )×SO(k )×SO(k ))/diag(SO(k )×SO(k )×SO(k )).
1 2 3 1 2 3
Consider left-invariant metrics on G which are determined by Ad(K)-invariant inner products
on so(k +k +k ) given by
1 2 3
h·,·i = x (−B)| +x (−B)| +x (−B)|
1 m1 2 m2 3 m3
(2.4)
+x (−B)| +x (−B)| +x (−B)| .
12 m12 13 m13 23 m23
By Lemma 2.1, we have the following formulae of the Ricci tensor.
Lemma 2.2 ([2]). The components of the Ricci tensor r of left-invariant metrics on G defined
by h·,·i of the form (2.4) are given as follows:
k −2 1 x x
1 1 1
r = + k +k ,
1 4(n−2)x 4(n−2) (cid:18) 2x2 3x2 (cid:19)
1 12 13
k −2 1 x x
2 2 2
r = + k +k ,
2 4(n−2)x 4(n−2) (cid:18) 1x2 3x2 (cid:19)
2 12 23
k −2 1 x x
3 3 3
r = + k +k ,
3 4(n−2)x 4(n−2) (cid:18) 1x2 2x2 (cid:19)
3 13 23
1 k x x x 1 (k −1)x (k −1)x
3 12 13 23 1 1 2 2
r = + − − − + ,
12 2x 4(n−2) (cid:18)x x x x x x (cid:19) 4(n−2) (cid:18) x2 x2 (cid:19)
12 13 23 12 23 12 13 12 12
1 k x x x 1 (k −1)x (k −1)x
2 13 12 23 1 1 3 3
r = + − − − + ,
13 2x 4(n−2) (cid:18)x x x x x x (cid:19) 4(n−2) (cid:18) x2 x2 (cid:19)
13 12 23 13 23 12 13 13 13
1 k x x x 1 (k −1)x (k −1)x
1 23 13 12 2 2 3 3
r = + − − − + ,
23 2x 4(n−2) (cid:18)x x x x x x (cid:19) 4(n−2) (cid:18) x2 x2 (cid:19)
23 13 12 12 23 23 13 23 23
(2.5)
where n =k +k +k .
1 2 3
Definition 2.3. A Riemannian homogeneous space (M = G/K,h·,·i) with a reductive comple-
ment m of k in g is called naturally reductive if
h[X,Y] ,Zi+hY,[X,Z] i= 0, ∀X,Y,Z ∈ m.
m m
Based on the criterion for a left-invariant metric on a Lie group to be naturally reductive
given in [9], we have the following lemma.
Lemma 2.4 ([2]). If a left-invariant metric h·,·i of the form (2.4) on G is naturally reductive
with respect to G×L, where L is a closed subgroup of G = SO(k +k +k ), then one of the
1 2 3
following holds:
NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON SO(n) 5
(1) x = x = x , x = x ;
1 2 12 13 23
(2) x = x = x , x = x ;
2 3 23 12 13
(3) x = x = x , x = x ;
1 3 13 12 23
(4) x = x = x .
12 13 23
Conversely, if one of the above conditions is satisfied, then there exists a closed subgroup L of
G such that the metric h·,·i of the form (2.4) is naturally reductive with respect to G×L.
3. The proof of Theorem 1.1
In [2], the authors prove that, for any n ≥ 9, the Lie group SO(n) admits at least one left-
invariant Einstein metric determined by the Ad(SO(n − 6) × SO(3) × SO(3))-invariant inner
product of the form (2.4), which is non-naturally reductive. In this section, we first prove the
following theorem.
Theorem 3.1. For any l > k ≥ 3, SO(2k+l) admits at least two left-invariant Einstein metrics
determined by Ad(SO(k)×SO(k)×SO(l))-invariant inner products of the form (2.4), which are
non-naturally reductive.
We will prove Theorem 3.1 by solving homogeneous Einstein equations
r = r , r = r , r = r , r = r , r = r
1 2 2 3 3 12 12 13 13 23
under the assumption k = k = k and l = k . Furthermore, we consider the metric (2.4)
1 2 3
with x = x . Then we have x = x . Standard the metric with x = x = 1. Therefore
1 2 13 23 13 23
homogeneous Einstein equations are equivalent to the following system of equations:
f = −2kx x 2x 2+lx 2x x 2+kx 2x +kx x 2−lx x 2+2x x 2−2x x 2 = 0,
1 2 3 12 2 3 12 2 3 3 12 2 12 2 12 3 12

f = lx 2x 2−lx x 3+3kx 2−4x x k+x 2k−2x 2+4x x −2x 2 = 0,
 2 2 12 2 12 2 2 12 12 2 2 12 12


f = kx x +2kx 2+kx x +lx 2−4kx −2x l−x x −x 2+l+4x −2 = 0.
3 2 3 3 3 12 3 3 3 2 3 3 3




