- Email: [email protected]
Homogeneous Einstein–Randers metrics on Aloff–Wallach spaces
Journal of Geometry and Physics 98 (2015) 196–200
Contents lists available at ScienceDirect
Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Homogeneous Einstein–Randers metrics on Aloff–Wallach spaces✩ Xingda Liu a , Shaoqiang Deng b,∗ a
School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, P.R. China
b
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
article
abstract
info
Article history: Received 6 June 2013 Accepted 10 August 2015 Available online 20 August 2015
Aloff–Wallach spaces are important in the study of positively curved homogeneous Riemannian manifolds. In this paper, we find some homogeneous Einstein–Randers metrics on such spaces. © 2015 Elsevier B.V. All rights reserved.
MSC: 22E46 53C30 Keywords: Aloff–Wallach spaces Einstein–Randers spaces Homogeneous manifolds
1. Introduction The study of Einstein metrics has important applications in physics, see [1] for an excellent survey. Generally, it is rather difficult to find explicit examples of Einstein manifolds. In the case that the manifold is homogeneous, some methods have been found to construct examples and to classify some special type of invariant Einstein metrics, see for example [1–3]. However, even in the homogeneous case, a complete classification of invariant Einstein metrics still seems to be unreachable. Recently the second author and some collaborators have initiated the study of homogeneous Einstein–Finsler metrics, see for example [4–6]. The main results are the construction of some types of invariant Einstein–Randers metrics on spheres as well as other homogeneous manifolds. However, up to now, known examples of homogeneous Einstein–Finsler metrics are rather rare. Therefore, it is important to find new method to construct new examples of Einstein metrics. In this paper, we find some homogeneous Einstein–Randers metrics on Aloff–Wallach spaces. Aloff–Wallach spaces were constructed in [7]. It was proved that there exists an infinite series of such spaces which are not diffeomorphism to each other and that on each of such spaces there exists homogeneous Riemannian metrics which have positive sectional curvature. These results eventually led to a complete classification of homogeneous Riemannian manifold of positive sectional curvature, see [8]. The main results of this paper can be stated as the following.
✩ Supported by NSFC (no. 11271198, 10971104) and SRFDP of China.
∗
Corresponding author. E-mail address: [email protected] (S. Deng).
http://dx.doi.org/10.1016/j.geomphys.2015.08.009 0393-0440/© 2015 Elsevier B.V. All rights reserved.
X. Liu, S. Deng / Journal of Geometry and Physics 98 (2015) 196–200
197
Theorem 1.1. 1. On the Aloff–Wallach space W1,0 there exists one family of homogeneous Einstein–Randers metrics defined by (3.2). 2. On the Aloff–Wallach space W1,1 there are two nonisometric families of homogeneous Einstein–Randers metrics defined by (3.3). 3. On the Aloff–Wallach space Wk,l , k > l > 0, there are one family of homogeneous Einstein metrics defined by (3.4). The proof of the above theorem will be completed by a case by case argument in Section 3. 2. Preliminaries In this section we recall some known results on homogeneous Einstein–Randers metrics and the structure of Aloff–Wallach spaces. 2.1. Einstein–Randers metrics Let (M , F ) be a n-dimensional connected Finsler space. Then (M , F ) is called Einstein if there is a smooth function K (x) on M such that Ric (x, y) = (n − 1)K (x),
x ∈ M , y ∈ Tx (M )\{0}.
It is still an open problem whether the function K (x) in the above equation must be a constant. However, if the Finsler space (M , F ) is homogeneous, then K (x) must be a constant. A Randers metric is defined to be a Finsler metric of the form F = α + β , where α is a Riemannian metric and β is a smooth one form on M whose length with respect to α is everywhere less than 1. Randers metrics were introduced by G. Randers [9] in the context of general relativity and have important applications to physics. There is another version of such metrics by the so-called Zermelo navigation data (see [10]):
F (x, y) =
h(y, W )2 + λh(y, y)
λ
−
h(y, W )
λ
,
(2.1)
where h is a Riemannian metric, W is a vector field on M with h(W , W ) < 1 and λ = 1 − h(W , W ). The pair (h, W ) is called the navigation data of the Randers metric F . Sometimes we say that the Randers metric F solves Zermelo’s problem of navigation on the Riemannian manifold (M , h) under the external influence W. In [10], the authors obtained a characterization of Einstein–Randers metrics as follows: Lemma 2.1. Suppose the Randers metric F solves Zermelo’s problem of navigation on the Riemannian manifold (M , h) under the external influence W, where h(W , W ) < 1. Then (M , F ) is Einstein with Ricci scalar Ric (x) = (n − 1)K (x) if and only if there exists a constant σ such that 1 σ 2 ), that is, (1) h is a Einstein with Ricci scalar (n − 1)(K (x) + 16 1 h 2 Ricik = (n − 1)(K (x) + 16 σ )hik and (2) W is an infinitesimal homothety of h, namely,
(LW h)ik = Wi:k + Wk:i = −σ hik . Furthermore, σ must vanish whenever h is not Ricci-flat. Based on this result and the description of invariant Randers metrics on homogeneous manifolds, the authors of [4] obtained a characterization of homogeneous Einstein–Randers metrics: Lemma 2.2. Let G be a connected Lie group and H a closed subgroup of G such that G/H is a reductive homogeneous space with a decomposition g = h ⊕ m. Suppose h is a G-invariant Riemannian metric on G/H and W ∈ m is invariant under H with be the corresponding G-invariant vector field on G/H with W |o = W . Then the Randers metric F with h(W , W ) < 1. Let W ) is Einstein with Ricci constant K if and only if h is Einstein with Ricci constant K and W satisfies navigation data (h, W h([W , X ]m , Y ) + h([W , Y ]m , X ) = 0,
∀X , Y ∈ m .
