Einstein–Randers metrics on some homogeneous manifolds

Nonlinear Analysis 91 (2013) 114–120

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Einstein–Randers metrics on some homogeneous manifolds Zhiqi Chen ∗ , Shaoqiang Deng, Ke Liang School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, PR China

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Article history: Received 7 February 2013 Accepted 27 June 2013 Communicated by Enzo Mitidieri MSC: 53C25 53C30 17B20 22E46

abstract In this paper we get some new examples of non-Riemannian Einstein–Randers metrics on homogeneous manifolds G/H when G is either the compact simple Lie group F4 or G2 . Based on the discussions associated with Kähler C -spaces, we obtain some new Einstein Riemannian metrics on some homogeneous manifolds and prove that they admit nonRiemannian Einstein–Randers metrics. © 2013 Elsevier Ltd. All rights reserved.

Keywords: Einstein metric Einstein–Randers metric Homogeneous manifold

1. Introduction The goal of this article is to give some new examples of Einstein–Randers metrics on homogeneous manifolds G/H when G is either the compact simple Lie group F4 or G2 . Randers metrics were introduced by Randers in the context of general relativity and named after him by Ingarden. Thus Randers metrics are very useful in physics. Moreover, they have proven to be useful in other fields; see Ingarden’s account on [1,2] for their application in the study the Lagrangian of relativistic electrons. A Randers metric F on M is built from a Riemannian metric and a 1-form, i.e., F = α + β, where α is a Riemannian metric and β is a 1-form whose length with respect to the Riemannian metric α is less than 1 everywhere. Obviously, a Randers metric is Riemannian if and only if it is reversible, i.e., if and only if F (x, y) = F (x, −y) for any x ∈ M and y ∈ Tx M. Sometimes it is convenient to use the following presentation of a Randers metric in [3], i.e.,

[h(W , y)]2 + h(y, y)λ h(W , y) − ; (1.1) λ λ here λ = 1 − h(W , W ) > 0. The pair (h, W ) is called the navigation data of the corresponding Randers metric F . The Ricci scalar Ric(x, y) of a Finsler metric is defined by the sum of those n − 1 flag curvatures K (x, y, ev ), where {ev : v = 1, 2, . . . , n − 1} is any collection of n − 1 orthonormal transverse edges perpendicular to the flagpole, i.e., 

F (x, y) =

Ric(x, y) =

n−1 

Rvv .

v=1

∗

Corresponding author. Tel.: +86 022 23506423; fax: +86 022 23506423. E-mail addresses: [email protected] (Z. Chen), [email protected] (S. Deng), [email protected] (K. Liang).

0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.06.014

(1.2)

Z. Chen et al. / Nonlinear Analysis 91 (2013) 114–120

115

The Ricci tensor is defined by

 Ricij =

1 2



.

F 2 Ric

(1.3)

yi yj

Obviously, the Ricci scalar depends on the position x and the flagpole y, but does not depend on the specific n − 1 flags with transverse edges orthogonal to y (see [3,4]). In the Riemannian case, it is a well known fact that the Ricci scalar depends only on x. Thus it is quite interesting to study a Finsler manifold whose Ricci scalar does not depend on the flagpole y. Generally, a Finsler metric with such a property is called an Einstein metric, i.e.,

Ric(x, y) = (n − 1)K (x)

(1.4)

for some function K (x) on M. In particular, for a Randers manifold (M , F ) with dim M ≥ 3, F is an Einstein metric if and only if there is a constant K such that (1.4) holds (see [3]). The following lemma is an important result on Einstein–Randers metrics. Lemma 1.1 ([3]). Suppose (M , F ) is a Randers space with the navigation data (h, W ). Then (M , F ) is Einstein with Ricci scalar Ric(x) = (n − 1)K (x) if and only if there exists a constant σ satisfying the following conditions: 1 σ 2 , and 1. h is Einstein with Ricci scalar (n − 1)K (x) + 16 2. W is an infinitesimal homothety of h, i.e. LW h = −σ h.

