33 (2017), 373–385 www.emis.de/journals ISSN 1786-0091
EINSTEIN METRICS ON WARPED PRODUCT FINSLER SPACES
M. KHAMEFOROUSH YAZDI, Y. ALIPOUR FAKHRI AND M. M. REZAII
Abstract. In this paper, we prove that a warped product Finsler metric is an Einstein metric if and only if some partial differential equations are satisfied. Several results are obtained in special cases, for example, the case of Riemannian, Locally Minkowski, and Berwald spaces. Moreover, we present the static vacuum Einstein equations on Finsler manifold.
1. Introduction
The warped product is an important concept in geometry and physics. This concept was first introduced by Bishop and O’Neill to construct Riemannian manifolds with negative curvature [11]. It has been applied for the construction of Einstein metrics on noncompact complete Riemannian manifolds and other important examples in relativity and differential geometry [8],[11]. Besse pro- duced a non-trivial Einstein warped product on a compact Riemannian man- ifold [10]. S. Kim established compact base manifolds with a positive scalar curvature which do not admit any non-trivial Einstein warped product [15].
One of the important problems in Finsler geometry is to characterize and construct the Einstein metrics, constant Ricci curvature metrics and, as a special case, constant flag curvature metrics. Many valuable results have been achieved, most of which are related to a special class of Finsler metrics named (α, β)-metrics due to its computability. In 2004, D. Bao, C. Robles and Shen classified Randers metrics with constant flag curvature [7]. Also with the help of the navigation problem, D. Bao and C. Robles give a characterization for Einstein metrics of Randers type [6].
Warped product extended for Finslerian metrics by the work of Kozma et al [16]. Some objects of Riemannian manifolds are expanded to the warped product Finsler manifolds in [4], [5]. Two authors of this paper presented some necessary and sufficient conditions which the spray manifold is projectively equivalent to the warped product Finsler manifolds [18]. Also, they found
2010Mathematics Subject Classification. 53C60, 53C20, 53C25.
Key words and phrases. Finsler space, Warped product, Einstein metric.
373
the necessary and sufficient conditions for the Sasaki-Finsler metric of the warped product Finsler manifold to be bundle like for the vertical foliation [2]. In [19] and [20] , Tayebi and his collaborates studied the warped and doubly warped product structure in Finsler geometry. They consider warped product Finsler metrics with scalar flag curvature and some well-known non- Riemannian curvature properties such as Berwald, Landsberg, and relatively isotropic (mean) Landsberg curvatures. Inspired by the mentioned works, we consider Einstein warped product Finsler metrics and present necessary and sufficient conditions for the warped product Finsler space to be Einstein . Moreover, several results are obtained in the special cases, for example, the case of Riemannian, Locally Minkowski and Berwald spaces are considered.
Also, we present the static vacuum Einstein equations on Finsler manifold.
2. Preliminaries
LetF1 = (M1, F1) andF2 = (M2, F2) be two Finsler manifolds andf: M1 → R be a non-negative smooth function. Consider F = (M, F) where M is the product manifoldM1×M2 and the function F is defined as
F2(x1, x2, y1, y2) = F12(x1, y1) +f(x1)2F22(x2, y2).
(1)
The functionF is smooth onT M0 and is obviously positively homogeneous of degree 1 with respect to (y1, y2). For the function F,
(gab(x, y)) = (1 2
∂2F2
∂ya∂yb) =
gˆij(x1, y1) 0
0 f2(x1)ˇgαβ(x2, y2) (2)
are the components of a positive definite quadratic form at every point (x, y).
ThereforeF= (M, F) defines a Finsler manifold and is called a warped product ofF1 andF2. The warped product Finsler metricF is denoted byF =F1×fF2 and the function f is called warping function [2],[16].
Notation. Lowercase Latin letters like {i, j, k, l, . . .} , {α, β, γ, . . .} and {a, b, c, d, . . .} are used in the upper position for variable indices. They be- long to the set {1, . . . , m1}, {1, . . . , m2} and {1, . . . , m1 +m2} respectively, according to the spaces F1,F2 orF=F1×f F2 they represent. Variables of F1
and F2 have lower indices 1 and 2 respectively, like xi1, y1j and xα2, yβ2.