In particular,
f = (x −x ) lx x 2+3kx −kx −2x +2x .
2 2 12 2 12 2 12 2 12
(cid:0) (cid:1)
If x = x , by Proposition 2.4, the left-invariant metric is naturally reductive. Assume that
2 12
x 6= x . Then we have
2 12
x (k−2)
12
x = .
2 lx 2+3k−2
12
6 HUIBINCHEN,ZHIQICHEN,ANDSHAOQIANGDENG
Substitute it into equations f = 0 and f = 0, we have the following equations:
1 3
g =2k2lx 2x 3−kl2x x 4−4klx 2x 3+2l2x x 4+6k3x 2x −7k2lx x 2+kl2x 3
1 3 12 3 12 3 12 3 12 3 12 3 12 12

 −16k2x32x12+20klx3x122−2klx123−2l2x123−10k3x3+3k2lx12+8kx32x12




 −12lx3x122+4lx123+34k2x3−6k2x12−8klx12 −32kx3 +16kx12 +4x12l+8x3




 −8x = 0,
 12


g =2klx 2x 2+klx x 3+l2x 2x 2−4klx x 2−2l2x x 2−lx 2x 2+6k2x 2
2 3 12 3 12 3 12 3 12 3 12 3 12 3




 +4k2x x +3klx 2+l2x 2+4lx x 2−12k2x −6klx −7kx 2−5kx x −2lx 2
 3 12 3 12 3 12 3 3 3 3 12 3



