On a Projective Class of Finsler Metrics∗ B. Najafi, Z. Shen† and A. Tayebi
1
Introduction
The Douglas (projective) curvature Dji kl and the Weyl (projective) curvature W ik are two most important quantities in projective Finsler geometry. Finsler metrics with Dj i kl = 0 are called Douglas metrics and Finsler metrics with W ik = 0 are called Weyl metrics. 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. Thus Weyl metrics are called metrics of scalar (flag) curvature, and being of scalar flag curvature is a projective property. Equations Dji kl = 0 and W ik = 0 are projectively invariant, namely, if a Finsler metric F satisfies one of the equations, then any Finsler metric projectively equivalent to F must satisfy the same equation. There is another projective invariant equation in Finsler geometry, that is, for some tensor Tjkl , Dj i kl;m ym = Tjklyi ,
(1)
where Dji kl;m denotes the horizontal covariant derivatives of Dj i kl with respect to the Berwald connection of F . Equation (1) is equivalent to that for any linearly parallel veector fields U = U (t), V = V (t) and W = W (t) along a geodesic c(t), there is a function T = T (t) such that i dh ˙ Dc˙ (U, V, W ) = T c. dt The geometric meaning of the above identity is that the rate of change of the Douglas curvature along a geodesic is tangent to the geodesic. For a manifold M , let GDW(M ) denote the class of all Finsler metrics satisfying (1) for some tensor Tjkl (Tjkl not fixed). In [2], Bacso-Papp show that GDW(M ) is closed under projective changes. More precisely, if F is projectively equivalent to a Finsler metric in GDW(M ), then F ∈ GDW(M ). A natural question is: how large is GDW(M ) and what kind of interesting metrics does it have? It is obvious that all Douglas metrics belong to this class. ∗ 2000
Mathematics Subject Classification. Primary 53B40 support in part by a National Natural Science Foundation of China (10371138) and a NSF grant on IR/D † Research is
1
On the other hand, all Weyl metrics (metrics of scalar flag curvature) also belong to this class. The later is really a surprising result, due to Sakaguchi [4]. In this sense, we shall call Finsler metrics in GDW(M ) GDW-metrics (generalized Douglas-Weyl metrics). In this paper, we are going to study and characterize GDW-metrics of Randers type on a manifold M . A Randers metric on a manifold M is a Finsler metric in the following form F = α + β, p i j where α = aij (x)y y is a Riemannian metric and β = bi (x)yi is a 1-form on M . One of the reasons why we would like to study Randers metrics for the above problem is because that Randers metrics are “computable”. Let 1 ∂bi ∂bj sij := . − 2 ∂xj ∂xi Clearly, β is closed if and only if sij = 0. It is known that F = α+β is a Douglas metric if and only if β is closed (see [1]). On the other hand, Shen-Yildirim [6] recently find a sufficient and necessary condition for F = α + β to be of scalar flag curvature, that is, o 1 m n 2 i i j ¯i R α = λ − δ − a y y t jk k k n−1 m 2 i i i i +α t k + t00δk − tk0y − t 0 yk − 3si 0 sk0, (2) o n 1 sij|k = (3) aik smj|m − ajk smi|m , n−1 ¯ i denotes the Riemann curvature tensor of α and λ = where tij := sim smj , R k λ(x) is a scalar function on M . Here sij|k denote the coefficients of the covariant derivative of sij with respect to α. We use aij and aij to lower or lift the indices of a tensor. Theorem 1.1 Let F = α + β be a Randers metric on an n-dimensional manifold M . F is a GDW metric if and only if (3) holds. Thus any Randers metrics of scalar curvature belongs to GDW(M ). This verifies Sakaguchi’s theorem for Randers metrics. The following Randers metric actually satisfies both (2) and (3). n Example 1.1 ([5]) p Let a ∈ R be a constant vector. Define F = α + β on an open ball Bn (1/ |a|) in Rn by p (1 − |a|2|x|4)|y|2 + (|x|2ha, yi − 2ha, xihx, yi)2 F : = 1 − |a|2|x|4 2 |x| ha, yi − 2ha, xihx, yi − . 1 − |a|2|x|4