When there is no appropriate position to place indices 1 and 2, objects of F1 and F2 will be hat and check respectively, like ˆgij and ˇgαβ, to indicate their relevant spaces.
The inverse gab of gab is given by (gab(x,y)) =
ˆgij(x1, y1) 0
0 f−2(x1)ˇgαβ(x2, y2)
. (3)
The local coordinates (x1, x2, y1, y2) on T M0 are transformed by the rules
˜
xi1 = ˜xi1(x11, . . . , xm11), y˜i1 = ∂x˜i1
∂xj1yj1,
˜
xα2 = ˜xα2(x12, . . . , xm22), y˜α2 = ∂x˜α2
∂xβ2y2β. And for ∂y∂a, we have
∂
∂y1i = ∂x˜j1
∂xi1
∂
∂y˜1j, ∂
∂y2α = ∂x˜j2
∂xi2
∂
∂y˜j2. (4)
So, the vertical distribution VT M0 is spanned by {∂y∂i 1
,∂y∂α
2} and the hori- zontal distribution HT M0 is spanned by{δxδi
1,δxδα
2}which
δ
δxi1 = ∂x∂i 1
−Gji ∂
∂yj1 −Gβi ∂
∂yβ2,
δ
δxα2 = ∂x∂α
2 −Gjα ∂
∂y1j −Gβα ∂
∂yβ2. (5)
The geodesic coefficients Ga = (Gi, Gα) are local defined functions as
Gi = ˆGi− 1
4gˆih∂f2
∂xh1F22, Gα = ˇGα+ 1 2
1
f2y2αy1h∂f2
∂xh1. (6)
Now, the coefficientsGab = (Gij, Giβ, Gαj, Gαβ) of the non-linear connection are given by
Gij = ˆGij− 1 4
∂ˆgih
∂y1j
∂f2
∂xh1F22, Giβ =−1
4gˆih∂f2
∂xh1
∂F22
∂yβ2 , Gαj = 1
2 1
f2y2α∂f2
∂xj1, Gαβ = ˇGαβ +1
2 1
f2yh1∂f2
∂xh1δβα. (7)
Corollary 2.1. The coefficients Gabc = (Gijk, Giβk, Giβγ, Gαjk, Gαjγ, Gαβγ) of the nonlinear connection on the warped product Finsler manifold are gotten as
Gijk = ˆGijk − 1 4
∂2ˆgih
∂yj1∂yk1
∂f2
∂xh1F22 =Gikj, Giβk =−1
4
∂gˆih
∂yk1
∂f2
∂xh1
∂F22
∂y2β =Gikβ, Giβγ =−1
2ˆgih∂f2
∂xh1gˇβγ =Giγβ , Gαjk = 0,
Gαjγ = 1 2
1 f2
∂f2
∂xj1δγα =Gαγj, Gαβγ = ˇGαβγ =Gαγβ.
(8)
Proof. Using Gabc= ∂G∂ycab, for detailed see [2].
3. The Laplacian of the Sasaki Finsler metrics
It is well known that various kinds of Laplace operators play a very im- portant role in differential geometry and physics, especially in the theory of harmonic integral and Bochner technique. In [14], Chunping and Tongde gen- eralized the Laplace operator in Riemannian manifolds to Finsler vector bun- dles as such bundles arise naturally in Finsler geometry. Using the h-Laplace operator, they proved some integral formulas for horizontal Finsler vector fields and scalar fields on vector bundles.
Let (M, F) be a Finsler manifold. Then the tangent bundle T M endowed with the Sasaki-type metric constructed from the given Finsler metric F is a Riemannian vector bundle. Consider
dV = det(gij)dx1∧. . .∧dxm∧δy1∧. . .∧δym (9)
be the volume form associated with the Riemannian structure, G = gijdxi⊗ dxj + gijδyi ⊗ δyj, on T M and LX be the Lie derivative with respect to X ∈ X(T M), then the notations as gradient and divergent on T M can be introduced. The divergence ofX =Xi δδxi + ¯Xi ∂∂yi is defined by
LXdV = (divX)dV.
(10)
We denote divhX =: div(Xi δδxi) and divvX =: div( ¯Xi ∂∂yi), then we have the following Lemma.
