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On a projective class of Finsler metrics with orthogonal invariance
Differential Geometry and its Applications 52 (2017) 167–180
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Differential Geometry and its Applications www.elsevier.com/locate/difgeo
On a projective class of Finsler metrics with orthogonal invariance Xiaohuan Mo a,1 , Hongmei Zhu b,2 a
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China b College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, PR China
a r t i c l e
i n f o
Article history: Received 16 January 2017 Available online xxxx Communicated by Z. Shen MSC: 58E20 53B40
a b s t r a c t In this paper, we study a class of Finsler metrics, called generalized Douglas– Weyl (GDW) metrics, which includes Douglas metrics and Weyl metrics. We find a sufficient and necessary condition for an orthogonally invariant Finsler metric to be a GDW-metric. As its application, we show that a certain class of Finsler metrics with orthogonal invariance are Douglas metrics if they are GDW-metrics, generalized a theorem previously only known in the case of Weyl metrics. © 2017 Elsevier B.V. All rights reserved.
Keywords: Finsler metric Douglas curvature Generalized Douglas–Weyl metric Orthogonal invariance
1. Introduction The Douglas curvature is one of the important quantities in projective Finsler geometry. There are two class of Finsler metrics with important Douglas curvature properties. The first one is Douglas metrics [5]. The second one is generalized Douglas–Weyl metrics [7,10]. A Finsler metric is called a Douglas metric if it has vanishing Douglas curvature. A Finsler metric is said to be of generalized Douglas–Weyl type (GDW type for short) if the rate of change of the Douglas curvature along a geodesic is tangent to the geodesic. Note that every Douglas metric must be of GDW type. Why do we call the second class the generalized Douglas–Weyl metrics? According to Sakaguchi’s surprising result, all Weyl metrics (metrics of vanishing Weyl curvature) must be of GDW type [8]. It is well-known that a Finsler metric is a Weyl metric if and only if it is of scalar flag curvature, namely, the flag curvature K(P, y) = K(x, y) is independent of the section P containing y. E-mail addresses: [email protected] (X. Mo), [email protected] (H. Zhu). Supported by the National Natural Science Foundation of China 11371032. 2 Supported by the National Natural Science Foundation of China 11626091 and Youth Science Fund of Henan Normal University 2015QK01. 1
http://dx.doi.org/10.1016/j.difgeo.2017.03.014 0926-2245/© 2017 Elsevier B.V. All rights reserved.
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In 2007, Najafi–Shen–Tayebi found an equation that characterize GDW-metrics of Randers type [6]. For example, the following Randers metric on Bn (ν) F =
f (|x|)|y|2 + τ 2 f 2 (|x|)x, y2 + τ f (|x|)x, y
(1.1)
is a generalized Douglas–Weyl metric, where f is any differentiable function and κ is a constant. Finsler metrics in (1.1) satisfy F (Ax, Ay) = F (x, y) for all A ∈ O(n), equivalently, the orthogonal group O(n) acts as isometries of F . Such metrics are said to be orthogonally invariant (spherically symmetric in an alternative terminology in [2, 5]). y Any orthogonally invariant Finsler metric F = F (x, y) can be expressed by F (x, y) = |y|φ |x|, x, |y| [3]. Hence all orthogonally invariant Finsler metrics are general (α, β)-metrics [11]. They include many Finsler metrics with nice curvature properties [2, 3, 4, 5]. A lot of non-trivial spherically symmetric metrics of scalar flag curvature had been constructed explicitly in [2]. In this paper, we find a sufficient and necessary condition for a spherically symmetric Finsler metric F to be a GDW-metric. Precisely, we show the following: y Theorem 1.1. Let F = |y|φ |x|, x, be an orthogonally invariant Finsler metric on Bn (ν) ⊂ Rn . Then |y| F is a GDW-metric if and only if 1 6Q(Qs − sQss ) + (Qrs − sQrss ) − 1 − 2Q(r2 − s2 ) Qsss = 0 r where Q is given in the first equation of (2.2). Note that spherically symmetric Finsler metric F = |y|φ |x|, R2 = 0 [2,4]. On the other hand, we have
x, y |y|
is of scalar curvature if and only if
∂R2 1 = 6Q(Qs − sQss ) + (Qrs − sQrss ) − 1 − 2Q(r2 − s2 ) Qsss . ∂s r This verifies Sakaguchi’s theorem for Finsler metrics with orthogonal invariance [8]. Very recently, H. Zhu has discussed a certain class of orthogonally invariant Finsler metrics. She showed that these metrics are Douglas metrics if they are of scalar flag curvature [12]. As an application of Theorem 1.1, we show Zhu’s result holds for GDW-metrics with orthogonal invariance. Theorem 1.2. Let F = |y|φ (r, s) be an orthogonally invariant Finsler metric on Bn (ν), where r = |x| and y s = x, |y| . Assume that Q = Q(r, s) is a polynomial function of k degree with respect to s defined by Q(r, s) = f0 (r) + f1 (r)s + · · · + fk (r)sk . Then F is a GDW-metric if and only if it is a Douglas metric. According to T. Sakaguchi’s result, all Weyl metrics are GDW-metrics [8]. Here we weaken Zhu’s condition. Finally, we should point out that Mo–Solorzano–Tenenblat obtain all the Douglas metrics with orthogonal invariance [6].
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2. Preliminaries In this section we will give some definitions and lemmas. Let M be a manifold. Let T M = ∪x∈M Tx M be the tangent bundle of M , where Tx M is the tangent space at x ∈ M . We set T Mo := T M \ {0} where {0} stands for {(x, 0)| x ∈ M, 0 ∈ Tx M }. A Finsler metric on M is a function F : T M → [0, ∞) with the following properties (a) F is C ∞ on T Mo ; (b) At each point x ∈ M , the restriction Fx := F |Tx M is a Minkowski norm on Tx M . Let F be a Finsler metric on Bn (ν) := {v ∈ Rn | |v| < ν}. F is said to be orthogonally invariant if it satisfies F (Ax, Ay) = F (x, y) for all x ∈ Bn (ν), y ∈ Tx Bn (ν) and A ∈ O(n). Let | , | and , be the standard Euclidean norm and inner product on Rn . In [3], Huang–Mo showed the following: Lemma 2.1. A Finsler metric F on Bn (ν) is orthogonally invariant if and only if there is a function φ : [0, ν) × R → R such that
x, y F (x, y) = |y|φ |x|, |y|
where (x, y) ∈ T Bn (ν) := T Bn (ν)\{0}. By a straightforward computation one obtains the following [11, Proof of Theorem 1.2]: y Lemma 2.2. Let F = |y|φ |x|, x, be an orthogonally invariant Finsler metric on Bn (ν) ⊂ Rn . Then its |y| geodesic coefficients are given by Gi = uP y i + u2 Qxi
(2.1)
where Q :=
rφss − φr + sφrs 1 , 2r φ − sφs + (r2 − s2 )φss
u = |y|,
r := |x|,
s :=
x, y , |y|
(2.2)
and Q rφs + sφr − sφ − (r2 − s2 )φs . 2rφ φ
(2.3)
1 j 1 s y , [u2 ]yj = 2y j , syj = (xj − y j ). u u u
(2.4)
P := Direct computations yield uy j = Let
¯ i = u2 Qxi . G
(2.5)
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¯ := y i ∂ i − 2G ¯ i ∂ i is a spray on Bn (ν). It follows that the Finsler spray Then G ∂x ∂y G := y i
∂ ∂ − 2Gi i i ∂x ∂y
¯ [9]. We denote the corresponding objects with respect to G ¯ by adding a bar ¯. is projectively equivalent to G Then ¯i ¯ji = ∂ G , N ∂y j
¯i = Γ jk
¯i ∂N j ∂y k
(2.6)
¯ Note that N ¯ i (resp. G ¯ i ) is homogeneous of degree with respect to the Berwald connection induced by G. j one (resp. two) with respect to y. From (2.6) we get ¯ i yk = N ¯ i. Γ jk j
¯i, ¯ i y j = 2G N j
(2.7)
By definition, the Douglas curvature is given by Dj
i
kl
∂3 := ∂y j ∂y k ∂y l
1 ∂Gm i y G − n + 1 m ∂y m
i
.
¯ Then Let “ ” denote the covariant differentiation with respect to G. Dj i klm =
i ∂Dj i kl p ∂Dj kl − N + Dj p kl Γipm − Dp i kl Γpjm − Dj i pl Γpkm − Dj i kp Γplm . m ∂xm ∂y p
Together with (2.7) we obtain ¯ i − Dp i kl N ¯ p − Dj i pl N ¯ p − Dj i kp N ¯p Dj i klm y m = (I) + Dj p kl N p j k l
(2.8)
where (I) :=
ym
∂ ¯m ∂ − 2G m ∂x ∂y m
¯ j i kl ). Dj i kl = G(D
(2.9)
By (2.5), (2.4) and the first equation of (2.6) ¯ji = ∂ (u2 Qxi ) = (u2 )yj Qxi + u2 Qs syj xi = W j xi , N ∂y j
(2.10)
W j := (2Q − sQs )y j + uQs xj .
(2.11)
where
Lemma 2.3. Let f = f (r, s) be a function on a domain U ⊆ R2 . Then s fxi y i = u( fr + fs ), r
fyi xi = i
i
fs 2 (r − s2 ). u
(2.12)
Proof. Simple calculations give the following: fxi = xr fr + yu fs . It follows that the first equation of (2.12) holds. By the third equation of (2.4), we have the second equation of (2.12). 2
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Corollary 2.4. Let f = f (r, s) be a function on a domain U ⊆ R2 . Then ¯ ) = u s fr + Ψfs G(f r
(2.13)
Ψ := 1 − 2Q(r2 − s2 ).