Then F is of scalar fuction. Thus it satisfies (2) and (3). See more examples in [3]. So far, we have not found a Randers metric satisfying (3), but not (2). We conjecture that such examples exist. 2
2
Randers Metrics
p Let F = α + β be a Randers metric on a manifold M , where α = aij (x)yi yj and β = bi(x)yi . Let bi|j denote the coefficients of the covariant derivative of β wirth respect to α. Let rij :=
1 (bi|j + bj|i), 2
sij :=
1 (bi|j − bj|i), 2
where bi|j denote the coefficients of the covariant derivative of β with respect to α. The spray coefficients of F are given by ¯ i + r00 − 2s0 α yi + αsi . Gi = G 0 2F Let Πi := Gi − Observe that
1 ∂Gm , n + 1 ∂ym
¯ i := G ¯i − Π
¯m 1 ∂G . n + 1 ∂ym
∂(αsm0) ym m = s + αsmm = 0. ∂ym α 0
Thus we have ¯ i + αsi . Πi = Π 0
(4)
By definition, the Douglas curvature is given by Dj i kl :=
∂ 3Πi . ∂yj ∂yk ∂yl
¯ i are always quadratic in y, we get Since Π Dji kl =
∂3 ∂yj ∂yk ∂yl
αsi 0 = αjklsi 0 + αjksi l + αjlsi k + αkl si j ,
(5)
where αj αjk
= α−1 yj = α−3 Ajk
αjkl
= −α−5 {Ajk yl + Ajl yk + Akl yj },
where Aij := α2aij − yi yj . It is easy to show that F is a Douglas metric if and only if si 0 = 0. This fact is due to Bacso-Matsumoto [1]. ˜ = yi ∂ i − 2G ˜ i ∂ i where Let G ∂x ∂y ˜ i := G ¯ i + αsi . G 0
3
(6)
˜ and G ¯ Let “k00 and “|00 denote the covariant differentiations with respect to G respectively. Then Dji klkm ym
=
∂ (D i )sp ∂yp j kl 0 +α−1[α2si p + yp si 0 ]Djpkl − α−1[α2spj + sp0yj ]Dpi kl
Dji kl|m ym − 2α
−α−1[α2spk + sp0yk ]Dji pl − α−1[α2spl + sp0 yl ]Dji kp . (7) Since “|00 is a differentiation with respect to α, aij|m = 0. Thus α|m = 0,
αj|m = 0,
αjk|m = 0,
αjkl|m = 0.
We obtain Dji kl|m ym
= =
αjklsi 0|0 + αjksi l|0 + αjl si k|0 + αkl si j|0 n o −α−5 Ajk yl + Ajl yk + Akl yj si 0|0 n o +α−3 Ajk si l|0 + Ajl si k|0 + Akl si j|0
Differentuating (5) yields ∂ (D i ) = αjklpsi 0 + αjklsi p + αjkpsi l + αjlp si k + αklpsi j , ∂yp j kl where αjklp
3α−5yp {ajkyl + ajl yk + akl yj } n o −α−3 ajk alp + ajl akp + akl ajp n o +3α−5 yk yl ajp + yj yl akp + yj yk alp
=
−15α−7yj yk yl yp , We get αjklpsp0
αjkpsp0 αjlp sp0 αklp sp0
n o = −α−5 Ajk sl0 + Ajl sk0 + Akl sj0 n o +2α−5 sj0 yk yl + sk0 yi yl + sl0 yj yk n o = −α−3 yk sj0 + yj sk0 n o = −α−3 yl sj0 + yj sl0 n o = −α−3 yk sl0 + yl sk0 .
Then we obtain −2α
∂ (D i )sp ∂yp j kl 0
=
n o 2α−4 Ajk sl0 + Ajl sk0 + Akl sj0 si 0 4
+2α−4{Ajk yl + Ajl yk + Akl yj }ti 0 n o −4α−4 yk yl sj0 + yj yl sk0 + yj yk sl0 si 0 n o +2α−2 yk sj0 + yj sk0 si l n o +2α−2 yl sj0 + yj sl0 si k n o +2α−2 yk sl0 + yl sk0 si j , By (5), we can also easily get α−1[α2si p + yp si 0 ]Djpkl
i −α−1 [α2spj + sp0yj ]Dpkl
=
=
−α−2 {Ajkyl + Ajl yk + Akl yj }ti 0 n o +α−2 Ajk ti l + Ajl ti k + Akl ti j n o −α−4 Ajk sl0 + Ajl sk0 si 0 + Akl sj0 si 0 . n o α−2 yk slj + yl skj + 2α−4yk yl sj0 si 0 −α−2(α2 skj − s0j yk )si l −α−2(α2 slj − s0j yl )si k n o +α−4 yj yk sl0 + yj yl sk0 si 0 −α−2yj sk0si l − α−2yj sl0 si k −α−4yj Akl ti 0 − α−4 Akl sj0si 0 − α−2Akl tij .
i −α−1[α2spk + sp0 yk ]Dplj
=
n o α−2 yl sjk + yj slk + 2α−4yl yj sk0si 0 −α−2(α2 slk − s0k yl )si j −α−2(α2 sjk − s0k yj )si l n o +α−4 yk yl sj0 + yk yj sl0 si 0 −α−2yk sl0 si j − α−2 yk sj0 si l −α−4yk Ajl ti 0 − α−4Alj sk0 si 0 − α−2 Ajl ti k
i −α−1[α2spl + sp0 yl ]Dpkj
=
n o α−2 yk sjl + yj skl + 2α−4yk yj sl0 si 0 −α−2 (α2skl − s0l yk )si j −α−2 (α2sjl − s0l yj )si k n o +α−4 yl yk sj0 + yl yj sk0 si 0 −α−2 yl sk0si j − α−2yl sj0 si k −α−4 yl Ajk ti 0 − α−4Akj sl0 si 0 − α−2Ajk ti l . 5
Plugging the above identities into (7), we get n o Dji klkm ym = α−5 Ajk H il + Ajl H ik + Akl H ij ,
(8)
where H ij := α2 si j|0 − yj si 0|0.