Lemma 3.1. Let X =Xi δδxi + ¯Xi ∂∂yi ∈ X(T M) then divX = div
h X+ div
v X, (11)
where
div
h X =∇ δ
δxi
Xi−PikkXi, div
v X =∇ ∂
∂yi
X¯i+CikkX¯i, which Pijk =Gkij −Fijk.
Proof. Applying proposition (3.1) of [14] on Finsler spaces.
Now if we define grad(f) =∇f by
G(∇f, X) =Xf, ∀X ∈ X(T M), f ∈C∞(T M) then in the adapted frame {δxδi,∂y∂i}, we have
∇f =∇hf +∇vf, (12)
where
∇hf =gij δf δxj
δ
δxi, ∇vf =gij ∂f
∂yj
∂
∂yi. Iff ∈C∞(M) then we have
G(∇hf2,∇hf2) = gij∂f2
∂xi
∂f2
∂xj. (13)
Now the h(v)-Laplace operator onT M is defined by
∆h := div
h ◦∇h, ∆v := div
v ◦∇v. (14)
Lemma 3.2. Let (M, F) be Finsler space and f ∈C∞(T M). Then
∆f = ∆hf + ∆vf, (15)
where
∆hf = δgij δxi
δf
δxj +gij δ δxi(δf
δxj)−Pikkgij δf δxj,
∆vf = ∂gij
∂yi
∂f
∂yj +gij ∂
∂yi(∂f
∂yj) +gij ∂f
∂yjCikk.
Proof. See [14].
Consider the Finsler manifold to be Riemannian space and f: M → R, so the horizontal Laplacian of f2 is given by
∆hf2 = ∂gij
∂xi
∂f2
∂xj +gij ∂2f2
∂xi∂xj +Gkikgij∂f2
∂xj. (16)
When the Finsler space is locally Minkowski space, the horizontal Laplacian of f ∈C∞(M) is gotten by
∆hf2 =gij ∂2f2
∂xi∂xj. (17)
4. Einstein Metrics
The importance of the Ricci tensor can be seen from the Bonnet-Myers theorem. The Riemannian version of this result is one of the most useful comparison theorems in differential geometry [13]. It was first extended to the Finsler manifolds by the work of Auslander [3]. Akbar-Zadeh generalized the concept of Ricci tensor on Finsler geometry [1]. In this section, we perform the concept of Ricci tensor on warped product Finsler manifolds and present some conditions for the warped product Finsler metric to be Einstein.
Let us begin by introducing Ricci scalar for the warped product Finsler space F= (M, F) as follows
<ic := Raa =Rii+Rαα, (18)
where
Rii := 1
F2(Rijikyj1yk1 +Riβiγy2βy2γ), Rαα := 1
F2(Rαjαkyj1y1k+Rαβαγy2βyγ2),
and Rabcd are theh-curvature tensor field of the Cartan connection of F. We define Ricci tensor from the Ricci scalar as follows
Ricbc := ∂2(12F2<ic)
∂yb∂yc . (19)