(2.14)
where
Proof. By (2.5) and (2.12), we have ¯ ) = fxi y i − 2u2 Qfyi xi = u s fr + Ψfs . G(f r From (2.5) and the first equation of (2.4), we have ¯ k ) = y k , G(y ¯ k ) = −2u2 Qxk . ¯ G(u) = −2u2 sQ, G(x
(2.15)
For an orthogonally invariant Finsler metric F = |y|φ(|x|, x,|y|y ) on Bn (ν) ⊆ Rn , the Douglas curvature of F is given by [5] Dj i kl =
1 [Rss δji xk xl + (R − sRs )δji δkl ](j → k → l → j) u sRss − 2 [δji (xk y l + xl y k ) + y i δjk xl ](j → k → l → j) u 1 + 3 (sRss + sRs − R)(δji y k y l + y i δjk y l )(j → k → l → j) u 1 Rsss + 5 (3R − s2 Rsss − 6s2 Rss − 3sRs )y i y j y k y l + 2 y i xj xk xl u u s 1 + 4 (sRsss + 3Rss )y i y j y k xl (j → k → l → j) + [(Qs − sQss )xi xl δjk u u 1 i j k l − 2 (Rss + sRsss )y y x x ](j → k → l → j) u 1 + 3 (s2 Qsss + sQss − Qs )xi xj y k y l (j → k → l → j) u s + 2 [(sQss − Qs )xi y j δkl − Qsss xi xj xl y k ](j → k → l → j) u xi + 4 [u3 Qsss xj xk xl + s(3Qs − 3sQss − s2 Qsss )y j y k y l ] u
(2.16)
where j → k → l → j denotes cyclic permutation, Q is given by (2.2) and R := −
1 2sQ + (r2 − s2 )Qs . n+1
By using (2.16) and Lemma 4.1–Lemma 4.9 below, we have ¯ j i kl ) = Akl δ i (j → k → l → j) + Bjkl xi (j → k → l → j) G(D j 2 Q(3R − 3sRs − 6s2 Rss − s3 Rsss )xi y j y k y l − 2QRsss xi xj xk xl u3 + Cjkl y i (j → k → l → j) + Dy i y j y k y l
1 s + 4sQRsss + Rrsss + ΨRssss ) + 6Q(Rss + sRsss y i xj xk xl u r −
(2.17)
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+ Ejkl xi (j → k → l → j) + F xi y j y k y l (j → k → l → j) s + (8sQQsss + Qrsss + ΨQssss )xj xk xl xi + Gjkl y i (j → k → l → j) r s Qsss j k l i + x x x y + 4 (3Qs − 3sQss − s2 Qsss )y j y k y l y i , u u
(2.18)
where s 1 Akl : = (6sQRss + Rrss + ΨRsss )xk xl + s[2Q(R − sRs ) + (Rr − sRrs ) − ΨRss ]δkl r r 2 1 s + [(1 − 4s2 Q)Rss − Rrss − (Rss + sRsss )Ψ − 2Q(s2 Rss + sRs − R)](xl y k + xk y l ) u r s 1 + 2 [6Q(s2 Rss + sRs − R) + (s2 Rrss + sRrs − Rr ) + Ψ(3Rss + sRsss ) − 2Rss ]y k y l , u r (2.19) Bjkl : = 2sQRss δjk xl − + Cjkl : =
2Q 2 2sQ (s Rss + sRs − R)δjk y l − 2 (sRsss + 3Rss )xl y j y k u u
2Q (Rss + sRsss )xk xl y j , u
(2.20)
1 s [6sQ(s2 Rss + sRs − R) + (s2 Rrss + sRrs − Rr ) − sRss + Ψ(3sRss + s2 Rsss )]δjk y l u2 r 1 s2 − [4s2 QRss + Rrss + (Rss + sRsss )Ψ + 2Q(s2 Rss + sRs − R)]δjk xl u r 1 s2 + 3 [8s2 Q(3Rss + sRsss ) + (sRrsss + 3Rrss ) − 2Q(3R − 3sRs − 6s2 Rss − s3 Rsss ) u r + Ψ(3Rss + 5sRsss + s2 Rssss ) − 2(Rss + sRsss )]y j y k xl
1 [Rsss − 4sQ(sRsss + 3Rss ) − 6sQ(Rss + sRsss ) u2 s − (Rrss + sRrsss ) − Ψ(2Rsss + sRssss )]y j xk xl , r s 1 D : = 4 [10Q(3R − 3sRs − 6s2 Rss − s3 Rsss ) + (3Rr − 3sRrs − 6s2 Rrss − s3 Rrsss ) u r 2 − Ψ(9sRsss + 15Rss + s Rssss ) + 3(sRsss + 3Rss )], +
1 Ejkl : = s[4Q(Qs − sQss ) + (Qrs − sQrss ) − ΨQsss ]xl δjk r 1 s2 + [(1 − 4s2 Q)(Qs − sQss ) + (sQrss − Qrs ) + Ψ(s2 Qsss + sQss − Qs )]y l δjk u r s 2 + 2 [12Q(sQss − Qs ) + 2(4s Q − 1)Qsss u 1 2 + (s Qrsss + sQrss − Qrs ) + Ψ(3Qsss + sQssss )]xj y k y l r 1 s2 − [(8s2 Q − 1)Qsss + 4Q(sQss − Qs ) + Qrsss + Ψ(Qsss + sQssss )]xj xk y l , u r 1 F : = 3 [3(s2 Qsss + sQss − Qs ) + 8s2 Q(3Qs − 3sQss − s2 Qsss ) u s2 + (3Qrs − 3sQrss − s2 Qrsss ) + Ψ(3Qs − 3sQss − 6s2 Qsss − s3 Qssss ), r 1 s 1 s Gjkl : = (Qs − sQss )(xl − y l )δjk + 3 (s2 Qsss + sQss − Qs )xj y k y l − 2 Qsss xj xl y k , u u u u
(2.21)
(2.22)
(2.23)
(2.24) (2.25)
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where Ψ is given in (2.14). By using (2.11), we have Σj xj W j = 2suQ + u(r2 − s2 )Qs ,
Σj y j W j = 2Qu2 .
Together with (2.10) and (2.16), we obtain ¯pi = Σp (Dj p kl W p )xi Dj p kl N s s = [Qs (R − sRs ) − 2sQRss ](xl − y l )δjk − (3Qs Rss + 2QRsss )xj xk y l u u xi + 2 [Qs (s2 Rss + sRs − R) + 2s2 Qs Rss + 2sQ(sRsss + Rss )]xk y j y l (j → k → l → j) u 1 + 3 [2Q(3R − s3 Rsss − 6s2 Rss − 3sRs ) + 3(2Q − sQs )(s2 Rss + sRs − R)]y j y k y l xi u + (3Qs Rss + 2QRsss )xj xk xl xi + [2sQ + Qs (r2 − s2 )]xi s s2 Qsss + sQss − Qs j k l sQsss j l k x x y (j → k → l → j) × (Qs − sQss )(xj − y j )δlk + x y y − u u2 u 1 + 3 [2sQ + Qs (r2 − s2 )] s(3Qs − 3sQss − s2 Qsss )y j y k y l + u3 Qsss xj xk xl xi . (2.26) u By (2.10), we obtain ¯ p = W j (Dp i kl xp ). Dp i kl N j
(2.27)
From (2.16), we get Dp i kl xp =
1 s [ΦRss + R − sRs ](xl − y l )δ i k (k ↔ l) u u sRss Rss k l 1 1 x x + 3 (s2 Rss + sRs − R)y k y l ]xi + [ (R − sRs )δkl − 2 xk y l (k ↔ l) + u u u u s 1 + [− 2 (R − sRs + ΦRss )δlk + 3 (3s2 Rss + sRs − R − Φ(Rss + sRsss ))xk y l (k ↔ l) u u s 1 + 2 (ΦRsss − 3sRss )xk xl + 4 (3(R − sRs − s2 Rss ) + Φ(3Rss + sRsss ))y k y l ]y i u u 1 s + [ (Qs − sQss )Φδkl + 2 [2(sQss − Qs ) − ΦQsss ]xk y l (k ↔ l) u u 2(Qs − sQss ) + ΦQsss k l 1 + x x + 3 [Φ(s2 Qsss + sQss − Qs ) + 2s2 (Qs − sQss ))y k y l ]xi u u
where k ↔ l denotes symmetrization and Φ = r2 − s2 . Plugging this into (2.27) and using total symmetry of Dj i kl with respect to j, k, l we obtain ¯ p (j → k → l → j) = W j (Dp i kl xp )(j → k → l → j) Dp i kl N j 1 s [ΦRss + R − sRs ](xl − y l )δ i k (k ↔ l) = (2Q − sQs )y j + uQs xj u u sRss k l Rss k l 1 1 x x + 3 (s2 Rss + sRs − R)y k y l ]xi + [ (R − sRs )δkl − 2 x y (k ↔ l) + u u u u s 1 + [− 2 (R − sRs + ΦRss )δlk + 3 (3s2 Rss + sRs − R − Φ(Rss + sRsss ))xk y l (k ↔ l) u u s 1 + 2 (ΦRsss − 3sRss )xk xl + 4 (3(R − sRs − s2 Rss ) + Φ(3Rss + sRsss ))y k y l ]y i u u
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1 s 2(Qs − sQss ) + ΦQsss k l x x + [ (Qs − sQss )Φδkl + 2 [2(sQss − Qs ) − ΦQsss ]xk y l (k ↔ l) + u u u 1 + 3 [Φ(s2 Qsss + sQss − Qs ) + 2s2 (Qs − sQss ))y k y l ]xi (j → k → l → j) u ΦRss + R − sRs 2s [2uQs xk xl + 2(Q − sQs )(xk y l + xl y k ) − (2Q − sQs )y k y l ]δ i j (j → k → l → j) = u u 1 1 (R − sRs )W l δjk + 2 [Qs (s2 Rss + sRs − R) − 2sRss (2Q − sQs )]xj y k y l + u u Rss (2Q − 3sQs )xj xk y l xi (j → k → l → j) + u 3 + 3 (2Q − sQs )(s2 Rss + sRs − R)y j y k y l xi + 3Qs Rss xk xl xj xi u s − 2 [R − sRs + ΦRss ]W j δjk u 1 − 3 [(4Q − 5sQs )(s2 Rss + sRs − R) + 4s2 (2Q − sQs )Rss + ΦsQs (3Rss + sRsss ) u − 2Φ(2Q − sQs )(Rss + sRsss )]xj y k y l 1 − 2 [Φ(2Q − sQs )Rsss − 3s(2Q − sQs )Rss + 2Qs (3s2 Rss + sRs − R) u − 2Qs Φ(Rss + sRsss )]xj xk y l y i (j → k → l → j) 3Qs 3s(2Q − sQs ) [ΦRsss − 3sRss ]xk xl xj y i [3(R − sRs − s2 Rss ) + Φ(3Rss + sRsss )]y j y k y l y i + 4 u u 1 1 (Qs − sQss )ΦW l δjk + 2 [4s(2Q − sQs )(sQss − Qs ) − 2sΦQsss (2Q − sQs ) + u u 1 + Qs Φ(s2 Qsss + sQss − Qs ) + 2s2 Qs (Qs − sQss )]xj y k y l + [2(2Q − sQs )(Qs − sQss ) u + ΦQsss (2Q − sQs ) + 4sQs (sQss − Qs ) − 2sQs ΦQsss ]xj xk y l xi (j → k → l → j) +
+
3 (2Q − sQs )[Φ(s2 Qsss + sQss − Qs ) + 2s2 (Qs − sQss )]y j y k y l xi u3
+ 3Qs [2(Qs − sQss ) + ΦQsss ]xk xl xj xi .