 −2lx 2+3kl+20kx +4x l+2x 2+2x x −6k−2l−8x +4 = 0
 12 3 3 3 12 3 3



ConsidertheidealI generated by{g ,g ,zx x −1}andthepolynomialringR withcoefficients
1 2 3 12
in Q, and take a lexicographic order > with z > x > x for a monomial ordering on R. By
3 12
the computer software, we get two polynomials containing in the Gro¨bner basis for the ideal I.
h(x ) = l2(k+l)(2k2 +2kl+l2−l)x 8
12 12
−2l2(2k+l−2)(4k2 +4kl+l2−l)x 7
12
+l(16k4+76k3l+71k2l2+22kl3+l4−20k3−119k2l−81kl2−16l3+8k2+51kl+19l2−4l)x 6
12
−4l(2k+l−2)(14k3 +20k2l+5kl2 −14k2 −21kl−4l2+4k+4l)x 5
12
+(32k5+344k4l+368k3l2+117k2l3+6kl4−80k4−842k3l−686k2l2−168kl3−4l4+82k3+
713k2l+406kl2 +60l3 −40k2 −236kl−76l2 +8k+20l)x 4
12
−2(2k+l−2)(48k4+124k3l+31k2l2−92k3−215k2l−46kl2+64k2+110kl+16l2−16k−
16l)x 3
12
+(448k5+608k4l+212k3l2+9k2l3−1424k4−1550k3l−448k2l2−12kl3+1714k3+1427k2l+
304kl2 +4l3−956k2 −556kl−64l2 +232k+76l−16)x 2
12
−4(k−1)(5k−2)(−2+3k)(2k+l−2)(4k+l−1)x
12
+4(5k−2)2(k−1)2(l−1+2k)
and
h(x ,x ) = −l2(k+l)(k2 +4kl+2l2−2k−2l)(2k2 +2kl+l2−l)x 7
12 3 12
+2l2(2k+l−2)(k2 +4kl+2l2−2k−2l)(4k2 +4kl+l2−l)x 6
12
−l(16k6 +120k5l+357k4l2+408k3l3+206k2l4+43kl5 +2l6 −52k5 −365k4l−841k3l2−
714k2l3−252kl4−32l5+48k4+357k3l+622k2l2+355kl3+66l4−16k3−122k2l−154kl2−
44l3 +8kl+8l2)x 5
12
+2l(2k+l−2)(28k5+112k4l+166k3l2+90k2l3+15kl4−84k4−254k3l−278k2l2−114kl3−
14l4 +64k3+172k2l+131kl2 +28l3−16k2 −32kl−14l2)x 4
12
+(−32k7 −372k6l −1118k5l2 −1337k4l3 −710k3l4 −153k2l5 −7kl6 +144k6 +1664k5l +
NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON SO(n) 7
3882k4l2 +3620k3l3 +1499k2l4 +247kl5 +6l6 −242k5 −2641k4l −4816k3l2 −3415k2l3 −
1003kl4 −98l5 +204k4 +1882k3l+2625k2l2 +1293kl3 +210l4 −88k3 −588k2l−586kl2 −
150l3 +16k2 +56kl+32l2)x 3
12
+2(2k +l −2)(48k6 +156k5l +263k4l2 +162k3l3 +27k2l4 −188k5 −583k4l −706k3l2 −
342k2l3−48kl4+248k4+688k3l+631k2l2+216kl3+20l4−144k3−316k2l−216kl2−40l3+
32k2 +48kl+20l2)x 2
12
+(−368k7 −740k6l−850k5l2−513k4l3−120k3l4−3k2l5+1908k6 +3454k5l+3330k4l2+
1642k3l3+321k2l4+8kl5−3740k5−5875k4l−4684k3l2−1787k2l3−256kl4−4l5+3546k4+
4678k3l+2943k2l2+784kl3 +60l4 −1676k3 −1786k2l−800kl2 −116l3 +344k2 +280kl+
68l2 −16k−8l)x
12
+2(k−1)(k−2)(5k−2)(3k2 +2kl+l2−2k−l)(8k2 +4kl−8k)
+2(l−1)(k−1)(k−2)(5k−2)(2k+l−1)(3k2 +(2k+l)(l−1))x .
3
By h(x ,x ) = 0, x can be expressed by a polynomial of x of degree 7 with coefficients in
12 3 3 12
Q for k ≥ 3 and l ≥ 2. It means if x = s ∈ R is a solution of h(x ) = 0, then there exists
12 12
x = t ∈ R such that h(s,t) = 0. Moreover, we have
3
h(0) = 4(5k−2)2(k−1)2(l−1+2k),
h(1) = (k−1)(l−1+2k)(k−l)(2k+l)2
h(+∞) → +∞.
For l > k ≥ 2, we have h(0) > 0 and h(1) < 0. as a result, h(x ) = 0 has at least two solutions,
12
one of which is between 0 and 1 and the other is more than 1. As is shown above, there exists
x ∈ R with respect to each solution of h(x ) = 0.
3 12
The following is to check when x ∈ R+. For this, take a lexicographic order > with z >
3
x > x for a monomial ordering on R. Similarly, we have the following polynomial containing
12 3
in the Gro¨bner basis for the ideal I:
p(x ) = 4(l−1+2k)(2k2 +2kl+l(l−1))(3k2 +2kl+l2−2k−l)2x8
3 3
−16(2k +l −2)(3k2 +(2k +l)(l −1))(3k4 +(12l −2)k3 +(14l2 −10l)k2 +(l2(8l −10)+
2l)k+2l3(l−2)+2l2)x7
3
+(160k7+(1928l−1000)k6+(5492l2−6764l+1408)k5+(7644l3−15830l2+7818l−736)k4+
(6201l4−18266l3+15881l2−3848l+128)k3+(3072l5−11752l4+14848l3−6912l2+808l)k2+
(880l6−4208l5+7012l4−4824l3+1204l2−64l)k+112l7−664l6+1432l5−1352l4+504l3−
32l2)x6
3
−4(2k +l−2)((128l −108)k5 +(582l2 −725l +144)k4 +(921l3 −1857l2 +800l −48)k3 +
(736l4−2072l3+1600l2−324l)k2+(312l5−1152l4+1292l3−484l2+48l)k+56l6−268l5+
8 HUIBINCHEN,ZHIQICHEN,ANDSHAOQIANGDENG
424l4 −252l3 +40l2)x5
3
+((200l−320)k6+(2448l2−4584l+1712)k5+(6812l3−18568l2+13190l−2216)k4+(8618l4−
32023l3+36747l2−13648l+1056)k3+(5781l5−27998l4+45785l3−29060l2+6104l−160)k2+
(2000l6−12260l5+27136l4−25936l3+9932l2−1120l)k+280l7−2120l6+6124l5−8336l4+
5284l3 −1280l2 +64l)x4
3
−(2(l −2))(2k +l −2)((128l −108)k4 +(582l2 −725l +144)k3 +(842l3 −1694l2 +716l −
48)k2 +(512l4 −1560l3 +1232l2 −244l)k+112l5 −480l4 +632l3 −280l2 +32l)x3
3
+2(l−2)2(20k5+(241l−125)k4+(659l2−800l+158)k3+(703l3−1593l2+820l−68)k2+
(328l4 −1156l3 +1168l2 −336l+8)k+56l5−276l4 +448l3 −268l2 +48l)x2
3
−2(l−2)3(2k+l−2)(−2+3k+2l)(3k2 +2k(6l−1)+4l(2l−1))x
3
+(l−2)4(k+l)(−2+3k+2l)2.
It is not hard to verify that the coefficients of polynomial p(x ) is positive for even degree
3
and negative for odd degree whenever l > k ≥ 3. That is, all the solution of x for homogeneous
3
Einstein equations are positive (if exist) whenever l > k ≥3.
In summary, for l > k ≥ 3, we get two solutions of the form
x (k−2)
12
{x = x = ,x = α(x ),x 6=1,x = x = 1},
1 2 lx 2+3k−2 3 12 12 13 23
12
where α(x ) is a rational polynomial of x with positive values. By Proposition 2.4, every
12 12
Einstein metric induced by the solution of this form is non-naturally reductive. Thus, we have
proved Theorem 3.1.
By Theorem 3.1, for any 3 ≤ k ≤ [n−1], SO(n) admits at least two non-naturally reductive
3
left-invariant Einstein metrics which are Ad(SO(k)×SO(k)×SO(n−2k))-invariant. That is,
Theorem 1.1 follows.
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School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China
E-mail address: [email protected]
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China
E-mail address: [email protected]
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China
E-mail address: [email protected]