In this paper, we study projectively flat Finsler metrics de- fined by the Euclidean metric and related 1-forms

33 (2017), 349–365 www.emis.de/journals ISSN 1786-0091

A CLASS OF BERWALDIAN FINSLER METRICS

BAHMAN REZAEI AND MEHRAN GABRANI

Abstract. In this paper, we study projectively flat Finsler metrics de- fined by the Euclidean metric and related 1-forms. For this class of Finsler metrics, we find the necessary and sufficient condition to be Berwaldian.

Then, we obtain the differential equations that characterize these metrics with vanishing Douglas curvature.

1. Introduction

The Berwald metrics are very important in Finsler geometry. They were first investigated by L. Berwald. The geodesics of a Finsler metricF(x, y) on a smooth manifoldM are determined by the systems of second order differential equations

d²xⁱ

dt² + 2Gⁱ

x,dx dt

= 0, (1)

where Gⁱ = Gⁱ(x, y) are scalar functions on T M₀ and called by spray coeffi- cients. They define a global vector field G=y^{i ∂}_∂xi −2G^{i ∂}_∂yi on T M0, which is called spray. By definition, F is called a Berwald metric if Gⁱ = Gⁱ(x, y) are quadratic in y∈T_xM at every point x, i.e.

Gⁱ = 1

2Γⁱ_kh(x)y^hy^k. (2)

In [7], Peyghan-Tayebi considered a class of Finsler metrics called generalized Berwald metrics which contains the class of Berwald metrics as a special case.

They find some interesting curvature properties of generalized Berwald met- rics. Very recently, Tayebi-Barzegari study generalized Berwald manifold with (α, β)-metrics and showed that a Finsler manifold with (α, β)-Finsler function of sign property is a generalized Berwald manifold if and only if there exists a covariant derivative such that it is compatible with α and β and equivalently if and only if the dual vector field β^] is of constant Riemannian length [9].

2010Mathematics Subject Classification. 53C60, 53C25.

Key words and phrases. Projectively flat metric, Berwald metric, Douglas metric.

349

A Finsler metric is said to be locally projectively equivalent to a Riemannian metricg if at every pointx, there is a local coordinate neighborhood in which the geodesics ofF coincide with that ofg as point sets. In this case, the spray coefficients Gⁱ are in the following form

Gⁱ = 1

2Γⁱ_kh(x)y^hy^k+P(x, y)yⁱ. (3)

Finsler metrics with this property are called Douglas metrics. Obviously, the Douglas metrics are more generalized than Berwald metrics.

In [8], Shen constructed a group of projectively flat metrics with K= 0 as the following

F(x, y) = n

1+ < a, x >+ (1− |x|²)< a, y >

p|y|²−(|x|²|y|²−< x, y >²)+< x, y >

o

×(p

|y|²−(|x|²|y|²−< x, y >²)+ < x, y >)² (1− |x|²)²(p

|y|²−(|x|²|y|²−< x, y >²) . Let us put

r=|y|, u=|x|², s= < x, y >

|y| , v =< a, x >, t = < a, y >

|y| , |a|<1.

(4)

Then, the above metric can be written as F =rn

1 +v+ (1−u)t

√1−u+s²+s o (√

1−u+s²+s)² (1−u)²√

1−u+s².

In [11], Tayebi-Shahbazi Nia find found a group of projectively flat Finsler metrics composed by u, v, s and t with double square roots. Then it is natu- ral to ask if there exist more projectively flag Finsler metrics defined by the Euclidean metric |y| and the 1−forms< x, y >, < a, y >?

These motivate us to study the following Finsler metric F =rφ(u, s, v, t),

(5)

where x∈Rⁿ,y ∈T_xRⁿ, a=a_iyⁱ is a constant 1−form, <, > is the standard inner product of Rⁿ and φ is a C^∞ function [2, 3]. When a = 0, then the metric F in (5) becomes a spherically symmetric. When a 6= 0, F in (5) is neither spherically symmetric nor general-(α, β) metric [4]. In this paper, we prove the following:

Theorem 1.1. Let F = rφ(u, s, v, t) be a Finsler metric on an open subset U ⊂Rⁿ with dimensionn ≥3in (5). Then F is a Berwald metric if and only if the following PDE’s hold

P −sP_s−tP_t= 0, P_ss=P_tt =P_st = 0, Q_s−sQ_ss−tQ_st = 0,

Q_t−tQ_tt−sQ_st = 0, Q_sss=Q_sst =Q_stt =Q_ttt= 0, R_s−sR_ss−tR_st = 0, Rt−tRtt−sRst = 0, Rsss=Rsst =Rstt =Rttt= 0.