(2.28)
By using (2.8), (2.9), (2.18)–(2.28) we have s s T1 (xk y l + xl y k ) + 2 (sT1 − T2 )y k y l δ i j (j → k → l → j) u u s 1 1 + T3 δjk xl − (T2 + T3 )δjk y l + 2 (2sT1 − T3 + s2 T4 )xl y j y k u u u 1 − (T1 + sT4 )xj xk y l y i (j → k → l → j) u T4 s + 4 [3(T2 + T3 − sT1 ) − s2 T4 ]y j y k y l y i + xk xl xj y i u u s 1 1 + s T5 (xl − y l )δjk − 2 (T5 − sT6 )xj y l y k − T6 xj xk y l xi (j → k → l → j) u u u 2 s + 3 (3T5 − sT6 )y j y k y l xi + T6 xk xl xj xi + Gjkl y i (j → k → l → j) u s Qsss j k l i x x x y + 4 (3Qs − 3sQss − s2 Qsss )y j y k y l y i , (2.29) + u u
Dj i klm y m = [T1 xk xl + sT2 δkl −
where
X. Mo, H. Zhu / Differential Geometry and its Applications 52 (2017) 167–180
s T1 := 2[3sQ − Qs (r2 − s2 )]Rss + Rrss + ΨRsss − 2Qs (R − sRs ), r 1 T2 := 2Q(R − sRs ) + (Rr − sRrs ) − ΨRss , r 2 2 T3 := −sT1 − [Ψ + sQs (r − s )]Rss + (2Q − sQs )(R − sRs ), s T4 := 3(2Q + 3sQs )Rss + [10sQ − 3Qs (r2 − s2 )]Rsss + Rrsss + ΨRssss , r 1 T5 := 6Q(Qs − sQss ) + (Qrs − sQrss ) − ΨQsss , r s T6 := 6Qs (sQss − Qs ) + 2[5sQ − Qs (r2 − s2 )]Qsss + Qrsss + ΨQssss . 2 r
175
(2.30) (2.31) (2.32) (2.33) (2.34) (2.35)
3. Proof of theorems First we are going to prove the following y Theorem 3.1. Let F = |y|φ |x|, x, be an orthogonally invariant Finsler metric on Bn (ν) ⊂ Rn . Then |y| F is a GDW-metric if and only if s s 1 T1 (xk y l + xl y k ) + 2 (sT1 − T2 )y k y l (δ i j − 2 y i y j )(j → k → l → j) u u u s 1 1 + s T5 (xl − y l )δjk − 2 (T5 − sT6 )xj y l y k − T6 xj xk y l (j → k → l → j) u u u 2 s s + 3 (3T5 − sT6 )y j y k y l + T6 xk xl xj (xi − y i ) = 0, u u
[T1 xk xl + sT2 δkl −
(3.1)
where T1 , · · · , T6 are defined in (2.30), (2.31), (2.32), (2.33), (2.34) and (2.35) respectively. y Proof. Let F = |y|φ |x|, x, be an orthogonally invariant Finsler metric on Bn (ν). Let G denote the |y| ¯ the spray defined in (2.5). Since G ¯ and G are projectively equivalent, the following spray of F and G conditions are equivalent [1,7] (i) F is a GDW-metric, i.e. there is a tensor Tjkl such that Dji kl;m y m = Tjkl y i ,
(3.2)
Dji klm y m = Djkl y i ,
(3.3)
(ii) there is a tensor Djkl such that
where Dji kl;m denotes the covariant derivatives of Dji kl with respect to the Berwald connections of G. This equivalence is essentially proved in [1]. Assume that F is a GDW-metric. Then (3.3) holds for some tensor Djkl . By (3.3) and (2.29), we have s s T1 (xk y l + xl y k ) + 2 (sT1 − T2 )y k y l δ i j (j → k → l → j) u u s 1 1 l l + T3 δjk x − (T2 + T3 )δjk y + 2 (2sT1 − T3 + s2 T4 )xl y j y k u u u 1 − (T1 + sT4 )xj xk y l y i (j → k → l → j) u s T4 + 4 [3(T2 + T3 − sT1 ) − s2 T4 ]y j y k y l y i + xk xl xj y i u u
Djkl y i = [T1 xk xl + sT2 δkl −
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s 1 1 + s T5 (xl − y l )δjk − 2 (T5 − sT6 )xj y l y k − T6 xj xk y l xi (j → k → l → j) u u u s2 + 3 (3T5 − sT6 )y j y k y l xi + T6 xk xl xj xi + Gjkl y i (j → k → l → j) u Qsss j k l i s + x x x y + 4 (3Qs − 3sQss − s2 Qsss )y j y k y l y i . u u
(3.4)
It follows that Djkl = =
1
(Djkl y i )y i u2 i 1 s s [T1 xk xl + sT2 δkl − T1 (xk y l + xl y k ) + 2 (sT1 − T2 )y k y l y j (j → k → l → j) 2 u u u 1 s 1 + T3 δjk xl − (T2 + T3 )δjk y l + 2 (2sT1 − T3 + s2 T4 )xl y j y k u u u 1 − (T1 + sT4 )xj xk y l (j → k → l → j) u s T4 + 4 [3(T2 + T3 − sT1 ) − s2 T4 ]y j y k y l + xk xl xj u u s2 s 1 1 T5 (xl − y l )δjk − 2 (T5 − sT6 )xj y l y k − T6 xj xk y l (j → k → l → j) + u u u u sT s3 6 x k xl xj + 4 (3T5 − sT6 )y j y k y l + u u 1 s 1 (Qs − sQss )(xl − y l )δjk + 3 (s2 Qsss + sQss − Qs )xj y k y l + u u u s − 2 Qsss xj xl y k (j → k → l → j) u Qsss j k l s + x x x + 4 (3Qs − 3sQss − s2 Qsss )y j y k y l . u u
(3.5)
Plugging (3.5) into (3.4) yields (3.1). Conversely, if (3.1) holds, it follows from (2.29) that Dji klm y m = Djkl y i where Djkl are given by (3.5). Thus F is a GDW-metric. 2 Lemma 3.2. (i). T5 =
∂R2 ∂s
(3.6)
where 1 R2 := 2Q(2Q − sQs ) + (2Qr − sQrs − rQss ) + Φ(2QQss − Q2s ) r
(3.7)
where Q is given in the first equation of (2.2). (ii). (3.1) is equivalent to the following 1 T5 := 6Q(Qs − sQss ) + (Qrs − sQrss ) − ΨQsss = 0. r Proof. (i). A simple calculation gives (3.6). (ii). From (2.17) we get
(3.8)
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R − sRs = −
177
Φ (Qs − sQss ), n+1
(3.9)
1 [2r(Qs − sQss ) + Φ(Qrs − sQrss )] , n+1 1 [2(Qs − sQss ) + ΦQsss ] . =− n+1
Rr − sRrs = − Rss
(3.10) (3.11)
Substituting these into (2.31) and using (2.34), we have T2 = −
r2 − s2 T5 . n+1
(3.12)
From (2.30), (2.31), (2.14), (2.34) and (2.35) we get (T2 )s = −T1 ,
(T5 )s = −T6 .