2.2. Aloff–Wallach spaces The Aloff–Wallach spaces are defined as Wk,l = SU(3)/ik,l (U(1)), where the embeddings of U(1) into SU(3) are as the following: √
√
√
√
ik,l : e2π −1θ → diag(e2kπ −1θ , e2lπ −1θ , e2mπ −1θ ), here k, l, m are integers with greatest common divisor 1, k + l + m = 0, k ≥ l ≥ 0 and k > 0. Consider the metric (X , Y ) = − 12 Retr(XY ) on the Lie algebra su(3) (we will use this convention in the following). Let h be the Lie algebra of √ ik,l (U(1)) and g = h ⊕ m be the reductive decomposition, where h = R −1diag(k, l, m).
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X. Liu, S. Deng / Journal of Geometry and Physics 98 (2015) 196–200
Denote L = k2 + l2 + m2 , and consider the following vectors:
√
k −1 Z = 0 0
√ −2 √3L (l − m) X0 = 0
√0 l −1
0 , √0 m −1
0
0
√ −2 √ (m − k)
X1 =
0
1 0 0
−1 0
0 0 , 0
0 X4 = √0
0 0 0
−1
√ −1
√0 X2 = −1 0
√ −1 0 ,
X5 =
0
0 0 0
0
0 0
0 x , 0
0 0 −1
0 1 , 0
, 0 √ −2 √ (k − l)
3L
0
0
X3 =
0 0 −1
0 X6 = 0 0
3L
0 0 0 0 √0
−1
1 0 , 0
√0 −1 . 0
Then the set {Xi | 0 ≤ i ≤ 6} form an orthogonal basis of m with respect to the above inner product. Now we consider the Ad H-modules m1 = span(X1 , X2 ), m2 = span(X3 , X4 ), m3 = (spanX5 , X6 ), and m4 = span(X0 ). Then we have the following decomposition:
g = h ⊕ m = h ⊕ m1 ⊕ m2 ⊕ m3 ⊕ m4 . A direct calculation shows that:
[Z , X0 ] = 0, [Z , X1 ] = (k − l)X2 , [Z , X2 ] = (l − k)X1 , [Z , X3 ] = (k − m)X4 ; [Z , X4 ] = (m − k)X3 , [Z , X5 ] = (l − m)X6 , [Z , X6 ] = (m − l)X5 and
√
√ 6m
[X 0 , X 1 ] = − √ X 2 , L √ 6l
[X0 , X4 ] = − √ X3 , L
√
6m
[X 0 , X 3 ] = √ X 4 ; L √
6k
[X0 , X6 ] = √ X5 .
[X 0 , X 2 ] = √ X 1 , L √ [X0 , X5 ] = − √ X6 , L
6l
6k L
It was shown in [11] that m1 is Ad H-isomorphic to m3 iff (k, l, m) = (1, 0, −1); that m2 is Ad H-isomorphic to m3 iff (k, l, m) = (1, 1, −2); and that m1 is never Ad H-isomorphic to m2 . In particular, none of the above Ad H-modules are pairwise isomorphic iff k > l > 0. 3. Homogeneous Einstein–Randers metrics on Aloff–Wallach spaces In this section we give a case by case proof of Theorem 1.1. 3.1. W1,0 From the discussion in Section 2, we know that all the Ad H-invariant elements in m must be the elements in m4 . On the other hand, One can verify the follows:
[X 0 , X 1 ] =
√
3X2 ,
√ [X0 , X5 ] = − 3X6 ,
√ [X0 , X2 ] = − 3X1 , √ [X0 , X6 ] = 3X5 .