2. Preliminaries

AFinsler metricon a manifoldM is a functionF :T M →[0,∞) which has the following properties: (i)F isC^∞onT M0; (ii)F(x, λy) = λF(x, y) λ >0;

and (iii) For any tangent vector y∈ TxM, the vertical Hessian of F²/2 given by

g_ij(x, y) = 1

2F²

yⁱy^j

is positive definite.

Every Finsler metric F induces a spray G =y^{i ∂}_∂xi −2Gⁱ(x, y)_∂y^∂i is defined by

Gⁱ(x, y) := 1

4g^il(x, y)n 2∂g_jl

∂x^k(x, y)− ∂g_jk

∂x^l (x, y)o y^jy^k, where the matrix (g^ij) means the inverse of matrix (g_ij).

From [4], we have the Hessian matrix gij(x, y) := ¹₂[F²]_yⁱ_y^j of F in (5) as follows

g_ij =C₀δ_ij+C₁a_ia_j +C₂y_i r

y_j

r +C₃ a_jy_i

r +a_iy_j

r ) +C₄(x_jy_i

r +x_iy_j r

+C₅

a_jx_i+a_ix_j

+C₆x_ix_j, where

C0 =φ²−sφφs−tφφt, C₁ =φ²_t +φφ_tt,

C₂ =s²(φ²_s+φφ_ss) +t²(φ²_t +φφ_tt) + 2ts(φ_sφ_t+φφ_st)−sφφ_s−tφφ_t, C₃ =φφ_t−s(φ_sφ_t+φφ_st)−t(φ²_s+φφ_ss),

C₄ =φφ_s−s(φ²_s+φφ_ss)−t(φ_sφ_t+φφ_st), C₅ =φ_sφ_t+φφ_st,

C₆ =φ²_s+φφ_ss.

In order to compute the geodesic spray coefficients ofF in (5), let us denote g_ij =C₀

F_ij +γy_i r

y_j r

, where

γ =−C₃ C0

, F_ij =E_ij +θN_iN_j, θ = C₅ C0

, N_i =a_i+x_i, E_ij =D_ij +ξM_iM_j, ξ = C₃

C0

, M_i =a_i+y_i r, D_ij =B_ij +a_ia_j, = C₁−C₃ −C₅

C₀ , Bij =Aij +λLiLj, λ= C₂

C₀, Li = C₄

C₂xi+y_i r,

A_ij =δ_ij +µx_ix_j, µ= C2C6−C₄²−C2C5

C₀C₂ . Therefore, the inverse of the metric tensor is given by g^ij =C₀⁻¹n

δ^ij −ζxⁱx^j −τ LⁱL^j−ν(B^ij)²a_ia_j −σMⁱM^j −κNⁱN^j−α(F^ij)²y_iy_jo , where

ζ = µ

1 +µu, τ = λ

1 +λL², Lⁱ =ωxⁱ+ yⁱ r,

ν =

1 +a², B^ija_j =b₁xⁱ+b₂yⁱ r +aⁱ,

σ = ξ

1 +ξM², Mⁱ =d₁xⁱ+d₂yⁱ

r +d₃aⁱ,

κ= θ

1 +θN², Nⁱ =e₁xⁱ+e₂yⁱ

r +e₃aⁱ, α= γ

1 +γy², F^ijy_j =f₁xⁱ+f₂yⁱ

r +f₃aⁱ.

Since ω, L², b₁, b₂, a², d₁, d₂, d₃, M², e₁, e₂, e₃, N², f₁, f₂, f₃ and y² are too long, they are listed in Appendix.

On the other hand, by the definition of the geodesic spray coefficients, we have

Gⁱ := 1 4g^iln

F²

x^ky^ly^k− F²

x^l

o

= F_x^ky^k

2F yⁱ+F 2g^il

F_x^k_y^ly^k−F_x^l . Since F_xk = 2rφ_ux_k+φ_sy_k+rφ_va_k, one can write the first part as

F_x^ky^k

2F yⁱ = r 2φ

2sφ_u +φ_s+tφ_v y_i. (6)