(3.13)
First suppose that (3.1) holds. We take j = k = l = i, then T6 (xi )4 + 3(T1 + sT5 )(xi )2 + 3sT2 ≡ 0,
mod y i .
This gives T2 = 0,
T1 + sT5 = 0,
T6 = 0.
(3.14)
Take x and y with x ∧ y = 0. Then Φ = r2 − s2 =
|x|2 |y|2 − x, y2 > 0. |y|2
Together with (3.12) and the first equation of (3.14) we have T5 = 0. Conversely, suppose that T5 = 0. Together with (3.12) and (3.13) we get T2 = 0, Hence (3.1) holds. 2
T1 = 0,
T6 = 0.
Proof of Theorem 1.1. Combining Lemma 3.2 with Theorem 3.1. 2 y It is known that F = |y|φ |x|, x, is a Douglas metric if and only if Qs − sQss = 0 [5]. On the other |y| y hand, Huang–Mo recently have found a sufficient and necessary condition for F = |y|φ |x|, x, to be of |y| scalar curvature, that is, 1 R2 := 2Q(2Q − sQs ) + (2Qr − sQrs − rQss ) + Φ(2QQss − Q2s ) = 0. r Theorem 1.1 tell us any spherically symmetric metrics of scalar curvature belongs to GDW (Bn (ν)) where GDW (Bn (ν)) denotes the class of all GDW-metrics on Bn (ν). Proof of Theorem 1.2. In [6], authors showed that a spherically symmetric Finsler metric F is a Douglas metric if and only if Q = Q(r, s) is given by Q(r, s) = f0 (r) + f2 (r)s2 . Case 1. k ≥ 1. By (3.15), we have [4] R2 ≡ (4 − k2 )fk2 s2k ,
mod s0 , s1 , · · · , s2k−1
(3.15)
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where R2 is given in (3.7). A simple calculation gives the following formula: ∂R2 ≡ 2k(4 − k2 )fk2 s2k−1 , ∂s
mod
s0 , s1 , · · · , s2k−2 .
First suppose that F is a GDW-metric. By Lemma 3.2 we get k = 2. It follows that [4] 2 1 R2 = 4f02 + f0 − r2 f12 − 2f2 + 4r2 f0 f2 + (6f0 f1 + f1 )s + 3f12 s2 + 2f1 f2 s3 . r r ∂R2 1 2 2 2 It follows that ∂R ∂s = 6f0 f1 + r f1 + 6f1 s + 6f1 f2 s . Note that T5 = ∂s = 0. Hence we obtain that f1 = 0. It follows that Q satisfies (3.15). Thus F is a Douglas metric. Conversely, we suppose that F is a Douglas metric, then it is easy to see from (3.2) that F is a GDWmetric. Case 2. k = 0. This is an immediate conclusion of (3.15). 2
4. Appendix We establish the Lemmas required in the computation of (2.18). By using (2.13) and (2.15), the following Lemmas can be obtained by straightforward calculations. ¯ on Bn (ν) we have Lemma 4.1. For the spray G ¯ G
1 i k l i [Rss δj x x + (R − sRs )δj δkl ](j → k → l → j) u s = 2sQδji [Rss xk xl + (R − sRs )δkl ] + ( Rrss + Rsss Ψ)δji xk xl (j → k → l → j) r 1 s i k l k l i + [ (Rr − Rrs ) − ΨsRss ]δj δkl + Rss (y x + x y )δj (j → k → l → j). r u
¯ on Bn (ν) we have Lemma 4.2. For the spray G ¯ G
sRss i k l l k i l [δ (x y + x y ) + y δjk x ](j → k → l → j) u2 j (y k xl + xk y l )δji + y i δjk xl s2 2 4s QRss + Rrss + Ψ(Rss + sRsss ) (j → k → l → j) = u r sRss 2δji (y k y l − 2u2 Qxk xl ) + δjk (y i y l − 2u2 Qxi xl ) (j → k → l → j). + u
¯ on Bn (ν) we have Lemma 4.3. For the spray G ¯ G
1 (sRss + sRs − R)(δji y k y l + y i δjk y l )(j → k → l → j) u3
=
6sQ(s2 Rss + sRs − R) + rs (s2 Rrss + sRrs − Rr ) + Ψ(3sRss + s2 Rsss ) (j → k → l → j) u2 2Q 2 (s Rss + sRs − R) (y k xl + xk y l )δji + (y l xi + xl y i )δjk (j → k → l → j). − u
¯ on Bn (ν) we have Lemma 4.4. For the spray G
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¯ G
1 Rsss i j k l 2 2 i j k l (3R − s Rsss − 6s Rss − 3sRs )y y y y + 2 y x x x u5 u s 1 = 4 10QΘ + Θr − Ψ(9sRsss + 15Rss + s2 Rssss y i y j y k y l u r 1 2Q s − 3 Θ xi y j y k y l + y i xj y k y l (j → k → l → j) + (4sQRsss + Rrsss + ΨRssss )y i xj xk xl u u r 1 i j k l 2 i j k l + 2 y y x x (j → k → l → j) − 2u Qx x x x Rsss , u
where Θ := 3R − 3sRs − 6s2 Rss − s3 Rsss . ¯ on Bn (ν) we have Lemma 4.5. For the spray G ¯ s (sRsss + 3Rss )y i y j y k xl (j → k → l → j)) G( u4 =
8s2 Q(3Rss + sRsss ) +
s2 r (3Rrss
+ sRrsss ) + Ψ(3Rss + 5sRsss + s2 Rssss ) i j k l y y y x (j → k → l → j) u3 s(3Rss + sRsss ) i j k l + 3y y y y − 2u2 Q(y j y k xl xi + 2xj y k xl y i )(j → k → l → j) . 4 u
¯ on Bn (ν) we have Lemma 4.6. For the spray G ¯ 1 [(Qs − sQss )xi xl δjk − 1 (Rss + sRsss )y i y j xk xl ](j → k → l → j)) G( u u2 6sQ(Rss + sRsss ) + rs (Rrss + sRrsss ) + Ψ(2Rsss + sRssss ) i j k l =− y y x x (j → k → l → j) u2 Rss + sRsss i j k l − 2(y y y x − u2 Qy j xk xl xi )(j → k → l → j) − 6u2 Qy i xj xk xl 3 u 1 + s 2Q(Qs − sQss ) + (Qrs − sQrss ) − ΨQsss xi xl δjk (j → k → l → j) r +
1 (Qs − sQss )δjk (y i xl + xi y l )(j → k → l → j). u
¯ on Bn (ν) we have Lemma 4.7. For the spray G ¯ G
1 2 i j k l (s Q + sQ − Q )x x y y (j → k → l → j) sss ss s u3 s 1 = 2 6QΞ + Ξr + Ψ(3Qsss + sQssss ) xi xj y k y l (j → k → l → j) u r Ξ + 3 (xj y k y l y i − 4u2 Qxj xk y l xi )(j → k → l → j) + 3xi y k y j y l , u
where Ξ := s2 Qsss + sQss − Qs . ¯ on Bn (ν) we have Lemma 4.8. For the spray G ¯ G
s i j i j l k [(sQ − Q )x y δ − Q x x x y ](j → k → l → j ss s kl sss u2 s2 1 2 4s Q(sQss − Qs ) + (sQrss − Qrs ) + ΨΞ xi y j δkl (j → k → l → j) = u r
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s (sQss − Qs )δkl (y i y j − 2u2 Qxi xj )(j → k → l → j) u2 s2 1 2 4s QQsss + Qrsss + Ψ(Qsss + sQssss ) xi xj y k xl (j → k → l → j) − u r s − 2 Qsss (xj xl y k y i + 2xj y l y k xi )(j → k → l → j) − 6u2 Qxi xj xk xl . u
+
¯ on Bn (ν) we have Lemma 4.9. For the spray G xi 3 j k l 2 j k l [u Q x x x + s(3Q − 3sQ − s Q )y y y ] sss s ss sss u4 s = (2sQQsss + Qrsss + ΨQssss )xi xj xk xl r 1 j l k i x x x y + y j xl xk xi (j → k → l → j) Qsss + u 1 s2 + 3 8s2 QΛ + Λr + Ψ(Λ − 5s2 Qsss − s3 Qssss ) xi y j y k y l u r sΛ + 4 y i y j y k y l − 2u2 Qxi xj y k y l (j → k → l → j) , u
¯ G
where Λ := 3Qs − 3sQss − s2 Qsss . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
S. Béacso, I. Papp, A note on a generalized Douglas space, Periodica Math. Hungaria 48 (2004) 181–184. L. Huang, X. Mo, On spherically symmetric Finsler metrics of scalar curvature, J. Geom. Phys. 62 (2012) 2279–2287. L. Huang, X. Mo, Projectively flat Finsler metrics with orthogonal invariance, Ann. Pol. Math. 107 (2013) 259–270. H. Liu, X. Mo, Examples of Finsler metrics with special curvature properties, Math. Nachr. 288 (2015) 1527–1537. M. Maleki, N. Sadeghzadeh, T. Rajabi, On conformally related spherically symmetric Finsler metrics, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1650118, 16 pp. X. Mo, N.M. Solorzano, K. Tenenblat, On spherically symmetric Finsler metrics with vanishing Douglas curvature, Differ. Geom. Appl. 31 (2013) 746–758. B. Najafi, Z. Shen, A. Tayebi, On a projective class of Finsler metrics, Publ. Math. (Debr.) (2007) 211–219. T. Sakaguchi, On Finsler spaces of scalar curvature, Tensor N.S. 38 (1982) 211–219. Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, 2001, 258 pages. A. Tayebi, H. Sadeghi, On generalized Douglas–Weyl (α, β)-metrics, Acta Math. Sin. Engl. Ser. 31 (2015) 1611–1620. C. Yu, H. Zhu, On a new class of Finsler metrics, Differ. Geom. Appl. 29 (2011) 244–254. H. Zhu, A class of Finsler metrics of scalar flag curvature, Differ. Geom. Appl. 40 (2015) 321–331.