[X0 , X3 ] = [X0 , X4 ] = 0;
Denote Y1 = cos(α)X1 + sin(α)X5 , Y3 = − sin(α)X1 + cos(α)X5 ,
Y2 = cos(α)X2 + sin(α)X6 ; Y4 = − sin(α)X2 + cos(α)X6 .
Consider the subspaces p1 = span(Y1 , Y2 ), p2 = span(Y3 , Y4 ), p3 = span(X3 , X4 ) = m2 , p4 = m4 . It is not difficult to verify that all the pi ’s are AdH-irreducible modules. Thus invariant metrics on W1,0 must be of the form:
⟨, ⟩1 = x1 (, ) |p1 ⊥ x2 (, ) |p2 ⊥ x3 (, ) |p3 ⊥ x4 (, ) |p4 where xi (1 ≤ i ≤ 4) are some positive real numbers. From [11], we know that the invariant metrics ⟨, ⟩1 are homogeneous Einstein metrics when (x1 , x2 , x3 , x4 ) = (8(1 − q), 8q, 24 , 128 ) for the parameter a = sin2 2α = 0, where q = 12 ± √1 ; and (x1 , x2 , x3 , x4 ) ≈ (5.67352, 1.09220, 5.50695, 5 45 2 5
5.72906) or (x1 , x2 , x3 , x4 ) ≈ (1.09220, 5.67352, 5.50695, 5.72906) for the parameter a = sin2 2α = 1.
X. Liu, S. Deng / Journal of Geometry and Physics 98 (2015) 196–200
199
For any of the above homogeneous Einstein metrics, one can verify that
⟨[X0 , Yi ]m , Yj ⟩ + ⟨[X0 , Yj ]m , Yi ⟩ = 0,
1 ≤ i, j ≤ 4.
On the other hand, one can also verify that
[X0 , p1 ⊕ p2 ] ⊆ p1 ⊕ p2 ; and that
[X0 , p3 ⊕ p4 ] = 0. Then we have
⟨[X0 , X ]m , Y ⟩1 + ⟨[X0 , Y ]m , X ⟩1 = 0,
∀X , Y ∈ m .
Therefore by Lemma 2.2, the following metrics are homogeneous SU(3)-invariant Einstein–Randers metrics on W1,0 :
[⟨W , Y ⟩1 ]2 + λ⟨Y , Y ⟩1 ⟨W , Y ⟩1 F (o, Y ) = − , λ λ here W = cX0 and ⟨W , W ⟩1 < 1.
(3.2)
3.2. W1,1 One can verify the following identities:
[X0 , X1 ] = 2X2 , [X0 , X4 ] = −X3 , [X1 , X2 ] = 2X0 , [X 1 , X 5 ] = X 3 , [X2 , X4 ] = −X5 ,
[X0 , X2 ] = −2X1 , [X0 , X3 ] = X4 ; [X0 , X5 ] = −X6 , [X0 , X6 ] = X5 ; [X1 , X3 ] = −X5 , [X1 , X4 ] = −X6 ; [X 1 , X 6 ] = X 4 , [X2 , X3 ] = X6 ; [X 2 , X 5 ] = X 4 , [X2 , X6 ] = −X3 .
Denote Y1 = cos φ X1 + sin φ X2 ; Y2 = − sin φ X1 + cos φ X2 ; Y3 = X0 , here φ is the angle between the vector X1 and the orthogonal projection of the vector Y1 on the plane span(X1 , X2 ) (See[11]). Then we have
[Y1 , Y2 ] = 2Y3 ,
[Y1 , Y2 ] = −2Y2 ,
[Y2 , Y3 ] = 2Y1 .