At the same time, it can be computed that F_x^k_y^l =

2rφ_ux_k+φ_sy_k+rφ_va_k

y^l

= 2 r

φ_u−sφ_us−tφ_ut

x_ky_l+ 1 r

φ_v−sφ_vs −tφ_vt a_ky_l

−1 r²

sφss+tφst

ykyl+1 r

φstalyk+φssxlyk

+2

φ_usx_lx_k+φ_uta_lx_k

+φ_vta_ka_l+φ_sδ_lk+φ_vsa_kx_l. Hence

F_x^k_y^ly^k−F_x^l =r

2sφus+φss+tφsv −2φu

xl−sy_l r

+r

2sφ_ut+φ_st+tφ_vt−φ_v

a_l−ty_l r

. (7)

Combining (6) and (7), the geodesic spray coefficients become Gⁱ = r

2φ

2sφu+φs+tφv

yi+r²φ 2 g^il

(

2sφus+φss+tφsv−2φu

xl−sy_l r

+

2sφ_ut+φ_st+tφ_vt−φ_v

a_l−ty_l r

)

. So we only need to compute

g^il

x_l−sy_l r

=C₀⁻¹

Cxⁱ+Dyⁱ

r +Eaⁱ

, (8)

and

g^il

a_l−tyl

r

=C₀⁻¹

Gxⁱ+Hyⁱ r +Iaⁱ

, (9)

whereC, D, E, G, H and I are again too long, they are listed in Appendix. By putting (8) and (9) intoGⁱ and simplifying the result it, one will finally come to the formula

Gⁱ =rP yⁱ+r²Qxⁱ+r²Raⁱ, (10)

where P, Q and R being long, they are again listed in Appendix. When a = 0, then the geodesics spray coefficients in (10) become the geodesics spray coefficients of spherically symmetric Finsler metrics [6].

2.1. Berwald curvature. The Berwald curvature of a Finsler metric is a tensor defined in local coordinates as follows

B :=Bⁱ_jkldx^j ⊗dx^k⊗dx^l⊗ ∂

∂xⁱ, where

Bⁱ_jkl= ∂³Gⁱ

∂y^j∂y^k∂y^l.

For a Finsler metric in (5), we already know its geodesic spray coefficients can be written as Gⁱ =rP yⁱ+r²Qxⁱ+r²Raⁱ.

Proposition 2.1. LetF =rφ(u, s, v, t)be a Finsler metric on an open subset U ⊂ Rⁿ with dimension n ≥3 in (5). Then Berwald curvature of F is given by

Bⁱ_jkl= +1 r{δ_jⁱ

Pssx^kx^l+Ptta^ka^l+Pst x^ka^l+x^la^k +(P −sP_s−tP_t)δⁱ_jδ_kl}(j →k →l →j)

−1

r²[(sP_ss+tP_st)δⁱ_jx^ky^l+ (sP_ss+tP_st)δ_jⁱy^kx^l

+(sP_ss+tP_st)yⁱδ_jkx^l+ (tP_tt+sP_st)δ_jⁱa^ky^l+ (tP_tt+sP_st)δ_jⁱy^ka^l

+(tP_tt+sP_ts)yⁱδ_jka^l](j →k →l →j) + 1

r³[(s²P_ss+t²P_tt+sP_s +tP_t+stP_st+stP_ts−P) δ_jⁱy^ky^l+yⁱδ_jky^l

](j →k →l →j) +1

r⁵(3P −s³P_sss−t³P_ttt−s²tP_tss−st²P_stt−s²tP_sst−st²P_tts

−s²tP_sts−st²P_tst−6s²P_ss−6t²P_tt−6stP_st −6stP_ts−3sP_s

−3tP_t)yⁱy^jy^ky^l+ yⁱ

r² P_sssx^jx^kx^l+P_ttta^ja^ka^l +1

r⁴[ s²P_sss+t²P_stt+stP_sst+stP_sts+ 3sP_ss+ 3tP_st

yⁱy^jy^kx^l +(s²P_sst+t²P_ttt+stP_stt+stP_tst+ 3tP_tt

+3sP_st)yⁱy^jy^ka^l](j →k →l →j)

−1

r³[(P_ss+sP_sss+tP_sst)yⁱy^jx^kx^l

+(P_st +sP_sst+tP_stt)yⁱy^ja^kx^l+ (P_st+sP_sts+tP_stt)yⁱy^jx^ka^l +(P_tt+tP_ttt+sP_stt)yⁱy^ja^ka^l](j →k→l →j)