3
Proof of Theorem 1.1
First we are going to prove the following Theorem 3.1 Let F = α + β be a Randers metric on an n-dimensional manifold M . F isa GDW-metric if and only if α2 sij|0 = si0|0yj − sj0|0yi .
(9)
Proof: F = α + β be a Randers metric on a manifold. Let G denote the spray ˜ the spray defined in (6). Since G ˜ and G are projectively equivalent, of F and G the following conditions are equivalent (i) there is a tensor Tjkl such that i Djkl;m ym = Djklyi ,
(ii) there is a tensor Djkl such that i Djklkm ym = Djkl yi ,
(10)
where Dj ikl;m and Dji klkm denote the covariant derivatives of Dji kl with re˜ respectively. This equivalence is spect to the Berwald connections of G and G, essentially proved in [2]. Thus the argument is omitted here. Assume that F is a GDW-metric. Then (10) holds for some tensor Djkl . By (8), we have n o Djklyi = α−5 Ajk H il + Ajl H ik + Akl H ij . (11) Contracting (11) with yi yields n o Djkl = −α−5 Ajk sl0|0 + Ajl sk0|0 + Akl sj0|0 .
(12)
Plugging (12) into (11), we get n o n o n o Ajk H il + sl0|0 yi + Ajl H ik + sk0|0yi + Akl H ij + sj0|0yi = 0.
(13)
Contracting (13) with akl we obtain H ij + sj0|0yi = 0. 6
(14)
This is obviously equivalent to (9). Conversely, if (9) holds, or equivalently, (14) holds, it follows from (8) that i Djklkm ym = Djkl yi ,
where Djkl are given by (12). Thus F is a GDW-metric.
Q.E.D.
To Prove Theorem 1.1, one just needs to prove the equivalence between (3) and (9). Lemma 3.2 (3) is equivalent to (9). Proof: Supoose that (3) holds. Then o n sij|k = λ aik smj|m − ajk smi|m ,
(15)
where λ = 1/(n − 1) (in fact, λ can be any scalar function). Contracting it with yk yields o n sij|0 = λ yi smj|m − yj smi|m . (16) Contracting (16) with yj yields
Thus
o n si0|0 = λ yi sm0|m − α2smi|m .
(17)
o n sj0|0 = λ yj sm0|m − α2smj|m .
(18)
By (17)-(18), si0|0yj − sj0|0yi
n o = λα2 smj|m yi − smi|m yj n o = λα2 aik smjm − ajksmm|i yk = α2 sij|0.
The last identity follows from (16). Then we obtain (9). Conversely, assume that (9) holds. First differentiating (9) with respect to yk , yl and ym yields 2akl sij|m + 2akmsij|l + 2alm sij|k = sik|l ajm + sik|m ajl + sil|k ajm + sim|k ajl + sil|m ajk + sim|l ajk −sjk|l aim − sjk|m ail − sjl|k aim − sjm|k ail − sjl|m aik − sjm|l aik . Contracting it with alm , we get nsij|k = sik|j − sjk|i + aik smj|m − ajk smim .
7
(19)
It follows from (19) that nsik|j nsjk|i
= sij|k + sjk|i + aij smkm − ajk smim = −sij|k + sik|j + aij smkm − aik smjm
(20) (21)
Subtracting (21) from (20), we get sik|j − sjk|i =
o 2 1 n sij|k + aik smjm − ajk smim . n+1 n+1
(22)
Plugging (22) back into (19) yields sij|k =
o 1 n aik smjm − ajk smim . n−1
We are done.
Q.E.D.
References [1] S. B´ acs´ o and M. Matsumoto, On Finsler spaces of Douglas type. A generalization of the notion of Berwald space, Publ. Math. Debrecen, 51(1997), 385-406. [2] S. Bacso and I. Papp, A note on a generalized Douglas space, Periodica Math. Hungarica 48(2004), 181-184. [3] X. Cheng and Z. Shen, Randers metrics of scalar flag curvature, preprint, 2005. [4] T. Sakaguchi, On Finsler spaces of scalar curvature, Tensor N. S. 38 (1982), 211-219. [5] Z. Shen, Landsberg Curvature, S-Curvature and Riemann Curvature, In “A Sampler of Finsler Geometry” MSRI series, Cambridge University Press, 2004. [6] Z. Shen and G.C. Yildirim, A characterization of Randers metrics of scalar flag curvature, preprint, 2005. B. Najafi No. 25, Ordibehesht 2 alley 13 Aban, Shahid Rajaei St. 1883748773, Tehran-Iran b
[email protected] Z. Shen Department of Mathematical Sciences Indiana University Purdue University Indianapolis (IUPUI) 402 N. Blackford Street
8
Indianapolis, IN 46202-3216, USA
[email protected] A. Tayebi Department of Mathematics and Computer Science Amirkabir University Tehran, Iran akbar
[email protected]
9