Furthermore, σ must be zero whenever h is not Ricci-flat. It is well known that K (x) is a constant if (M , F ) is a homogeneous Einstein Finsler manifold. Here a Finsler manifold (M , F ) is called homogeneous if its full group of isometries acts transitively on M. Based on Lemma 1.1, Deng–Hou obtained a characterization of homogeneous Einstein–Randers metrics. Lemma 1.2 ([5]). 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 the h(W , W ) < 1. Let W  ) is Einstein with the Ricci constant K if and only if h is Einstein with the Ricci constant K and W satisfies navigation data (h, W

⟨[W , X ]m , Y ⟩ + ⟨X , [W , Y ]m ⟩ = 0,

∀X , Y ∈ m,

(1.5)

 is a Killing vector field with respect to the Riemannian metric h. where ⟨, ⟩ is the restriction of h on To (G/H ) ≃ m. In this case, W Just as in the Riemannian case, it is a fundamental problem to classify homogeneous Einstein Finsler spaces. In particular, it is very important to know if a homogeneous manifold admits invariant Einstein Finsler metrics. In general, the problem is unreachable. For homogeneous Randers manifolds, it is shown in [6–8] that there are non-Riemannian Einstein–Randers metrics on S 4n−1 with n ≥ 2, SU (2n − 1)/Sp(n − 1) with n ≥ 3, Spin(7)/SU (3), Spin(8)/U (3), SO(n + 1)/SO(n − 1), some Stiefel manifolds SO(n)/SO(l) and their symplectic analogs Sp(n)/Sp(l). Up to now, every known homogeneous Randers manifold with an Einstein–Randers metric is of the form G/H for a classical group G. The main goal of this paper is to give some non-Riemannian Einstein–Randers metrics on homogeneous manifolds G/H for exceptional groups F4 and G2 . The paper is organized as follows. In Section 2, we recall some results on compact semi-simple Lie groups associated with Kähler C -spaces, and list the decompositions of F4 and G2 associated with Kähler C -spaces. Based on the above decompositions, in Section 3, we obtain some examples of Einstein metrics on homogeneous manifolds F4 /C3 , F4 /B3 and G2 /A1 . In Section 4, we prove that these homogeneous manifolds admit non-Riemannian Einstein–Randers metrics. 2. The decomposition associated with Kähler C -spaces Let G be a compact semi-simple Lie group, g the Lie algebra of G and t a maximal abelian subalgebra of g. We denote by gC and tC the complexification of g and ∆ of gC relative to the √t respectively. We identify an element of the root system C C Cartan subalgebra t with an element of −1t by the duality defined by the Killing form of g . Let Π = {α1 , . . . , αl } be a fundamental system of ∆ and {Λ1 , . . . , Λl } the fundamental weights of gC corresponding to Π , that is 2(Λi , αj )

(αj , αj )

= δij ,

∀1 ≤ i, j ≤ l.

Let Π0 be a subset of Π and Π − Π0 = {αi1 , . . . , αir }. We put [Π0 ] = ∆ ∩ {Π0 }Z where {Π0 }Z denotes the subspace of √ −1t generated by Π0 . Consider the root space decomposition of gC relative to tC :

gC = tC +

 α∈∆

gC α.

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Z. Chen et al. / Nonlinear Analysis 91 (2013) 114–120

Define a parabolic subalgebra u of gC by



u = tC +

gC α,

α∈[Π0 ]∩∆+

where ∆ is the set of all positive roots relative to Π . The nilradical n of u is given by +



n=

gC .

α∈∆+ −[Π0 ]

We denote by  α the highest root of gC . Let GC be a simply connected complex semi-simple Lie group whose Lie algebra is gC and U the parabolic subgroup of GC generated by u. Then the complex homogeneous manifold GC /U is compact simply connected and G acts transitively on GC /U. Note that H = G ∩ U is a connected closed subgroup of G, that GC /U = G/H as C ∞ -manifolds, and that GC /U admits a G-invariant Kähler metric. Let h be the Lie algebra of H and hC the complexification of h. Then we have a direct decomposition

u = hC ⊕ n,

hC = tC +

 α∈[Π0 ]

gC α.