+1

r² P_sttyⁱx^ja^ka^l+P_sstyⁱa^jx^kx^l

(j →k →l→j) +1

r[(Q_s−sQ_ss−tQ_st)xⁱδ_jkx^l

+(Q_t−tQ_tt−sQ_ts)xⁱδ_jka^l](j →k→l →j) +1

r³[(s²Q_sss+t²Q_stt+stQ_sst+stQ_sts+sQ_ss+tQ_st

−Q_s)xⁱx^jy^ky^l+ (t²Q_ttt+s²Q_sst+stQ_stt+stQ_tst+tQ_tt+sQ_st

−Q_t)xⁱy^jy^ka^l](j →k →l→j) + 1

r²(s²Q_ss+t²Q_tt +stQ_ts+stQ_st−sQ_s−tQ_t)xⁱy^jδ_kl(j →k→l →j)

−1

r² (tQ_sst+sQ_sss)xⁱx^jx^ly^k(j →k →l→j) +xⁱ

r Qsssx^jx^kx^l+Qttta^ja^ka^l +1

r⁴(3sQ_s+ 3tQ_t−3s²Q_ss−3t²Q_tt−3stQ_ts −3stQ_st−s³Q_sss

−t³Q_ttt−s²tQ_sst−st²Q_tts−s²tQ_sts−st²Q_tst

−s²tQtss−st²Qstt)xⁱy^jy^ky^l +1

r Q_sstxⁱx^jx^ka^l+Q_sttxⁱx^ja^ka^l

(j →k →l→j)

−1

r²[(sQ_sst+tQ_stt)xⁱx^jy^ka^l+ (sQ_sts+tQ_stt)xⁱx^ja^ky^l + (sQ_stt+tQ_ttt)xⁱy^ja^ka^l](j →k →l →j)

+1

r[(R_t−tR_tt−sR_ts)aⁱδ_jka^l

+(R_s−sR_ss−tR_st)aⁱδ_jkx^l](j →k →l→j) +1

r³[(t²R_ttt+s²R_sst+stR_stt+stR_tst+tR_tt+sR_ts−R_t)aⁱa^jy^ky^l +(s²R_sss+t²R_stt+stR_sst+stR_sts+sR_ss

+tR_st−R_s)aⁱy^jy^kx^l](j →k →l →j) + 1

r²(s²R_ss+t²R_tt +stRst +stRts−sRs−tRt)aⁱy^jδkl(j →k→l →j)

−1

r²(sR_stt+tR_ttt)aⁱa^ja^ky^l(j →k→l →j) +1

r R_tttaⁱa^ja^ka^l+R_sssaⁱx^jx^kx^l +1

r⁴(3sR_s+ 3tR_t−3s²R_ss−3t²R_tt−3stR_st−3stR_ts−s³R_sss

−t³R_ttt−s²tR_sst−st²R_stt−s²tR_sts−st²R_tst−s²tR_tss

−st²R_tts)aⁱy^jy^ky^l +1

r R_sttaⁱa^ja^kx^l+R_sstaⁱa^jx^kx^l

(j →k →l →j)

−1

r²[(tR_stt+sR_sts)aⁱa^jy^kx^l+ (tR_stt+sR_sst)aⁱa^jx^ky^l +(tRsst+sRsss)aⁱy^jx^kx^l](j →k→l →j).

(11)

Whena= 0, then the Berwald curvature in (11) becomes the Berwald curvature of spherically symmetric Finsler metrics [6].

Proof. Let F be a Finsler metric in (5). From (4) and (10), we have

∂Gⁱ

∂yⁱ =r_y^jP yⁱ+r(Pss_y^j+Ptt_y^j)yⁱ+rP δⁱ_j+ 2y^jQxⁱ

+r²(Q_ss_y^j +Q_tt_y^j)xⁱ+ 2y^jRaⁱ+r²(R_ss_y^j +R_tt_y^j)aⁱ, (12)

where we have used _∂y^∂ui = 0 and ^∂r_∂y²j = 2y^j. By (12), we obtain

∂²Gⁱ

∂y^j∂y^k = [(Pss_y^k +Ptt_y^k)yⁱr_y^j +P δ_kⁱr_y^j+r(Pss_y^j +Ptt_y^j)δ_kⁱ +2(Q_ss_y^j +Q_tt_y^j)y^kxⁱ+ 2(R_ss_y^j +R_tt_y^j)y^kaⁱ](j ↔k) +r(P_sss_yk +P_stt_yk)yⁱs_y^j+P_sryⁱs_yjy^k +P yⁱr_yjy^k