Let p1 = span(X3 , X4 ), p2 = span(X5 , X6 ), and p3 , p4 , p5 be the 1-dimensional modules spanned by the vectors Y1 , Y2 and Y3 , respectively. Then invariant Riemannian metrics on W1,1 have the form
⟨, ⟩2 = x1 (, ) |p1 ⊥ x2 (, ) |p2 ⊥ x3 (, ) |p3 ⊥ x4 (, ) |p4 ⊥ x5 (, ) |p5 . It was proved in [11] that there are two nonisometric and nonhomothetic (Riemannian) Einstein metrics defined by the parameters (t , t , 2t , 2t , 2t ) and (t , t , 2t5 , 2t5 , 2t5 ), where t > 0. By the definition of W1,1 , it is easily seen that X0 , X1 , and X2 are Ad H invariant. Then it follows that Y1 , Y2 , and Y3 are Ad H invariant, so they are linear combinations of X0 , X1 , and X2 . Set W = α Y1 + β Y2 + γ Y3 (α 2 + β 2 + γ 2 ̸= 0). Then one can verify that
[W , Y1 ] = −2β Y3 + 2γ Y2 , [W , Y2 ] = 2α Y3 − 2γ Y1 , [W , Y3 ] = −2α Y2 + 2β Y1 ; [W , X3 ] = γ X4 − (α cos φ − β sin φ)X5 + (α sin φ − β cos φ)X6 ; [W , X4 ] = −γ X3 − (α sin φ + β cos φ)X5 − (α cos φ − β sin φ)X6 ; [W , X5 ] = −γ X6 + (α cos φ − β sin φ)X3 + (α sin φ + β cos φ)X4 ; [W , X6 ] = γ X5 + (α cos φ − β sin φ)X4 − (α sin φ + β cos φ)X3 . For each of the Einstein metrics above, we can deduce that
⟨[W , Yi ]m , Yj ⟩2 + ⟨[W , Yj ]m , Yi ⟩2 = 0, that
⟨[W , Xk ]m , Xl ⟩2 + ⟨[W , Xl ]m , Xk ⟩2 = 0,
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X. Liu, S. Deng / Journal of Geometry and Physics 98 (2015) 196–200
and that
⟨[W , Yi ]m , Xk ⟩2 + ⟨[W , Xk ]m , Yi ⟩2 = 0, here 1 ≤ i, j ≤ 3 and 3 ≤ k, l ≤ 6. Then we have
⟨[W , X ]m , Y ⟩2 + ⟨[W , Y ]m , X ⟩2 = 0,
∀X , Y ∈ m.
Thus there are two nonisometric families of invariant Einstein–Randers metrics on W1,1 , as the following:
[⟨W , Y ⟩2 ]2 + λ⟨Y , Y ⟩2 ⟨W , Y ⟩2 − , λ λ here W = α Y1 + β Y2 + γ Y3 and ⟨W , W ⟩2 < 1.
F (o, Y ) =
(3.3)
3.3. Wk,l , k > l > 0 From the discussion in [11] and Section 2, we know that all the Ad H-invariant elements in m must be the elements in m4 , and the homogeneous Einstein metrics can be written as:
⟨., .⟩3 = x1 (., .) |m1 ⊥ x2 (., .) |m2 ⊥ x3 (., .) |m3 ⊥ x4 (., .) |m4 , where x1 , x2 , x3 , x4 are the positive solutions of the following system of equations: 6 x1 6 x2 6 x3
+ + +
3x4
L
x1 x2 x3 x2 x1 x3 x3 x1 x2 m2 x21
− − − +
x2 x1 x3 x1 x2 x3 x1 x2 x3 l2
x22
+
− − − k2 x23
x3 x1 x2 x3 x1 x2 x2 x1 x3
− − −
3m2 x4 L
=λ
x21
3l2 x4 L x22 3k2 x4 L x23
=λ =λ
= λ.
From Theorem 1 in [12], it follows that there are exactly two distinct homothety classes of homogeneous Einstein metrics. On the other hand, for each of the homogeneous Einstein metrics above, one can verify that:
⟨[X0 , Xi ]m , Xj ⟩3 + ⟨[X0 , Xj ]m , Xi ⟩3 = 0,
1 ≤ i, j ≤ 6 .
Therefore we have
⟨[X0 , X ]m , Y ⟩3 + ⟨[X0 , Y ]m , X ⟩3 = 0,
∀X , Y ∈ m.
Thus we obtain one family of homogeneous Einstein–Randers metrics on Wk,l , k > l > 0 by Lemma 2.2:
⟨W , Y ⟩3 [⟨W , Y ⟩3 ]2 + λ⟨Y , Y ⟩3 − , λ λ here W = cX0 and ⟨W , W ⟩3 < 1.