Proposition 2.1 ([9, Proposition 4.3]). Let z be the center of the nilpotent Lie algebra n. Then we have ad(hC )(z) ⊂ z and the C action of hC on z is irreducible. Moreover, the ad(hC )-module z is generated by the highest root space g α. Select a Weyl basis E−α ∈ gC α satisfying

[Eα , E−α ] = −α, ∀α ∈ ∆, [Eα , Eβ ] = 0 if α + β ̸∈ ∆, [Eα , Eβ ] = Nα,β Eα+β if α + β ∈ ∆ where Nα,β = N−α,−β ∈ R. Then we have

g=t+



√



R(Eα + E−α ) + R −1(Eα − E−α )

α∈∆

and the Lie subalgebra h is given by

 

h=t+

√



R(Eα + E−α ) + R −1(Eα − E−α ) .

α∈[Π0 ]

Let m be the orthogonal complement of h in g with respect to the negative of the Killing form. Then we have

g = h ⊕ m,

l

[h, m] ⊂ m.

From now on we assume that g is simple and Π0 = Π − αi0 . For a non-negative integer k, put ∆k = {α ∈ ∆+ |α =

j =1

mj αj , mi0 = k}. We define a subspace nk of n by

nk =



Eα .

α∈∆k



Set t = max mi0 |α =

l

j =1



mj αj ∈ ∆+ . Then nk , k = 1, . . . , t are ad(hC )-invariant subspaces, and n =

t

j =1

nj is an

irreducible decomposition of n (see [10,11]). In particular, by Proposition 2.1, z = nt . Define a subspace mk of m by

m=



√



R(Eα + E−α ) + R −1(Eα − E−α ) .

α∈∆k

Then mk , k = 1, . . . , t are Ad(H )-invariant subspaces of m and m = following proposition is well known.

t

j =1

mj is an irreducible decomposition of m. The

Proposition 2.2 ([12]). The Kähler C -space GC /U = G/H admits a G-invariant Kähler–Einstein metric given by B|m1 + 2B|m2 + · · · + tB|mt . Here B denote the negative of the Killing form. In the case that t = 2, we have a pair (Π , Π0 ) which has an irreducible decomposition

g = h0 ⊕ h1 ⊕ h2 ⊕ m1 ⊕ m2

(2.1)

Z. Chen et al. / Nonlinear Analysis 91 (2013) 114–120

117

as Ad(H )-modules, where h0 is the center of h and hi , i = 1, 2 are simple ideals of h. In such a decomposition of g, either one of h1 and h2 is zero or both are non-zero. If G = F4 or G = G2 , the decomposition is

g = h0 ⊕ h1 ⊕ m1 ⊕ m2 ,

(2.2)

and the classification is given as follows (see [13] for the details): Case (I): g = F4 , the diagram is Case (II): g = F4 , the diagram is

, where dim h1 = 21, dim m1 = 28 and dim m2 = 2; , where dim h1 = 21, dim m1 = 16 and dim m2 = 14;

, where dim h1 = 3, dim m1 = 8 and dim m2 = 2. Case (III): g = G2 , the diagram is In the above cases, the Dynkin diagram corresponding to Π0 = Π − {αi0 } obtained by removing the vertex component.

has one

3. Einstein Riemannian metrics on some homogeneous manifolds Consider the decomposition in the above section

g = h0 ⊕ h1 ⊕ m1 ⊕ m2 .