Contents lists available at ScienceDirect
Differential Geometry and its Applications www.elsevier.com/locate/difgeo
On a projective class of Finsler metrics with orthogonal invariance Xiaohuan Mo a,1 , Hongmei Zhu b,2 a
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China b College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, PR China
a r t i c l e
i n f o
Article history: Received 16 January 2017 Available online xxxx Communicated by Z. Shen MSC: 58E20 53B40
a b s t r a c t In this paper, we study a class of Finsler metrics, called generalized Douglas– Weyl (GDW) metrics, which includes Douglas metrics and Weyl metrics. We find a sufficient and necessary condition for an orthogonally invariant Finsler metric to be a GDW-metric. As its application, we show that a certain class of Finsler metrics with orthogonal invariance are Douglas metrics if they are GDW-metrics, generalized a theorem previously only known in the case of Weyl metrics. © 2017 Elsevier B.V. All rights reserved.
Keywords: Finsler metric Douglas curvature Generalized Douglas–Weyl metric Orthogonal invariance
1. Introduction The Douglas curvature is one of the important quantities in projective Finsler geometry. There are two class of Finsler metrics with important Douglas curvature properties. The first one is Douglas metrics [5]. The second one is generalized Douglas–Weyl metrics [7,10]. A Finsler metric is called a Douglas metric if it has vanishing Douglas curvature. A Finsler metric is said to be of generalized Douglas–Weyl type (GDW type for short) if the rate of change of the Douglas curvature along a geodesic is tangent to the geodesic. Note that every Douglas metric must be of GDW type. Why do we call the second class the generalized Douglas–Weyl metrics? According to Sakaguchi’s surprising result, all Weyl metrics (metrics of vanishing Weyl curvature) must be of GDW type [8]. It is well-known that a Finsler metric is a Weyl metric if and only if it is of scalar flag curvature, namely, the flag curvature K(P, y) = K(x, y) is independent of the section P containing y. E-mail addresses: [email protected] (X. Mo), [email protected] (H. Zhu). Supported by the National Natural Science Foundation of China 11371032. 2 Supported by the National Natural Science Foundation of China 11626091 and Youth Science Fund of Henan Normal University 2015QK01. 1
http://dx.doi.org/10.1016/j.difgeo.2017.03.014 0926-2245/© 2017 Elsevier B.V. All rights reserved.
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In 2007, Najafi–Shen–Tayebi found an equation that characterize GDW-metrics of Randers type [6]. For example, the following Randers metric on Bn (ν) F =
f (|x|)|y|2 + τ 2 f 2 (|x|)x, y2 + τ f (|x|)x, y
(1.1)
is a generalized Douglas–Weyl metric, where f is any differentiable function and κ is a constant. Finsler metrics in (1.1) satisfy F (Ax, Ay) = F (x, y) for all A ∈ O(n), equivalently, the orthogonal group O(n) acts as isometries of F . Such metrics are said to be orthogonally invariant (spherically symmetric in an alternative terminology in [2, 5]). y Any orthogonally invariant Finsler metric F = F (x, y) can be expressed by F (x, y) = |y|φ |x|, x, |y| [3]. Hence all orthogonally invariant Finsler metrics are general (α, β)-metrics [11]. They include many Finsler metrics with nice curvature properties [2, 3, 4, 5]. A lot of non-trivial spherically symmetric metrics of scalar flag curvature had been constructed explicitly in [2]. In this paper, we find a sufficient and necessary condition for a spherically symmetric Finsler metric F to be a GDW-metric. Precisely, we show the following: y Theorem 1.1. Let F = |y|φ |x|, x, be an orthogonally invariant Finsler metric on Bn (ν) ⊂ Rn . Then |y| F is a GDW-metric if and only if 1 6Q(Qs − sQss ) + (Qrs − sQrss ) − 1 − 2Q(r2 − s2 ) Qsss = 0 r where Q is given in the first equation of (2.2). Note that spherically symmetric Finsler metric F = |y|φ |x|, R2 = 0 [2,4]. On the other hand, we have
x, y |y|
is of scalar curvature if and only if
∂R2 1 = 6Q(Qs − sQss ) + (Qrs − sQrss ) − 1 − 2Q(r2 − s2 ) Qsss . ∂s r This verifies Sakaguchi’s theorem for Finsler metrics with orthogonal invariance [8]. Very recently, H. Zhu has discussed a certain class of orthogonally invariant Finsler metrics. She showed that these metrics are Douglas metrics if they are of scalar flag curvature [12]. As an application of Theorem 1.1, we show Zhu’s result holds for GDW-metrics with orthogonal invariance. Theorem 1.2. Let F = |y|φ (r, s) be an orthogonally invariant Finsler metric on Bn (ν), where r = |x| and y s = x, |y| . Assume that Q = Q(r, s) is a polynomial function of k degree with respect to s defined by Q(r, s) = f0 (r) + f1 (r)s + · · · + fk (r)sk . Then F is a GDW-metric if and only if it is a Douglas metric. According to T. Sakaguchi’s result, all Weyl metrics are GDW-metrics [8]. Here we weaken Zhu’s condition. Finally, we should point out that Mo–Solorzano–Tenenblat obtain all the Douglas metrics with orthogonal invariance [6].
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2. Preliminaries In this section we will give some definitions and lemmas. Let M be a manifold. Let T M = ∪x∈M Tx M be the tangent bundle of M , where Tx M is the tangent space at x ∈ M . We set T Mo := T M \ {0} where {0} stands for {(x, 0)| x ∈ M, 0 ∈ Tx M }. A Finsler metric on M is a function F : T M → [0, ∞) with the following properties (a) F is C ∞ on T Mo ; (b) At each point x ∈ M , the restriction Fx := F |Tx M is a Minkowski norm on Tx M . Let F be a Finsler metric on Bn (ν) := {v ∈ Rn | |v| < ν}. F is said to be orthogonally invariant if it satisfies F (Ax, Ay) = F (x, y) for all x ∈ Bn (ν), y ∈ Tx Bn (ν) and A ∈ O(n). Let | , | and , be the standard Euclidean norm and inner product on Rn . In [3], Huang–Mo showed the following: Lemma 2.1. A Finsler metric F on Bn (ν) is orthogonally invariant if and only if there is a function φ : [0, ν) × R → R such that
x, y F (x, y) = |y|φ |x|, |y|
where (x, y) ∈ T Bn (ν) := T Bn (ν)\{0}. By a straightforward computation one obtains the following [11, Proof of Theorem 1.2]: y Lemma 2.2. Let F = |y|φ |x|, x, be an orthogonally invariant Finsler metric on Bn (ν) ⊂ Rn . Then its |y| geodesic coefficients are given by Gi = uP y i + u2 Qxi
(2.1)
where Q :=
rφss − φr + sφrs 1 , 2r φ − sφs + (r2 − s2 )φss
u = |y|,
r := |x|,
s :=
x, y , |y|
(2.2)
and Q rφs + sφr − sφ − (r2 − s2 )φs . 2rφ φ
(2.3)
1 j 1 s y , [u2 ]yj = 2y j , syj = (xj − y j ). u u u
(2.4)
P := Direct computations yield uy j = Let
¯ i = u2 Qxi . G
(2.5)
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¯ := y i ∂ i − 2G ¯ i ∂ i is a spray on Bn (ν). It follows that the Finsler spray Then G ∂x ∂y G := y i
∂ ∂ − 2Gi i i ∂x ∂y
¯ [9]. We denote the corresponding objects with respect to G ¯ by adding a bar ¯. is projectively equivalent to G Then ¯i ¯ji = ∂ G , N ∂y j
¯i = Γ jk
¯i ∂N j ∂y k
(2.6)
¯ Note that N ¯ i (resp. G ¯ i ) is homogeneous of degree with respect to the Berwald connection induced by G. j one (resp. two) with respect to y. From (2.6) we get ¯ i yk = N ¯ i. Γ jk j
¯i, ¯ i y j = 2G N j
(2.7)
By definition, the Douglas curvature is given by Dj
i
kl
∂3 := ∂y j ∂y k ∂y l
1 ∂Gm i y G − n + 1 m ∂y m
i
.