F (o, Y ) =
(3.4)
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. Besse, Einstein Manifolds, Springer, Berlin, 1987. M. Wang, W. Ziller, On normal homogeneous Einstein manifolds, Ann. Sci. Ec. Norm. Super. 18 (1985) 563–633. M. Wang, W. Ziller, Existence and non-existence of homogeneous Einstein metrics, Invent. Math. 84 (1986) 177–194. S. Deng, Z. Hou, Homogeneous Einstein–Randers spaces of negative Ricci curvature, C. R. Math. Acad. Sci. Paris 347 (2009) 1169–1172. H. Wang, S. Deng, Some Einstein–Randers metrics on homogeneous spaces, Nonlinear Anal. 72 (2010) 4407–4414. H. Wang, L. Huang, S. Deng, Homogeneous Einstein–Randers metrics on spheres, Nonlinear Anal. 74 (2011) 6295–6301. S. Aloff, N. Wallach, An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975) 93–97. L. Bérard Bergery, Les variétés Riemannienes homogènes simplement connexes de dimension impair à courbure strictement positive, J. Math. Pures Appl. 55 (1976) 47–68. G. Randers, On an asymmetrical metric in the four-space of general relativity, Phys. Rev. 59 (1941) 195–199. D. Bao, C. Robles, Ricci and flag curvatures in Finsler geometry, in: D. Bao, R. Bryant, S.S. Chern, Z. Shen (Eds.), A Sample of Riemannian-Finsler Geometry, Cambridge University Press, 2004, pp. 197–260. Yu.G. Nikonorov, Compact Homogeneous Einstein 7-Manifolds, Geometriae Dedicata 109: 7C30, 2004, Kluwer Academic Publishers, 2004, Printed in the Netherlands. O. Kowalski, Z. Vlasek, Homogeneous Einstein metrics on Aloff–Wallach spaces, Differential Geom. Appl. 3 (1993) 157–167.
Contents lists available at ScienceDirect
Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
Homogeneous Einstein–Randers metrics on Aloff–Wallach spaces✩ Xingda Liu a , Shaoqiang Deng b,∗ a
School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, P.R. China
b
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
article
abstract
info
Article history: Received 6 June 2013 Accepted 10 August 2015 Available online 20 August 2015
Aloff–Wallach spaces are important in the study of positively curved homogeneous Riemannian manifolds. In this paper, we find some homogeneous Einstein–Randers metrics on such spaces. © 2015 Elsevier B.V. All rights reserved.
MSC: 22E46 53C30 Keywords: Aloff–Wallach spaces Einstein–Randers spaces Homogeneous manifolds
1. Introduction The study of Einstein metrics has important applications in physics, see [1] for an excellent survey. Generally, it is rather difficult to find explicit examples of Einstein manifolds. In the case that the manifold is homogeneous, some methods have been found to construct examples and to classify some special type of invariant Einstein metrics, see for example [1–3]. However, even in the homogeneous case, a complete classification of invariant Einstein metrics still seems to be unreachable. Recently the second author and some collaborators have initiated the study of homogeneous Einstein–Finsler metrics, see for example [4–6]. The main results are the construction of some types of invariant Einstein–Randers metrics on spheres as well as other homogeneous manifolds. However, up to now, known examples of homogeneous Einstein–Finsler metrics are rather rare. Therefore, it is important to find new method to construct new examples of Einstein metrics. In this paper, we find some homogeneous Einstein–Randers metrics on Aloff–Wallach spaces. Aloff–Wallach spaces were constructed in [7]. It was proved that there exists an infinite series of such spaces which are not diffeomorphism to each other and that on each of such spaces there exists homogeneous Riemannian metrics which have positive sectional curvature. These results eventually led to a complete classification of homogeneous Riemannian manifold of positive sectional curvature, see [8]. The main results of this paper can be stated as the following.
✩ Supported by NSFC (no. 11271198, 10971104) and SRFDP of China.
∗
Corresponding author. E-mail address: [email protected] (S. Deng).
http://dx.doi.org/10.1016/j.geomphys.2015.08.009 0393-0440/© 2015 Elsevier B.V. All rights reserved.
X. Liu, S. Deng / Journal of Geometry and Physics 98 (2015) 196–200
197
Theorem 1.1. 1. On the Aloff–Wallach space W1,0 there exists one family of homogeneous Einstein–Randers metrics defined by (3.2). 2. On the Aloff–Wallach space W1,1 there are two nonisometric families of homogeneous Einstein–Randers metrics defined by (3.3). 3. On the Aloff–Wallach space Wk,l , k > l > 0, there are one family of homogeneous Einstein metrics defined by (3.4). The proof of the above theorem will be completed by a case by case argument in Section 3. 2. Preliminaries In this section we recall some known results on homogeneous Einstein–Randers metrics and the structure of Aloff–Wallach spaces. 2.1. Einstein–Randers metrics Let (M , F ) be a n-dimensional connected Finsler space. Then (M , F ) is called Einstein if there is a smooth function K (x) on M such that Ric (x, y) = (n − 1)K (x),
x ∈ M , y ∈ Tx (M )\{0}.