(3.1)

We know that the Ad(H )-modules mi , i = 1, 2 are irreducible and mutually non-equivalent, that the ideals hi , i = 0, 1 of h are mutually non-isomorphic, and that dim h0 = 1. Consider the following left invariant metric on G which is Ad(H )invariant:

⟨ , ⟩ = u0 · B|h0 + u1 · B|h1 + x1 · B|m1 + x2 · B|m2 ,

(3.2)

where u0 , u1 , x1 , x2 ∈ R . Note that the space of left invariant symmetric covariant 2-tensors on G which are Ad(H )invariant is given by +

v0 · B|h0 + v1 · B|h1 + v2 · B|m1 + v3 · B|m2 ,

(3.3)

where v0 , v1 , v2 , v3 ∈ R. In particular, the Ricci tensor r of a left invariant Riemannian metric ⟨ , ⟩ on G is a left invariant symmetric covariant 2-tensor on G which is Ad(H )-invariant. Thus r is of the form (3.3). Let d1 = dim h0 = 1, d2 = dim h1 , d3 = dim m1 and d4 = dim m2 . By a technical computation, we have the following proposition. Proposition 3.1 ([13]). For the cases (I) and (III), the components of the Ricci tensor r of the metric (3.2) on G are given by

 d3 u0 d4 u0  + 2 , rh0 =  2  4x1 (d3+ 4d4 ) x2 (d3 + 4d4 )      d3 (d3 + 2) u1 d3 (d3 + 2) 1  rh1 = d2 − + ,  2 4d2 u1 2(d3 + 4d4 ) 4d  2 x1 2(d3 + 4d4 )  d4 1 1 d3 + 2 1 x2   − 2 u0 + u1 , rm1 = − 2    2x1 (d3 + 4d4 ) 2(d3 + 4d4 ) 2x1 (d3 + 4d4 ) 2x1    2d4 x2 d3 u0 2  r m = 1 + 2 − 2 . 2 x2 (d3 + 4d4 ) 4x1 (d3 + 4d4 ) x2 (d3 + 4d4 ) For the case (II), the components of the Ricci tensor r of the metric (3.2) on G are given by

 d3 u0 d4 u0  + 2 , rh0 =  2  ( d + 4d ) ( d + 4d4 ) 4x x  3 4 3 1 2    1 d4 (2d2 + 2 − d4 ) u1 d3 u1 d4 (d4 − 2)   + 2 + , rh1 = 2d2 u1 (d3 + 4d4 ) 4x1(d3 + 4d4 ) 2d2 x22 (d3 + 4d4 ) 1 x2 d4 1 1 d2   rm1 = − 2 − 2 u0 + u1 ,    2x1 (d3 + 4d4 ) (d3 + 4d4 ) 2x1 (d3 + 4d4 ) 2x1    2d4 x2 d3 u0 2 u1 d 4 − 2  r m = 1 + 2 − 2 − 2 . 2 x2 (d3 + 4d4 ) 4x1 (d3 + 4d4 ) x2 (d3 + 4d4 ) x2 (d3 + 4d4 ) On the other hand, the components of the Ricci tensor r of the metric

( , ) = x1 · B|m1 + x2 · B|m2 on the homogeneous space G/H are given as follows.

(3.4)

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Z. Chen et al. / Nonlinear Analysis 91 (2013) 114–120

Proposition 3.2 ([13]). For the cases (I), (II) and (III), the components of the Ricci tensor r of the metric (3.4) on G/H are given by

 1 x2   − 2 r m1 =   r m2 =

d4

2x1 (d3 + 4d4 ) 2d4 x2

2x1 1

x2 (d3 + 4d4 )

+

, d3

4x21 (d3 + 4d4 )

.

Let H1 be the connect Lie subgroup of G with the Lie algebra h1 . Consider the following left invariant metric on G/H1 which is Ad(H )-invariant:

⟨ , ⟩G/H1 = u0 · B|h0 + x1 · B|m1 + x2 · B|m2 ,

(3.5)

where u0 , x1 , x2 ∈ R . +

Remark 3.3. Here the metric of the form (3.5) is Ad(H )-invariant. As Ad(H )-modules, h0 , m1 and m2 are irreducible and mutually non-equivalent. Thus any left invariant metric on G/H1 which is Ad(H )-invariant is of the form (3.5). But they are not all the left invariant metrics on G/H1 since the left invariant metrics on G/H1 only need to be Ad(H1 )-invariant. But m1 is not necessary to be irreducible as an Ad(H1 )-module. Thus there exist left invariant metrics on G/H1 which are not of the form (3.5). By Propositions 3.1 and 3.2, and the fact that h0 , m1 and m2 are irreducible and mutually non-equivalent as Ad(H )modules, we have the following theorem. Theorem 3.4. For the cases (I), (II) and (III), the components of the Ricci tensor  r of the metric (3.5) on G/H1 are given by