¯ Then Let “ ” denote the covariant differentiation with respect to G. Dj i klm =
i ∂Dj i kl p ∂Dj kl − N + Dj p kl Γipm − Dp i kl Γpjm − Dj i pl Γpkm − Dj i kp Γplm . m ∂xm ∂y p
Together with (2.7) we obtain ¯ i − Dp i kl N ¯ p − Dj i pl N ¯ p − Dj i kp N ¯p Dj i klm y m = (I) + Dj p kl N p j k l
(2.8)
where (I) :=
ym
∂ ¯m ∂ − 2G m ∂x ∂y m
¯ j i kl ). Dj i kl = G(D
(2.9)
By (2.5), (2.4) and the first equation of (2.6) ¯ji = ∂ (u2 Qxi ) = (u2 )yj Qxi + u2 Qs syj xi = W j xi , N ∂y j
(2.10)
W j := (2Q − sQs )y j + uQs xj .
(2.11)
where
Lemma 2.3. Let f = f (r, s) be a function on a domain U ⊆ R2 . Then s fxi y i = u( fr + fs ), r
fyi xi = i
i
fs 2 (r − s2 ). u
(2.12)
Proof. Simple calculations give the following: fxi = xr fr + yu fs . It follows that the first equation of (2.12) holds. By the third equation of (2.4), we have the second equation of (2.12). 2
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Corollary 2.4. Let f = f (r, s) be a function on a domain U ⊆ R2 . Then ¯ ) = u s fr + Ψfs G(f r
(2.13)
Ψ := 1 − 2Q(r2 − s2 ).
(2.14)
where
Proof. By (2.5) and (2.12), we have ¯ ) = fxi y i − 2u2 Qfyi xi = u s fr + Ψfs . G(f r From (2.5) and the first equation of (2.4), we have ¯ k ) = y k , G(y ¯ k ) = −2u2 Qxk . ¯ G(u) = −2u2 sQ, G(x
(2.15)
For an orthogonally invariant Finsler metric F = |y|φ(|x|, x,|y|y ) on Bn (ν) ⊆ Rn , the Douglas curvature of F is given by [5] Dj i kl =
1 [Rss δji xk xl + (R − sRs )δji δkl ](j → k → l → j) u sRss − 2 [δji (xk y l + xl y k ) + y i δjk xl ](j → k → l → j) u 1 + 3 (sRss + sRs − R)(δji y k y l + y i δjk y l )(j → k → l → j) u 1 Rsss + 5 (3R − s2 Rsss − 6s2 Rss − 3sRs )y i y j y k y l + 2 y i xj xk xl u u s 1 + 4 (sRsss + 3Rss )y i y j y k xl (j → k → l → j) + [(Qs − sQss )xi xl δjk u u 1 i j k l − 2 (Rss + sRsss )y y x x ](j → k → l → j) u 1 + 3 (s2 Qsss + sQss − Qs )xi xj y k y l (j → k → l → j) u s + 2 [(sQss − Qs )xi y j δkl − Qsss xi xj xl y k ](j → k → l → j) u xi + 4 [u3 Qsss xj xk xl + s(3Qs − 3sQss − s2 Qsss )y j y k y l ] u
(2.16)
where j → k → l → j denotes cyclic permutation, Q is given by (2.2) and R := −
1 2sQ + (r2 − s2 )Qs . n+1
By using (2.16) and Lemma 4.1–Lemma 4.9 below, we have ¯ j i kl ) = Akl δ i (j → k → l → j) + Bjkl xi (j → k → l → j) G(D j 2 Q(3R − 3sRs − 6s2 Rss − s3 Rsss )xi y j y k y l − 2QRsss xi xj xk xl u3 + Cjkl y i (j → k → l → j) + Dy i y j y k y l
1 s + 4sQRsss + Rrsss + ΨRssss ) + 6Q(Rss + sRsss y i xj xk xl u r −
(2.17)
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+ Ejkl xi (j → k → l → j) + F xi y j y k y l (j → k → l → j) s + (8sQQsss + Qrsss + ΨQssss )xj xk xl xi + Gjkl y i (j → k → l → j) r s Qsss j k l i + x x x y + 4 (3Qs − 3sQss − s2 Qsss )y j y k y l y i , u u
(2.18)
where s 1 Akl : = (6sQRss + Rrss + ΨRsss )xk xl + s[2Q(R − sRs ) + (Rr − sRrs ) − ΨRss ]δkl r r 2 1 s + [(1 − 4s2 Q)Rss − Rrss − (Rss + sRsss )Ψ − 2Q(s2 Rss + sRs − R)](xl y k + xk y l ) u r s 1 + 2 [6Q(s2 Rss + sRs − R) + (s2 Rrss + sRrs − Rr ) + Ψ(3Rss + sRsss ) − 2Rss ]y k y l , u r (2.19) Bjkl : = 2sQRss δjk xl − + Cjkl : =
2Q 2 2sQ (s Rss + sRs − R)δjk y l − 2 (sRsss + 3Rss )xl y j y k u u
2Q (Rss + sRsss )xk xl y j , u
(2.20)
1 s [6sQ(s2 Rss + sRs − R) + (s2 Rrss + sRrs − Rr ) − sRss + Ψ(3sRss + s2 Rsss )]δjk y l u2 r 1 s2 − [4s2 QRss + Rrss + (Rss + sRsss )Ψ + 2Q(s2 Rss + sRs − R)]δjk xl u r 1 s2 + 3 [8s2 Q(3Rss + sRsss ) + (sRrsss + 3Rrss ) − 2Q(3R − 3sRs − 6s2 Rss − s3 Rsss ) u r + Ψ(3Rss + 5sRsss + s2 Rssss ) − 2(Rss + sRsss )]y j y k xl
1 [Rsss − 4sQ(sRsss + 3Rss ) − 6sQ(Rss + sRsss ) u2 s − (Rrss + sRrsss ) − Ψ(2Rsss + sRssss )]y j xk xl , r s 1 D : = 4 [10Q(3R − 3sRs − 6s2 Rss − s3 Rsss ) + (3Rr − 3sRrs − 6s2 Rrss − s3 Rrsss ) u r 2 − Ψ(9sRsss + 15Rss + s Rssss ) + 3(sRsss + 3Rss )], +
1 Ejkl : = s[4Q(Qs − sQss ) + (Qrs − sQrss ) − ΨQsss ]xl δjk r 1 s2 + [(1 − 4s2 Q)(Qs − sQss ) + (sQrss − Qrs ) + Ψ(s2 Qsss + sQss − Qs )]y l δjk u r s 2 + 2 [12Q(sQss − Qs ) + 2(4s Q − 1)Qsss u 1 2 + (s Qrsss + sQrss − Qrs ) + Ψ(3Qsss + sQssss )]xj y k y l r 1 s2 − [(8s2 Q − 1)Qsss + 4Q(sQss − Qs ) + Qrsss + Ψ(Qsss + sQssss )]xj xk y l , u r 1 F : = 3 [3(s2 Qsss + sQss − Qs ) + 8s2 Q(3Qs − 3sQss − s2 Qsss ) u s2 + (3Qrs − 3sQrss − s2 Qrsss ) + Ψ(3Qs − 3sQss − 6s2 Qsss − s3 Qssss ), r 1 s 1 s Gjkl : = (Qs − sQss )(xl − y l )δjk + 3 (s2 Qsss + sQss − Qs )xj y k y l − 2 Qsss xj xl y k , u u u u
(2.21)
(2.22)
(2.23)
(2.24) (2.25)
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where Ψ is given in (2.14). By using (2.11), we have Σj xj W j = 2suQ + u(r2 − s2 )Qs ,
Σj y j W j = 2Qu2 .