It is still an open problem whether the function K (x) in the above equation must be a constant. However, if the Finsler space (M , F ) is homogeneous, then K (x) must be a constant. A Randers metric is defined to be a Finsler metric of the form F = α + β , where α is a Riemannian metric and β is a smooth one form on M whose length with respect to α is everywhere less than 1. Randers metrics were introduced by G. Randers [9] in the context of general relativity and have important applications to physics. There is another version of such metrics by the so-called Zermelo navigation data (see [10]):
F (x, y) =
h(y, W )2 + λh(y, y)
λ
−
h(y, W )
λ
,
(2.1)
where h is a Riemannian metric, W is a vector field on M with h(W , W ) < 1 and λ = 1 − h(W , W ). The pair (h, W ) is called the navigation data of the Randers metric F . Sometimes we say that the Randers metric F solves Zermelo’s problem of navigation on the Riemannian manifold (M , h) under the external influence W. In [10], the authors obtained a characterization of Einstein–Randers metrics as follows: Lemma 2.1. Suppose the Randers metric F solves Zermelo’s problem of navigation on the Riemannian manifold (M , h) under the external influence W, where h(W , W ) < 1. Then (M , F ) is Einstein with Ricci scalar Ric (x) = (n − 1)K (x) if and only if there exists a constant σ such that 1 σ 2 ), that is, (1) h is a Einstein with Ricci scalar (n − 1)(K (x) + 16 1 h 2 Ricik = (n − 1)(K (x) + 16 σ )hik and (2) W is an infinitesimal homothety of h, namely,
(LW h)ik = Wi:k + Wk:i = −σ hik . Furthermore, σ must vanish whenever h is not Ricci-flat. Based on this result and the description of invariant Randers metrics on homogeneous manifolds, the authors of [4] obtained a characterization of homogeneous Einstein–Randers metrics: Lemma 2.2. Let G be a connected Lie group and H a closed subgroup of G such that G/H is a reductive homogeneous space with a decomposition g = h ⊕ m. Suppose h is a G-invariant Riemannian metric on G/H and W ∈ m is invariant under H with be the corresponding G-invariant vector field on G/H with W |o = W . Then the Randers metric F with h(W , W ) < 1. Let W ) is Einstein with Ricci constant K if and only if h is Einstein with Ricci constant K and W satisfies navigation data (h, W h([W , X ]m , Y ) + h([W , Y ]m , X ) = 0,
∀X , Y ∈ m .
2.2. Aloff–Wallach spaces The Aloff–Wallach spaces are defined as Wk,l = SU(3)/ik,l (U(1)), where the embeddings of U(1) into SU(3) are as the following: √
√
√
√
ik,l : e2π −1θ → diag(e2kπ −1θ , e2lπ −1θ , e2mπ −1θ ), here k, l, m are integers with greatest common divisor 1, k + l + m = 0, k ≥ l ≥ 0 and k > 0. Consider the metric (X , Y ) = − 12 Retr(XY ) on the Lie algebra su(3) (we will use this convention in the following). Let h be the Lie algebra of √ ik,l (U(1)) and g = h ⊕ m be the reductive decomposition, where h = R −1diag(k, l, m).
198
X. Liu, S. Deng / Journal of Geometry and Physics 98 (2015) 196–200
Denote L = k2 + l2 + m2 , and consider the following vectors:
√
k −1 Z = 0 0
√ −2 √3L (l − m) X0 = 0
√0 l −1
0 , √0 m −1
0
0
√ −2 √ (m − k)
X1 =
0
1 0 0
−1 0
0 0 , 0
0 X4 = √0
0 0 0
−1
√ −1
√0 X2 = −1 0
√ −1 0 ,
X5 =
0
0 0 0
0
0 0
0 x , 0
0 0 −1
0 1 , 0
, 0 √ −2 √ (k − l)
3L
0
0
X3 =
0 0 −1
0 X6 = 0 0
3L
0 0 0 0 √0
−1
1 0 , 0
√0 −1 . 0
Then the set {Xi | 0 ≤ i ≤ 6} form an orthogonal basis of m with respect to the above inner product. Now we consider the Ad H-modules m1 = span(X1 , X2 ), m2 = span(X3 , X4 ), m3 = (spanX5 , X6 ), and m4 = span(X0 ). Then we have the following decomposition:
g = h ⊕ m = h ⊕ m1 ⊕ m2 ⊕ m3 ⊕ m4 . A direct calculation shows that:
[Z , X0 ] = 0, [Z , X1 ] = (k − l)X2 , [Z , X2 ] = (l − k)X1 , [Z , X3 ] = (k − m)X4 ; [Z , X4 ] = (m − k)X3 , [Z , X5 ] = (l − m)X6 , [Z , X6 ] = (m − l)X5 and
√
√ 6m
[X 0 , X 1 ] = − √ X 2 , L √ 6l
[X0 , X4 ] = − √ X3 , L
√
6m
[X 0 , X 3 ] = √ X 4 ; L √
6k
[X0 , X6 ] = √ X5 .