+r(P_tss_y^k +P_ttt_y^k)yⁱt_y^j +P_tryⁱt_y^j_y^k + 2Qxⁱδ_jk

+r²(Q_sss_y^k +Q_stt_y^k)xⁱs_y^j+r²Q_sxⁱs_y^j_y^k+r²Q_txⁱt_y^j_y^k +r²(Q_tss_y^k+Q_ttt_y^k)xⁱt_y^j + 2Raⁱδ_jk+r²R_saⁱs_y^j_y^k +r²(Rsss_y^k +Rstt_y^k)aⁱs_y^j +r²(Rtss_y^k +Rttt_y^k)aⁱt_y^j

+r²R_taⁱt_y^j_y^k,

where j ↔k denotes symmetrization. By definition, we get Bⁱ_jkl= Pttyⁱt_y^jt_y^kr_y^l+Ptyⁱt_y^j_y^kr_y^l+Ptyⁱt_y^jr_y^k_y^l

(j →k →l →j) +[P_s s_y^jr_yk +s_ykr_y^j

δ_lⁱ+P r_yjy^kδ_lⁱ +P_ssrs_y^js_y^kδ_lⁱ](j →k →l→j)

+(Pssyⁱs_y^js_y^kr_y^l+Psyⁱs_y^j_y^kr_y^l+Psyⁱs_y^jr_y^k_y^l)(j →k →l→j) +

P_t t_y^jr_yk+t_ykr_y^j

δ_lⁱ+P_ttrt_y^jt_ykδ_lⁱ

(j →k →l→j) + P_srs_y^j_y^kδⁱ_l+P_ssryⁱs_y^js_y^k_y^l

(j →k →l→j) + P_trt_y^j_y^kδ_lⁱ+P_ttryⁱt_y^jt_y^k_y^l

(j →k →l →j) +[P_sstryⁱs_y^js_ykt_yl+P_sttryⁱs_y^jt_ykt_yl

+P_stryⁱ s_y^j_y^kt_y^l+s_y^jt_y^k_y^l

](j →k →l→j) +P_st{[yⁱ s_y^jt_yk+s_ykt_y^j

r_yl

+r s_y^jt_y^k +s_y^kt_y^j

δ_lⁱ]}(j →k →l →j) +2xⁱ Q_ss_y^jδ_kl+Q_ssy^js_y^ks_y^l+Q_sy^js_y^k_y^l

(j →k →l→j) +2xⁱ Q_tt_y^jδ_kl+Q_tty^jt_ykt_yl+Q_ty^jt_yky^l

(j →k →l→j) +[Qsstr²xⁱs_y^js_y^kt_y^l+Qsttr²xⁱs_y^jt_y^kt_y^l

+Q_str²xⁱ s_yjy^kt_yl+s_y^jt_yky^l

](j →k →l →j) +[2Q_stxⁱy^j s_y^kt_y^l+s_y^lt_y^k

+r²xⁱ Q_sss_y^js_y^k_y^l +Q_ttt_y^jt_y^k_y^l

](j →k →l →j) +2aⁱ R_ss_y^jδ_kl+R_ssy^js_yks_yl+R_sy^js_yky^l

(j →k→l →j) +2aⁱ R_tt_y^jδ_kl+R_tty^jt_y^kt_y^l+R_ty^jt_y^k_y^l

(j →k→l →j) +[R_sstr²aⁱs_y^js_y^kt_y^l +R_sttr²aⁱs_y^jt_y^kt_y^l

+R_str²aⁱ s_y^j_y^kt_y^l+s_y^jt_y^k_y^l

](j →k →l→j) +[2R_staⁱy^j s_y^kt_y^l+s_y^lt_y^k

+r²aⁱ R_sss_y^js_yky^l+R_ttt_y^jt_yky^l

](j →k→l →j)

+P_sssryⁱs_y^js_y^ks_y^l+P yⁱr_y^j_y^k_y^l+P_sryⁱs_y^j_y^k_y^l+P_tttryⁱt_y^jt_y^kt_y^l +Ptryⁱt_y^j_y^k_y^l+Qsssr²xⁱs_y^js_y^ks_y^l+Qsr²xⁱs_y^j_y^k_y^l

+Q_tttr²xⁱt_y^jt_ykt_yl +Q_tr²xⁱt_yjy^ky^l+R_sssr²aⁱs_y^js_yks_yl

+R_sr²aⁱs_y^j_y^k_y^l+R_tttr²aⁱt_y^jt_y^kt_y^l+R_tr²aⁱt_y^j_y^k_y^l, (13)

where j →k→l →j denotes cyclic permutation.