Together with (2.10) and (2.16), we obtain ¯pi = Σp (Dj p kl W p )xi Dj p kl N s s = [Qs (R − sRs ) − 2sQRss ](xl − y l )δjk − (3Qs Rss + 2QRsss )xj xk y l u u xi + 2 [Qs (s2 Rss + sRs − R) + 2s2 Qs Rss + 2sQ(sRsss + Rss )]xk y j y l (j → k → l → j) u 1 + 3 [2Q(3R − s3 Rsss − 6s2 Rss − 3sRs ) + 3(2Q − sQs )(s2 Rss + sRs − R)]y j y k y l xi u + (3Qs Rss + 2QRsss )xj xk xl xi + [2sQ + Qs (r2 − s2 )]xi s s2 Qsss + sQss − Qs j k l sQsss j l k x x y (j → k → l → j) × (Qs − sQss )(xj − y j )δlk + x y y − u u2 u 1 + 3 [2sQ + Qs (r2 − s2 )] s(3Qs − 3sQss − s2 Qsss )y j y k y l + u3 Qsss xj xk xl xi . (2.26) u By (2.10), we obtain ¯ p = W j (Dp i kl xp ). Dp i kl N j
(2.27)
From (2.16), we get Dp i kl xp =
1 s [ΦRss + R − sRs ](xl − y l )δ i k (k ↔ l) u u sRss Rss k l 1 1 x x + 3 (s2 Rss + sRs − R)y k y l ]xi + [ (R − sRs )δkl − 2 xk y l (k ↔ l) + u u u u s 1 + [− 2 (R − sRs + ΦRss )δlk + 3 (3s2 Rss + sRs − R − Φ(Rss + sRsss ))xk y l (k ↔ l) u u s 1 + 2 (ΦRsss − 3sRss )xk xl + 4 (3(R − sRs − s2 Rss ) + Φ(3Rss + sRsss ))y k y l ]y i u u 1 s + [ (Qs − sQss )Φδkl + 2 [2(sQss − Qs ) − ΦQsss ]xk y l (k ↔ l) u u 2(Qs − sQss ) + ΦQsss k l 1 + x x + 3 [Φ(s2 Qsss + sQss − Qs ) + 2s2 (Qs − sQss ))y k y l ]xi u u
where k ↔ l denotes symmetrization and Φ = r2 − s2 . Plugging this into (2.27) and using total symmetry of Dj i kl with respect to j, k, l we obtain ¯ p (j → k → l → j) = W j (Dp i kl xp )(j → k → l → j) Dp i kl N j 1 s [ΦRss + R − sRs ](xl − y l )δ i k (k ↔ l) = (2Q − sQs )y j + uQs xj u u sRss k l Rss k l 1 1 x x + 3 (s2 Rss + sRs − R)y k y l ]xi + [ (R − sRs )δkl − 2 x y (k ↔ l) + u u u u s 1 + [− 2 (R − sRs + ΦRss )δlk + 3 (3s2 Rss + sRs − R − Φ(Rss + sRsss ))xk y l (k ↔ l) u u s 1 + 2 (ΦRsss − 3sRss )xk xl + 4 (3(R − sRs − s2 Rss ) + Φ(3Rss + sRsss ))y k y l ]y i u u
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1 s 2(Qs − sQss ) + ΦQsss k l x x + [ (Qs − sQss )Φδkl + 2 [2(sQss − Qs ) − ΦQsss ]xk y l (k ↔ l) + u u u 1 + 3 [Φ(s2 Qsss + sQss − Qs ) + 2s2 (Qs − sQss ))y k y l ]xi (j → k → l → j) u ΦRss + R − sRs 2s [2uQs xk xl + 2(Q − sQs )(xk y l + xl y k ) − (2Q − sQs )y k y l ]δ i j (j → k → l → j) = u u 1 1 (R − sRs )W l δjk + 2 [Qs (s2 Rss + sRs − R) − 2sRss (2Q − sQs )]xj y k y l + u u Rss (2Q − 3sQs )xj xk y l xi (j → k → l → j) + u 3 + 3 (2Q − sQs )(s2 Rss + sRs − R)y j y k y l xi + 3Qs Rss xk xl xj xi u s − 2 [R − sRs + ΦRss ]W j δjk u 1 − 3 [(4Q − 5sQs )(s2 Rss + sRs − R) + 4s2 (2Q − sQs )Rss + ΦsQs (3Rss + sRsss ) u − 2Φ(2Q − sQs )(Rss + sRsss )]xj y k y l 1 − 2 [Φ(2Q − sQs )Rsss − 3s(2Q − sQs )Rss + 2Qs (3s2 Rss + sRs − R) u − 2Qs Φ(Rss + sRsss )]xj xk y l y i (j → k → l → j) 3Qs 3s(2Q − sQs ) [ΦRsss − 3sRss ]xk xl xj y i [3(R − sRs − s2 Rss ) + Φ(3Rss + sRsss )]y j y k y l y i + 4 u u 1 1 (Qs − sQss )ΦW l δjk + 2 [4s(2Q − sQs )(sQss − Qs ) − 2sΦQsss (2Q − sQs ) + u u 1 + Qs Φ(s2 Qsss + sQss − Qs ) + 2s2 Qs (Qs − sQss )]xj y k y l + [2(2Q − sQs )(Qs − sQss ) u + ΦQsss (2Q − sQs ) + 4sQs (sQss − Qs ) − 2sQs ΦQsss ]xj xk y l xi (j → k → l → j) +
+
3 (2Q − sQs )[Φ(s2 Qsss + sQss − Qs ) + 2s2 (Qs − sQss )]y j y k y l xi u3
+ 3Qs [2(Qs − sQss ) + ΦQsss ]xk xl xj xi .
(2.28)
By using (2.8), (2.9), (2.18)–(2.28) we have s s T1 (xk y l + xl y k ) + 2 (sT1 − T2 )y k y l δ i j (j → k → l → j) u u s 1 1 + T3 δjk xl − (T2 + T3 )δjk y l + 2 (2sT1 − T3 + s2 T4 )xl y j y k u u u 1 − (T1 + sT4 )xj xk y l y i (j → k → l → j) u T4 s + 4 [3(T2 + T3 − sT1 ) − s2 T4 ]y j y k y l y i + xk xl xj y i u u s 1 1 + s T5 (xl − y l )δjk − 2 (T5 − sT6 )xj y l y k − T6 xj xk y l xi (j → k → l → j) u u u 2 s + 3 (3T5 − sT6 )y j y k y l xi + T6 xk xl xj xi + Gjkl y i (j → k → l → j) u s Qsss j k l i x x x y + 4 (3Qs − 3sQss − s2 Qsss )y j y k y l y i , (2.29) + u u
Dj i klm y m = [T1 xk xl + sT2 δkl −
where
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s T1 := 2[3sQ − Qs (r2 − s2 )]Rss + Rrss + ΨRsss − 2Qs (R − sRs ), r 1 T2 := 2Q(R − sRs ) + (Rr − sRrs ) − ΨRss , r 2 2 T3 := −sT1 − [Ψ + sQs (r − s )]Rss + (2Q − sQs )(R − sRs ), s T4 := 3(2Q + 3sQs )Rss + [10sQ − 3Qs (r2 − s2 )]Rsss + Rrsss + ΨRssss , r 1 T5 := 6Q(Qs − sQss ) + (Qrs − sQrss ) − ΨQsss , r s T6 := 6Qs (sQss − Qs ) + 2[5sQ − Qs (r2 − s2 )]Qsss + Qrsss + ΨQssss . 2 r
175
(2.30) (2.31) (2.32) (2.33) (2.34) (2.35)
3. Proof of theorems First we are going to prove the following y Theorem 3.1. Let F = |y|φ |x|, x, be an orthogonally invariant Finsler metric on Bn (ν) ⊂ Rn . Then |y| F is a GDW-metric if and only if s s 1 T1 (xk y l + xl y k ) + 2 (sT1 − T2 )y k y l (δ i j − 2 y i y j )(j → k → l → j) u u u s 1 1 + s T5 (xl − y l )δjk − 2 (T5 − sT6 )xj y l y k − T6 xj xk y l (j → k → l → j) u u u 2 s s + 3 (3T5 − sT6 )y j y k y l + T6 xk xl xj (xi − y i ) = 0, u u
[T1 xk xl + sT2 δkl −
(3.1)
where T1 , · · · , T6 are defined in (2.30), (2.31), (2.32), (2.33), (2.34) and (2.35) respectively. y Proof. Let F = |y|φ |x|, x, be an orthogonally invariant Finsler metric on Bn (ν). Let G denote the |y| ¯ the spray defined in (2.5). Since G ¯ and G are projectively equivalent, the following spray of F and G conditions are equivalent [1,7] (i) F is a GDW-metric, i.e. there is a tensor Tjkl such that Dji kl;m y m = Tjkl y i ,
(3.2)
Dji klm y m = Djkl y i ,
(3.3)
(ii) there is a tensor Djkl such that
where Dji kl;m denotes the covariant derivatives of Dji kl with respect to the Berwald connections of G. This equivalence is essentially proved in [1]. Assume that F is a GDW-metric. Then (3.3) holds for some tensor Djkl . By (3.3) and (2.29), we have s s T1 (xk y l + xl y k ) + 2 (sT1 − T2 )y k y l δ i j (j → k → l → j) u u s 1 1 l l + T3 δjk x − (T2 + T3 )δjk y + 2 (2sT1 − T3 + s2 T4 )xl y j y k u u u 1 − (T1 + sT4 )xj xk y l y i (j → k → l → j) u s T4 + 4 [3(T2 + T3 − sT1 ) − s2 T4 ]y j y k y l y i + xk xl xj y i u u
Djkl y i = [T1 xk xl + sT2 δkl −
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s 1 1 + s T5 (xl − y l )δjk − 2 (T5 − sT6 )xj y l y k − T6 xj xk y l xi (j → k → l → j) u u u s2 + 3 (3T5 − sT6 )y j y k y l xi + T6 xk xl xj xi + Gjkl y i (j → k → l → j) u Qsss j k l i s + x x x y + 4 (3Qs − 3sQss − s2 Qsss )y j y k y l y i . u u
(3.4)
It follows that Djkl = =
1
(Djkl y i )y i u2 i 1 s s [T1 xk xl + sT2 δkl − T1 (xk y l + xl y k ) + 2 (sT1 − T2 )y k y l y j (j → k → l → j) 2 u u u 1 s 1 + T3 δjk xl − (T2 + T3 )δjk y l + 2 (2sT1 − T3 + s2 T4 )xl y j y k u u u 1 − (T1 + sT4 )xj xk y l (j → k → l → j) u s T4 + 4 [3(T2 + T3 − sT1 ) − s2 T4 ]y j y k y l + xk xl xj u u s2 s 1 1 T5 (xl − y l )δjk − 2 (T5 − sT6 )xj y l y k − T6 xj xk y l (j → k → l → j) + u u u u sT s3 6 x k xl xj + 4 (3T5 − sT6 )y j y k y l + u u 1 s 1 (Qs − sQss )(xl − y l )δjk + 3 (s2 Qsss + sQss − Qs )xj y k y l + u u u s − 2 Qsss xj xl y k (j → k → l → j) u Qsss j k l s + x x x + 4 (3Qs − 3sQss − s2 Qsss )y j y k y l . u u
(3.5)
Plugging (3.5) into (3.4) yields (3.1). Conversely, if (3.1) holds, it follows from (2.29) that Dji klm y m = Djkl y i where Djkl are given by (3.5). Thus F is a GDW-metric. 2 Lemma 3.2. (i). T5 =
∂R2 ∂s
(3.6)
where 1 R2 := 2Q(2Q − sQs ) + (2Qr − sQrs − rQss ) + Φ(2QQss − Q2s ) r
(3.7)
where Q is given in the first equation of (2.2). (ii). (3.1) is equivalent to the following 1 T5 := 6Q(Qs − sQss ) + (Qrs − sQrss ) − ΨQsss = 0. r Proof. (i). A simple calculation gives (3.6). (ii). From (2.17) we get
(3.8)
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R − sRs = −
177
Φ (Qs − sQss ), n+1
(3.9)
1 [2r(Qs − sQss ) + Φ(Qrs − sQrss )] , n+1 1 [2(Qs − sQss ) + ΦQsss ] . =− n+1
Rr − sRrs = − Rss
(3.10) (3.11)
Substituting these into (2.31) and using (2.34), we have T2 = −
r2 − s2 T5 . n+1
(3.12)
From (2.30), (2.31), (2.14), (2.34) and (2.35) we get (T2 )s = −T1 ,
(T5 )s = −T6 .