[X 0 , X 2 ] = √ X 1 , L √ [X0 , X5 ] = − √ X6 , L
6l
6k L
It was shown in [11] that m1 is Ad H-isomorphic to m3 iff (k, l, m) = (1, 0, −1); that m2 is Ad H-isomorphic to m3 iff (k, l, m) = (1, 1, −2); and that m1 is never Ad H-isomorphic to m2 . In particular, none of the above Ad H-modules are pairwise isomorphic iff k > l > 0. 3. Homogeneous Einstein–Randers metrics on Aloff–Wallach spaces In this section we give a case by case proof of Theorem 1.1. 3.1. W1,0 From the discussion in Section 2, we know that all the Ad H-invariant elements in m must be the elements in m4 . On the other hand, One can verify the follows:
[X 0 , X 1 ] =
√
3X2 ,
√ [X0 , X5 ] = − 3X6 ,
√ [X0 , X2 ] = − 3X1 , √ [X0 , X6 ] = 3X5 .
[X0 , X3 ] = [X0 , X4 ] = 0;
Denote Y1 = cos(α)X1 + sin(α)X5 , Y3 = − sin(α)X1 + cos(α)X5 ,
Y2 = cos(α)X2 + sin(α)X6 ; Y4 = − sin(α)X2 + cos(α)X6 .
Consider the subspaces p1 = span(Y1 , Y2 ), p2 = span(Y3 , Y4 ), p3 = span(X3 , X4 ) = m2 , p4 = m4 . It is not difficult to verify that all the pi ’s are AdH-irreducible modules. Thus invariant metrics on W1,0 must be of the form:
⟨, ⟩1 = x1 (, ) |p1 ⊥ x2 (, ) |p2 ⊥ x3 (, ) |p3 ⊥ x4 (, ) |p4 where xi (1 ≤ i ≤ 4) are some positive real numbers. From [11], we know that the invariant metrics ⟨, ⟩1 are homogeneous Einstein metrics when (x1 , x2 , x3 , x4 ) = (8(1 − q), 8q, 24 , 128 ) for the parameter a = sin2 2α = 0, where q = 12 ± √1 ; and (x1 , x2 , x3 , x4 ) ≈ (5.67352, 1.09220, 5.50695, 5 45 2 5
5.72906) or (x1 , x2 , x3 , x4 ) ≈ (1.09220, 5.67352, 5.50695, 5.72906) for the parameter a = sin2 2α = 1.
X. Liu, S. Deng / Journal of Geometry and Physics 98 (2015) 196–200
199
For any of the above homogeneous Einstein metrics, one can verify that
⟨[X0 , Yi ]m , Yj ⟩ + ⟨[X0 , Yj ]m , Yi ⟩ = 0,
1 ≤ i, j ≤ 4.
On the other hand, one can also verify that
[X0 , p1 ⊕ p2 ] ⊆ p1 ⊕ p2 ; and that
[X0 , p3 ⊕ p4 ] = 0. Then we have
⟨[X0 , X ]m , Y ⟩1 + ⟨[X0 , Y ]m , X ⟩1 = 0,
∀X , Y ∈ m .
Therefore by Lemma 2.2, the following metrics are homogeneous SU(3)-invariant Einstein–Randers metrics on W1,0 :
[⟨W , Y ⟩1 ]2 + λ⟨Y , Y ⟩1 ⟨W , Y ⟩1 F (o, Y ) = − , λ λ here W = cX0 and ⟨W , W ⟩1 < 1.
(3.2)
3.2. W1,1 One can verify the following identities:
[X0 , X1 ] = 2X2 , [X0 , X4 ] = −X3 , [X1 , X2 ] = 2X0 , [X 1 , X 5 ] = X 3 , [X2 , X4 ] = −X5 ,
[X0 , X2 ] = −2X1 , [X0 , X3 ] = X4 ; [X0 , X5 ] = −X6 , [X0 , X6 ] = X5 ; [X1 , X3 ] = −X5 , [X1 , X4 ] = −X6 ; [X 1 , X 6 ] = X 4 , [X2 , X3 ] = X6 ; [X 2 , X 5 ] = X 4 , [X2 , X6 ] = −X3 .
Denote Y1 = cos φ X1 + sin φ X2 ; Y2 = − sin φ X1 + cos φ X2 ; Y3 = X0 , here φ is the angle between the vector X1 and the orthogonal projection of the vector Y1 on the plane span(X1 , X2 ) (See[11]). Then we have
[Y1 , Y2 ] = 2Y3 ,
[Y1 , Y2 ] = −2Y2 ,
[Y2 , Y3 ] = 2Y1 .