The definition of the Ricci tensor is not practical if one wants to compute it. So we use the generalized Berwarld’s formula on warped product Finsler manifold that is defined as
Kaa:= 2∂G∂xaa − ∂G∂yba∂G∂yab −yb∂x∂Gb∂yaa + 2Gb∂y∂Gb∂yaa. (20)
The Ricci scalar is related to the generalized Berwald’s Formula in the fol- lowing manner:
F2Raa=Kaa. (21)
Then the Ricci tensor of the warped product Finsler space is given by
Rickl = ˆRickl+1
2F22[−1 2
∂3ˆgih
∂xi1∂yk1∂yl1
∂f2
∂xh1 +1
4y1j ∂4gˆih
∂xj1∂y1i∂y1k∂y1l
∂f2
∂xh1
−1 2
∂2gˆih
∂y1k∂y1l
∂2f2
∂xh1∂xi1 +1
4y1j ∂3ˆgih
∂y1i∂yk1∂y1l
∂2f2
∂xj1∂xh1 +1 4
1 f2
∂2ˆgih
∂y1k∂y1l
∂f2
∂xh1
∂f2
∂xi1
−1 2
1
f2yh1∂f2
∂xh1
∂f2
∂xs1
∂3ˆgis
∂y1i∂y1k∂y1l]− 1 2
∂2lnf2
∂xk1∂xl1 +1 2
1
f2Gˆjkl∂f2
∂xj1 +1
8
∂f2
∂xs1F22[ ˆGiklj∂gˆjs
∂yi1 + ˆGikj ∂2gˆjs
∂y1i∂y1l + ˆGilj ∂2gˆjs
∂y1i∂y1k + ˆGij ∂3ˆgjs
∂y1i∂y1k∂y1l] +1
8
∂f2
∂xh1F22[ ˆGjkli∂gˆih
∂y1j + ˆGjki ∂2gˆih
∂y1j∂y1l + ˆGjli ∂2gˆih
∂y1j∂y1k + ˆGji ∂3ˆgih
∂y1j∂y1k∂y1l]
−1 4
∂f2
∂xs1F22[ ˆGjkl ∂2gˆis
∂y1j∂y1i + ˆGjk ∂3ˆgis
∂yj1∂yi1∂yl1 + ˆGjl ∂3ˆgis
∂y1j∂y1i∂y1k (22)
+ ˆGj ∂4gˆis
∂y1j∂y1i∂y1k∂y1l]+1
8yβ2 ∂3gˆih
∂y1i∂y1k∂y1l
∂f2
∂xh1
∂F22
∂xβ2 −1 4
Gˇβ ∂3gˆis
∂y1i∂y1k∂y1l
∂f2
∂xs1
∂F22
∂yβ2
−1 4
∂f2
∂xh1F22[ ˆGiji ∂2ˆgjh
∂yk1∂y1l + ˆGilji∂gˆjh
∂yk1 + ˆGikji∂gˆjh
∂y1l + ˆGikljigˆjh] +1
2
∂f2
∂xh1
∂f2
∂xs1F24[− 1 16
∂3gˆih
∂y1j∂y1k∂y1l
∂gˆjs
∂y1i − 1 16
∂2gˆih
∂y1j∂y1k
∂2gˆjs
∂y1i∂y1l
− 1 16
∂2gˆih
∂y1j∂y1l
∂2gˆjs
∂yi1∂yk1 − 1 16
∂ˆgih
∂y1j
∂3gˆjs
∂yi1∂yk1∂yl1 +1 8
∂2gˆjh
∂y1k∂y1l
∂2ˆgis
∂yj1∂yi1 +1
8
∂ˆgjh
∂y1k
∂3ˆgis
∂yj1∂y1i∂yl1 +1 8
∂ˆgjh
∂y1l
∂3gˆis
∂y1j∂y1i∂y1k +1
8gˆjh ∂4gˆis
∂yj1∂yi1∂yk1∂yl1] Rickν = 1
2
∂F22
∂y2ν [−1 2
∂2ˆgih
∂xi1∂yk1
∂f2
∂xh1 −1 2
∂gˆih
∂y1k
∂2f2
∂xh1∂xi1 + 1 4
Gˆikj∂gˆjs
∂y1i
∂f2
∂xs1 +1
4
Gˆij ∂2gˆjs
∂yi1∂y1k
∂f2
∂xs1 + 1 4
Gˆjki∂gˆih
∂y1j
∂f2
∂xh1 + 1 4
Gˆji ∂2ˆgih
∂y1j∂y1k
∂f2
∂xh1 +1
4y1j ∂3gˆih
∂xj1∂y1i∂y1k
∂f2
∂xh1 + 1
4yj1 ∂2gˆih
∂yi1∂y1k
∂2f2
∂xj1∂xh1 −1 2
Gˆjk ∂2gˆis
∂y1j∂y1i
∂f2
∂xs1
−1 2
Gˆj ∂3gˆis
∂y1j∂y1i∂yk1
∂f2
∂xs1 − 1 2
∂ˆgjh
∂y1k
∂f2
∂xh1
Gˆiji− 1
2gˆjh∂f2
∂xh1 Gˆikji (23)
+1 4
1 f2
∂ˆgih
∂yk1
∂f2
∂xh1
∂f2
∂xi1 − 1 2
1
f2yh1∂f2
∂xh1
∂f2
∂xs1
∂2ˆgis
∂y1i∂y1k]+1
8yβ2 ∂2gˆih