(3.13)
First suppose that (3.1) holds. We take j = k = l = i, then T6 (xi )4 + 3(T1 + sT5 )(xi )2 + 3sT2 ≡ 0,
mod y i .
This gives T2 = 0,
T1 + sT5 = 0,
T6 = 0.
(3.14)
Take x and y with x ∧ y = 0. Then Φ = r2 − s2 =
|x|2 |y|2 − x, y2 > 0. |y|2
Together with (3.12) and the first equation of (3.14) we have T5 = 0. Conversely, suppose that T5 = 0. Together with (3.12) and (3.13) we get T2 = 0, Hence (3.1) holds. 2
T1 = 0,
T6 = 0.
Proof of Theorem 1.1. Combining Lemma 3.2 with Theorem 3.1. 2 y It is known that F = |y|φ |x|, x, is a Douglas metric if and only if Qs − sQss = 0 [5]. On the other |y| y hand, Huang–Mo recently have found a sufficient and necessary condition for F = |y|φ |x|, x, to be of |y| scalar curvature, that is, 1 R2 := 2Q(2Q − sQs ) + (2Qr − sQrs − rQss ) + Φ(2QQss − Q2s ) = 0. r Theorem 1.1 tell us any spherically symmetric metrics of scalar curvature belongs to GDW (Bn (ν)) where GDW (Bn (ν)) denotes the class of all GDW-metrics on Bn (ν). Proof of Theorem 1.2. In [6], authors showed that a spherically symmetric Finsler metric F is a Douglas metric if and only if Q = Q(r, s) is given by Q(r, s) = f0 (r) + f2 (r)s2 . Case 1. k ≥ 1. By (3.15), we have [4] R2 ≡ (4 − k2 )fk2 s2k ,
mod s0 , s1 , · · · , s2k−1
(3.15)
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where R2 is given in (3.7). A simple calculation gives the following formula: ∂R2 ≡ 2k(4 − k2 )fk2 s2k−1 , ∂s
mod
s0 , s1 , · · · , s2k−2 .
First suppose that F is a GDW-metric. By Lemma 3.2 we get k = 2. It follows that [4] 2 1 R2 = 4f02 + f0 − r2 f12 − 2f2 + 4r2 f0 f2 + (6f0 f1 + f1 )s + 3f12 s2 + 2f1 f2 s3 . r r ∂R2 1 2 2 2 It follows that ∂R ∂s = 6f0 f1 + r f1 + 6f1 s + 6f1 f2 s . Note that T5 = ∂s = 0. Hence we obtain that f1 = 0. It follows that Q satisfies (3.15). Thus F is a Douglas metric. Conversely, we suppose that F is a Douglas metric, then it is easy to see from (3.2) that F is a GDWmetric. Case 2. k = 0. This is an immediate conclusion of (3.15). 2
4. Appendix We establish the Lemmas required in the computation of (2.18). By using (2.13) and (2.15), the following Lemmas can be obtained by straightforward calculations. ¯ on Bn (ν) we have Lemma 4.1. For the spray G ¯ G
1 i k l i [Rss δj x x + (R − sRs )δj δkl ](j → k → l → j) u s = 2sQδji [Rss xk xl + (R − sRs )δkl ] + ( Rrss + Rsss Ψ)δji xk xl (j → k → l → j) r 1 s i k l k l i + [ (Rr − Rrs ) − ΨsRss ]δj δkl + Rss (y x + x y )δj (j → k → l → j). r u
¯ on Bn (ν) we have Lemma 4.2. For the spray G ¯ G
sRss i k l l k i l [δ (x y + x y ) + y δjk x ](j → k → l → j) u2 j (y k xl + xk y l )δji + y i δjk xl s2 2 4s QRss + Rrss + Ψ(Rss + sRsss ) (j → k → l → j) = u r sRss 2δji (y k y l − 2u2 Qxk xl ) + δjk (y i y l − 2u2 Qxi xl ) (j → k → l → j). + u
¯ on Bn (ν) we have Lemma 4.3. For the spray G ¯ G
1 (sRss + sRs − R)(δji y k y l + y i δjk y l )(j → k → l → j) u3
=
6sQ(s2 Rss + sRs − R) + rs (s2 Rrss + sRrs − Rr ) + Ψ(3sRss + s2 Rsss ) (j → k → l → j) u2 2Q 2 (s Rss + sRs − R) (y k xl + xk y l )δji + (y l xi + xl y i )δjk (j → k → l → j). − u
¯ on Bn (ν) we have Lemma 4.4. For the spray G
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¯ G
1 Rsss i j k l 2 2 i j k l (3R − s Rsss − 6s Rss − 3sRs )y y y y + 2 y x x x u5 u s 1 = 4 10QΘ + Θr − Ψ(9sRsss + 15Rss + s2 Rssss y i y j y k y l u r 1 2Q s − 3 Θ xi y j y k y l + y i xj y k y l (j → k → l → j) + (4sQRsss + Rrsss + ΨRssss )y i xj xk xl u u r 1 i j k l 2 i j k l + 2 y y x x (j → k → l → j) − 2u Qx x x x Rsss , u
where Θ := 3R − 3sRs − 6s2 Rss − s3 Rsss . ¯ on Bn (ν) we have Lemma 4.5. For the spray G ¯ s (sRsss + 3Rss )y i y j y k xl (j → k → l → j)) G( u4 =
8s2 Q(3Rss + sRsss ) +
s2 r (3Rrss
+ sRrsss ) + Ψ(3Rss + 5sRsss + s2 Rssss ) i j k l y y y x (j → k → l → j) u3 s(3Rss + sRsss ) i j k l + 3y y y y − 2u2 Q(y j y k xl xi + 2xj y k xl y i )(j → k → l → j) . 4 u
¯ on Bn (ν) we have Lemma 4.6. For the spray G ¯ 1 [(Qs − sQss )xi xl δjk − 1 (Rss + sRsss )y i y j xk xl ](j → k → l → j)) G( u u2 6sQ(Rss + sRsss ) + rs (Rrss + sRrsss ) + Ψ(2Rsss + sRssss ) i j k l =− y y x x (j → k → l → j) u2 Rss + sRsss i j k l − 2(y y y x − u2 Qy j xk xl xi )(j → k → l → j) − 6u2 Qy i xj xk xl 3 u 1 + s 2Q(Qs − sQss ) + (Qrs − sQrss ) − ΨQsss xi xl δjk (j → k → l → j) r +
1 (Qs − sQss )δjk (y i xl + xi y l )(j → k → l → j). u
¯ on Bn (ν) we have Lemma 4.7. For the spray G ¯ G
1 2 i j k l (s Q + sQ − Q )x x y y (j → k → l → j) sss ss s u3 s 1 = 2 6QΞ + Ξr + Ψ(3Qsss + sQssss ) xi xj y k y l (j → k → l → j) u r Ξ + 3 (xj y k y l y i − 4u2 Qxj xk y l xi )(j → k → l → j) + 3xi y k y j y l , u
where Ξ := s2 Qsss + sQss − Qs . ¯ on Bn (ν) we have Lemma 4.8. For the spray G ¯ G
s i j i j l k [(sQ − Q )x y δ − Q x x x y ](j → k → l → j ss s kl sss u2 s2 1 2 4s Q(sQss − Qs ) + (sQrss − Qrs ) + ΨΞ xi y j δkl (j → k → l → j) = u r
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s (sQss − Qs )δkl (y i y j − 2u2 Qxi xj )(j → k → l → j) u2 s2 1 2 4s QQsss + Qrsss + Ψ(Qsss + sQssss ) xi xj y k xl (j → k → l → j) − u r s − 2 Qsss (xj xl y k y i + 2xj y l y k xi )(j → k → l → j) − 6u2 Qxi xj xk xl . u
+
¯ on Bn (ν) we have Lemma 4.9. For the spray G xi 3 j k l 2 j k l [u Q x x x + s(3Q − 3sQ − s Q )y y y ] sss s ss sss u4 s = (2sQQsss + Qrsss + ΨQssss )xi xj xk xl r 1 j l k i x x x y + y j xl xk xi (j → k → l → j) Qsss + u 1 s2 + 3 8s2 QΛ + Λr + Ψ(Λ − 5s2 Qsss − s3 Qssss ) xi y j y k y l u r sΛ + 4 y i y j y k y l − 2u2 Qxi xj y k y l (j → k → l → j) , u
¯ G
where Λ := 3Qs − 3sQss − s2 Qsss . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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