Let p1 = span(X3 , X4 ), p2 = span(X5 , X6 ), and p3 , p4 , p5 be the 1-dimensional modules spanned by the vectors Y1 , Y2 and Y3 , respectively. Then invariant Riemannian metrics on W1,1 have the form
⟨, ⟩2 = x1 (, ) |p1 ⊥ x2 (, ) |p2 ⊥ x3 (, ) |p3 ⊥ x4 (, ) |p4 ⊥ x5 (, ) |p5 . It was proved in [11] that there are two nonisometric and nonhomothetic (Riemannian) Einstein metrics defined by the parameters (t , t , 2t , 2t , 2t ) and (t , t , 2t5 , 2t5 , 2t5 ), where t > 0. By the definition of W1,1 , it is easily seen that X0 , X1 , and X2 are Ad H invariant. Then it follows that Y1 , Y2 , and Y3 are Ad H invariant, so they are linear combinations of X0 , X1 , and X2 . Set W = α Y1 + β Y2 + γ Y3 (α 2 + β 2 + γ 2 ̸= 0). Then one can verify that
[W , Y1 ] = −2β Y3 + 2γ Y2 , [W , Y2 ] = 2α Y3 − 2γ Y1 , [W , Y3 ] = −2α Y2 + 2β Y1 ; [W , X3 ] = γ X4 − (α cos φ − β sin φ)X5 + (α sin φ − β cos φ)X6 ; [W , X4 ] = −γ X3 − (α sin φ + β cos φ)X5 − (α cos φ − β sin φ)X6 ; [W , X5 ] = −γ X6 + (α cos φ − β sin φ)X3 + (α sin φ + β cos φ)X4 ; [W , X6 ] = γ X5 + (α cos φ − β sin φ)X4 − (α sin φ + β cos φ)X3 . For each of the Einstein metrics above, we can deduce that
⟨[W , Yi ]m , Yj ⟩2 + ⟨[W , Yj ]m , Yi ⟩2 = 0, that
⟨[W , Xk ]m , Xl ⟩2 + ⟨[W , Xl ]m , Xk ⟩2 = 0,
200
X. Liu, S. Deng / Journal of Geometry and Physics 98 (2015) 196–200
and that
⟨[W , Yi ]m , Xk ⟩2 + ⟨[W , Xk ]m , Yi ⟩2 = 0, here 1 ≤ i, j ≤ 3 and 3 ≤ k, l ≤ 6. Then we have
⟨[W , X ]m , Y ⟩2 + ⟨[W , Y ]m , X ⟩2 = 0,
∀X , Y ∈ m.
Thus there are two nonisometric families of invariant Einstein–Randers metrics on W1,1 , as the following:
[⟨W , Y ⟩2 ]2 + λ⟨Y , Y ⟩2 ⟨W , Y ⟩2 − , λ λ here W = α Y1 + β Y2 + γ Y3 and ⟨W , W ⟩2 < 1.
F (o, Y ) =
(3.3)
3.3. Wk,l , k > l > 0 From the discussion in [11] and Section 2, we know that all the Ad H-invariant elements in m must be the elements in m4 , and the homogeneous Einstein metrics can be written as:
⟨., .⟩3 = x1 (., .) |m1 ⊥ x2 (., .) |m2 ⊥ x3 (., .) |m3 ⊥ x4 (., .) |m4 , where x1 , x2 , x3 , x4 are the positive solutions of the following system of equations: 6 x1 6 x2 6 x3
+ + +
3x4
L
x1 x2 x3 x2 x1 x3 x3 x1 x2 m2 x21
− − − +
x2 x1 x3 x1 x2 x3 x1 x2 x3 l2
x22
+
− − − k2 x23
x3 x1 x2 x3 x1 x2 x2 x1 x3
− − −
3m2 x4 L
=λ
x21
3l2 x4 L x22 3k2 x4 L x23
=λ =λ
= λ.
From Theorem 1 in [12], it follows that there are exactly two distinct homothety classes of homogeneous Einstein metrics. On the other hand, for each of the homogeneous Einstein metrics above, one can verify that:
⟨[X0 , Xi ]m , Xj ⟩3 + ⟨[X0 , Xj ]m , Xi ⟩3 = 0,
1 ≤ i, j ≤ 6 .
Therefore we have
⟨[X0 , X ]m , Y ⟩3 + ⟨[X0 , Y ]m , X ⟩3 = 0,
∀X , Y ∈ m.
Thus we obtain one family of homogeneous Einstein–Randers metrics on Wk,l , k > l > 0 by Lemma 2.2:
⟨W , Y ⟩3 [⟨W , Y ⟩3 ]2 + λ⟨Y , Y ⟩3 − , λ λ here W = cX0 and ⟨W , W ⟩3 < 1.
F (o, Y ) =
(3.4)
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