∂y1i∂y1k
∂f2
∂xh1
∂2F22
∂xβ2∂yν2 +1
2
∂f2
∂xh1
∂f2
∂xs1
∂F24
∂yν2 [− 1 16
∂2gˆih
∂y1j∂y1k
∂gˆjs
∂y1i − 1 16
∂ˆgih
∂y1j
∂2gˆjs
∂yi1∂yk1 +1 8
∂ˆgjh
∂y1k
∂2ˆgis
∂y1j∂y1i
+1
8gˆjh ∂3gˆis
∂yj1∂yi1∂yk1]− 1 4
∂2ˆgis
∂y1i∂y1k
∂f2
∂xs1[ ˇGβν∂F22
∂y2β + ˇGβ ∂2F22
∂y2β∂y2ν]
Ricµν = ˇRicµν+ 1 2
∂2F22
∂y2µ∂y2ν[−1 2
∂gˆih
∂xi1
∂f2
∂xh1 − 1
2ˆgih ∂2f2
∂xh1∂xi1 + 1
4y1j ∂2ˆgih
∂xj1∂y1i
∂f2
∂xh1 +1
4yj1∂ˆgih
∂y1i
∂2f2
∂xj1∂xh1 − 1
2Gˆj ∂2gˆis
∂yj1∂yi1
∂f2
∂xs1 − 1
2gˆjh∂f2
∂xh1
Gˆiji+1 4
1
f2gˆih∂f2
∂xh1
∂f2
∂xi1
−1 2
1
f2yh1∂f2
∂xh1
∂f2
∂xs1
∂ˆgis
∂y1i + 1 4
Gˆij∂gˆjs
∂y1i
∂f2
∂xs1 +1 4
Gˆji∂gˆih
∂yj1
∂f2
∂xh1] (24)
−1 4
∂ˆgis
∂yi1
∂f2
∂xs1[ ˇGβµν∂F22
∂y2β + ˇGβµ ∂F22
∂yβ2∂y2ν + ˇGβ ∂3F22
∂y2β∂y2µ∂y2ν + ˇGβν ∂F22
∂yβ2∂yµ2] +1
2
∂f2
∂xh1
∂f2
∂xs1
∂2F24
∂y2µ∂y2ν[− 1 16
∂ˆgih
∂y1j
∂ˆgjs
∂y1i +1
8gˆjh ∂2gˆis
∂yj1∂yi1] +1
8y2β∂gˆih
∂yi1
∂f2
∂xh1
∂3F22
∂xβ2∂yµ2∂y2ν +1 2
1
f2y1h∂f2
∂xh1 Gˇαµνα
Definition 4.1. Warped product Finsler metricF is called an Einstein metric if there exists a constant K ∈R such that
Ricab =Kgab. (25)
In this case, the warped product Finsler space F = (M, F) is called an Einstein manifold. In Riemannian geometry, the warped product manifold (M, g) = M1×fM2 is Einstein with Ric =Kgif and only if (M2,gˇ) is Einstein, i.e., ˇRic =K2gˇfor a constant K2 and the followings hold [17]:
Kˆg = ˆRic− d fHf, K = K2
f2 − ∆f
f −(d−1)|∇f f |2gˆ, (26)
where dimM1 ≥2, dimM2 =d≥2 and ∆f = trHf = tr(Hessf).
Now, we want to generalize this result on warped product Finsler manifolds.
In fact, we answer Chern’s question on warped product Finsler space. He asked, ”whenever a smooth manifold admits an Einstein Finsler metric?”
Theorem 4.2. Let F=F1×fF2 be a warped product Finsler space. Consider the warped product Finsler metric F is an Einstein metric of constant K then the following conditions hold:
Ricˆ kl =Kˆgkl−1 2
∂ ln f2
∂xj1
Gˆjkl+1 2
∂2ln f2
∂xk1∂xl1, (27)
∆ˆhf2− 1 2
1 f2
G( ˆˆ ∇hf2,∇ˆhf2) + 2Kf2 −2ˇgµνRicˇ µν +Pijjˆgih∂f2
∂xh1 =
∂f2
∂xs1(1 2
Gˆij∂gˆjs
∂yi1 −Gˆj ∂2gˆis
∂y1j∂y1i) +y1hgˇµν∂lnf2
∂xh1 Gˇαµνα
− ∂f2
∂xh1(1
2Gˆji∂gˆih
∂yj1 + ˆgjhGˆiji).