 u0 d3 d4 u0  rh0 = + 2 ,  2  ( d + 4d ) ( d +  4x1 3 x2 3 4d4 ) 4   x2 u0 d4 1 1  − 2 − 2 , rm1 = 2x ( d + 4d ) ( d + 4d4 )  2x 2x 1 3 4 3 1 1    1 x2 d3 u0 2 2d4   rm2 = + 2 − 2 . x2 (d3 + 4d4 ) 4x1 (d3 + 4d4 ) x2 (d3 + 4d4 ) Furthermore, the metric is Einstein if and only if there exists a positive solution {u0 , x1 , x2 , k} of the system of equations

 rh0 = k,

 rm1 = k,

 rm2 = k.

We normalize the system of equations by putting x1 = 1. For the case (I), we have

 u0 7u0    + , rh0 =  2  36 18x  2  1 x2 u0  rm1 = − − ,  2 36 72   1 7x2 u0    . rm2 = + − 2 9x2

36

18x2

By  rh0 =  rm1 and  rh0 =  rm2 , we have

  5 1 1 x2   +  24 18x2 u0 = 2 − 36 , 2    7 1 1 7x2   + 2 u0 = + . 36

9x2

9x2

36

By the second equation, we have u0 = x2 . Putting into the first equation, we have 17x22 − 36x2 + 4 = 0. Then we have x2 = 2 or x2 = For the case (II), we have

2 . 17

 u0 7u0    rh0 = + ,   18 36x22   1 7x2 u0  rm1 = − − ,  2 72 144   7 x u  2 0   rm2 = + − . 2 18x2

18

36x2

Z. Chen et al. / Nonlinear Analysis 91 (2013) 114–120

119

rm2 , we have rh0 =  rm1 and  By  rh0 = 

  1 7 7x2 1   + , u0 = −  2 16 2 72 36x2    x2 1 2 7   + . + 2 u0 = 18

18x2

9x2

18

By the above equation, we have

(23x3 − 26x2 + 95x − 98)(x − 2) = 0. It follows that x2 = 2 or x2 ≈ 1.0525. Then u0 = For the case (III), we have

11 4

or u0 ≈ 1.6707.

 u0 u0   + 2, rh0 =    8 8x  2  x2 u0 1  − , rm1 = −  2 16 32   1 x2 u0   rm2 = + − 2.  4x2

8

8x2

By  rh0 =  rm1 and  rh0 =  rm2 , we have

  1 5 1 x2    32 + 8x2 u0 = 2 − 16 , 2    1 1 x2 1   + 2 u0 = + . 8

4x2

4x2

8

By the second equation, we have u0 = x2 . Putting into the first equation, we have 7x22 − 16x2 + 4 = 0. Then we have x2 = 2 or x2 = 27 . In summary, we have the following theorem. Theorem 3.5. Let the notations be as above. Assume that G is either the compact simple Lie group F4 or G2 . Then every left invariant metric on G/H1 which is Ad(H )-invariant must be of the form

⟨ , ⟩G/H1 = u0 · B|h0 + x1 · B|m1 + x2 · B|m2 , where u0 , x1 , x2 ∈ R+ . Moreover, up to scaling, we have that the metric 1. ⟨ , ⟩G/H1 is Einstein when G = F4 and H1 = C3 if and only if

(u0 , x1 , x2 ) = (2, 1, 2),

or

( u0 , x 1 , x 2 ) =



2 17

, 1,



2 17

.

2. ⟨ , ⟩G/H1 is Einstein when G = F4 and H1 = B3 if and only if

( u0 , x 1 , x 2 ) =



11 4

 , 1, 2 ,

or

(u0 , x1 , x2 ) ≈ (1.6707, 1, 1.0525).