(28)
Proof. Using (2), (23) and (25), we obtain
1 2
∂F22
∂yν2 [−1 2
∂2gˆih
∂xi1∂y1k
∂f2
∂xh1 − 1 2
∂gˆih
∂y1k
∂2f2
∂xh1∂xi1 +1 4
Gˆikj∂gˆjs
∂yi1
∂f2
∂xs1 +1 4
Gˆij ∂2ˆgjs
∂y1i∂y1k
∂f2
∂xs1 +1
4
Gˆjki∂ˆgih
∂y1j
∂f2
∂xh1 +1 4
Gˆji ∂2ˆgih
∂yj1∂y1k
∂f2
∂xh1 +1
4y1j ∂3ˆgih
∂xj1∂yi1∂y1k
∂f2
∂xh1 + 1
4y1j ∂2gˆih
∂yi1∂yk1
∂2f2
∂xj1∂xh1
− 1
2Gˆjk ∂2gˆis
∂y1j∂y1i
∂f2
∂xs1 −1
2Gˆj ∂3gˆis
∂y1j∂y1i∂y1k
∂f2
∂xs1 − 1 2
∂ˆgjh
∂y1k
∂f2
∂xh1
Gˆiji− 1
2gˆjh∂f2
∂xh1 Gˆikji +1
4 1 f2
∂gˆih
∂y1k
∂f2
∂xh1
∂f2
∂xi1 −1 2
1
f2yh1∂f2
∂xh1
∂f2
∂xs1
∂2gˆis
∂yi1∂y1k] =−1
8y2β ∂2gˆih
∂yi1∂yk1
∂f2
∂xh1
∂2F22
∂xβ2∂yν2
− 1 2
∂f2
∂xh1
∂f2
∂xs1
∂F24
∂y2ν[− 1 16
∂2ˆgih
∂yj1∂yk1
∂gˆjs
∂y1i − 1 16
∂gˆih
∂yj1
∂2ˆgjs
∂y1i∂y1k + 1 8
∂gˆjh
∂yk1
∂2gˆis
∂yj1∂yi1 +1
8ˆgjh ∂3ˆgis
∂y1j∂y1i∂y1k] + 1 4
∂2gˆis
∂y1i∂yk1
∂f2
∂xs1[ ˇGβν∂F22
∂yβ2 + ˇGβ ∂2F22
∂y2β∂yν2] (29)
Differentiating (29) with respect to yl1 and then contracting it with 12y2ν. By inserting this result into (22), Eq. (25) can be written as
Ricˆ kl−Kgˆkl+ 1 4
Gˇβ ∂3ˆgis
∂y1i∂y1k∂y1l
∂f2
∂xs1
∂F22
∂y2β + 1 2
1 f2
Gˆjkl∂f2
∂xj1 −1 2
∂2ln f2
∂xk1∂xl1 = 1
2
∂f2
∂xh1
∂f2
∂xs1F24[− 1 16
∂3gˆih
∂y1j∂y1k∂yl1
∂gˆjs
∂y1i
− 1 16
∂2ˆgih
∂yj1∂yk1
∂2ˆgjs
∂yi1∂y1l − 1 16
∂2gˆih
∂y1j∂y1l
∂2gˆjs
∂yi1∂y1k
− 1 16
∂gˆih
∂yj1
∂3ˆgjs
∂y1i∂y1k∂y1l +1 8
∂2ˆgjh
∂yk1∂yl1
∂2gˆis
∂y1j∂y1i + 1 8
∂gˆjh
∂yk1
∂3gˆis
∂y1j∂yi1∂y1l + 1
8
∂gˆjh
∂y1l
∂3gˆis
∂yj1∂yi1∂y1k +1
8ˆgjh ∂4gˆis
∂y1j∂y1i∂y1k∂y1l] (30)