3. ⟨ , ⟩G/H1 is Einstein when G = G2 and H1 = A1 if and only if

(u0 , x1 , x2 ) = (2, 1, 2),

or

( u0 , x 1 , x 2 ) =



2 7

, 1,

2 7



.

4. Einstein–Randers metrics We now show that there exist invariant non-Riemannian Einstein–Randers metrics on the homogeneous manifolds appearing in Theorem 3.5. By the equivalence of the adjoint representation and the isotropy representation of H1 on h0 ⊕ m1 ⊕ m2 , the vector field

 |gH = d(τ (g ))|o (W ), W

∀g ∈ G, W ∈ h0

is well defined, and it is G-invariant (see [14]). For every metric given in Theorem 3.5, one can easily verify that the equation

⟨[W , X ]h0 ⊕m1 ⊕m2 , Y ⟩G/H1 + ⟨X , [W , Y ]h0 ⊕m1 ⊕m2 ⟩G/H1 = 0

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Z. Chen et al. / Nonlinear Analysis 91 (2013) 114–120

holds for any W ∈ h0 and X , Y ∈ h0 ⊕ m1 ⊕ m2 , using the fact that h0 ⊂ h and that the metric is Ad(H )-invariant. Then by Lemma 1.2, when W satisfies ⟨W , W ⟩G/H1 < 1, the homogeneous metric

[⟨W , y⟩G/H1 ]2 + ⟨y, y⟩G/H1 λ ⟨W , y⟩G/H1 − λ λ is a G-invariant Einstein–Randers metric on G/H1 , and F is Riemannian if and only if W = 0. 

F (x, y) =

We conclude the article with the following. Theorem 4.1. Let the notations be as above. Then there are at least two families of homogeneous G-invariant non-Riemannian Einstein–Randers metrics on G/H1 , where G/H1 is F4 /C3 , or F4 /B3 , or G2 /A1 . Acknowledgment This work was supported by National Natural Science Foundation of China (Nos. 10971104 and 11001133). References [1] P.L. Antonelli, R.S. Ingarden, M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic Publishers, Dordrecht, 1993. [2] R.S. Ingarden, On physical applications of Finsler geometry, Contemp. Math. 196 (1996) 213–223. [3] 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. [4] D. Bao, C. Robles, Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66 (2004) 377–435. [5] S. Deng, Z. Hou, Homogeneous Einstein–Randers metrics on symmetric space, C. R. Acad. Sci. Paris Ser. 1 367 (2009) 1169–1172. [6] H. Wang, S. Deng, Some Einstein–Randers metrics on homogeneous spaces, Nonlinear Anal. 72 (12) (2010) 4407–4414. [7] H. Wang, S. Deng, Invariant Einstein–Randers metrics on Stiefel manifolds, Nonlinear Anal. RWA 14 (1) (2013) 594–600. [8] H. Wang, L. Huang, S. Deng, Homogeneous Einstein–Randers metrics on spheres, Nonlinear Anal. 74 (17) (2011) 6295–6301. [9] F.E. Burstall, J.H. Rawnsley, Twistor Theory for Riemannian Symmetric Spaces, in: Lect. Notes Math., vol. 1424, Springer-Verlag, Heidelberg, 1990. [10] M. Kimura, Homogeneous Einstein metrics on certain Kähler C -spaces, recent topics in differential and analytic geometry, Adv. Stud. Pure Math. 18-I (1990) 303–320. [11] J.A. Wolf, Spaces of Constant Curvature, Publish or Perish, Wilmington, 1984. [12] A. Borel, F. Hirzebruch, Characteristic classes and homogeneous spaces I, Amer. J. Math. 80 (1958) 458–538. [13] A. Arvanitoyeorgos, K. Mori, Y. Sakane, Einstein metrics on compact Lie groups which are not naturally reductive, Geom. Dedicata 160 (2012) 261–285. [14] S. Deng, Z. Hou, Invariant Randers metric on homogeneous Riemannian manifolds, J. Phys. A: Math. Gen. 37 (2004) 4353–4360; Corrigendum in J. Phys. A: Math. Gen. 39 (2006) 5249–5250.