Observe that

r_y^j = y^j r , (14)

r_yjy^k = r²δ_jk −y^jy^k r³ , (15)

r_yjy^ky^l = 3y^jy^ky^l−r²δ_jky^l(j →k →l→j)

r⁵ .

(16)

where we have used (4). Direct computations yield s_y^j = rx^j −sy^j

r² , s_y^j_y^k = 3sy^jy^k−rx^jy^k−rx^ky^j−sr²δ_jk

r⁴ ,

(17)

s_yjy^ky^l = 1

r⁵(3x^jy^ky^l+ 3srδ_jky^l−r²x^jδ_kl)(j →k→l →j) (18)

−15

r⁶sy^jy^ky^l, t_y^j = ra^j−ty^j

r² , (19)

t_y^j_y^k = 3ty^jy^k−ra^jy^k−ra^ky^j −tr²δ_jk

r⁴ ,

(20)

t_y^j_y^k_y^l = 1

r⁵(3a^jy^ky^l+ 3trδ_jky^l−r²a^jδ_kl)(j →k →l →j) (21)

−15

r⁶ty^jy^ky^l.

From (13)-(21) we conclude the proof.

Proof of theorem 1.1. As we know, a Finsler metricF is called Berwald metric if Berwald curvature is zero. From (11) , a Finsler metricF in (5) is a Berwald metric if and only ifP, Q and R in its geodesic spray coefficients must satisfy

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







P −sP_s−tP_t= 0, P_ss=P_tt =P_st = 0, Q_s−sQ_ss−tQ_st = 0, Q_t−tQ_tt−sQ_st = 0,

Q_sss =Q_sst =Q_stt =Q_ttt = 0, Rs−sRss−tRst = 0,

R_t−tR_tt−sR_st = 0,

R_sss=R_sst =R_stt =R_ttt= 0.

From these equations, one can first solve P, Qand R, then completely deter-

mine the metric functionF.

2.2. Douglas curvature. In [1], Douglas introduced the local function D_{j kl}ⁱ as follows

D_{j kl}ⁱ := ∂³

∂y^j∂y^k∂y^l

Gⁱ− 1 n+ 1

∂G^m

∂y^myⁱ

, (22)

These functions are called Douglas curvature and a Finsler metric F is said to be a Douglas metric ifD_{j kl}ⁱ = 0 [10, 12].

Proposition 2.2. LetF =rφ(u, s, v, t)be a Finsler metric on an open subset U ⊂Rⁿ with dimensionn≥3in (5). ThenF has vanishing Douglas curvature if and only if the following hold



































Q_s−sQ_ss−tQ_st = 0, Qt−tQtt−sQst = 0,

Q_sss =Q_sst =Q_stt =Q_ttt = 0, R_s−sR_ss−tR_st = 0,

R_t−tR_tt−sR_st = 0,

R_sss=R_sst =R_stt =R_ttt= 0.

(23)

Proof. Let F be a Finsler metric in (5). From (4) and (10), we have

∂G^j

∂y^j =r_y^jP y^j+r(P_ss_y^j +P_tt_y^j)y^j+nrP + 2Q < x, y >

+r²(Q_ss_y^j +Q_tt_y^j)x^j + 2R < a, y >+r²(R_ss_y^j +R_tt_y^j)a^j

=r

(n+ 1)P + 2sQ+ 2tR+ (u−s²)Q_s+ (v−st)Q_t+ (v −st)R_s +(a²−t²)R_t

. It follows that

Gⁱ− 1 n+ 1

∂G^j

∂y^jyⁱ =rZyⁱ+r²Qxⁱ+r²Raⁱ, where

Z =− 1

n+ 1[2sQ+ 2tR+ (u−s²)Q_s+ (v−st)Q_t+ (v−st)R_s +(a²−t²)R_t].

(24)

SubstitutingGⁱ into (22) we get D_{j kl}ⁱ := ∂³

∂y^j∂y^k∂y^l rZyⁱ+r²Qxⁱ+r²Raⁱ .

In this case, to get Douglas curvature, one can replace Z and P in (11).

When a = 0, then the Douglas curvature becomes the Douglas curvature of spherically symmetric Finsler metrics [5].

Therefore, a Finsler metric F in (5) is a Douglas metric if, and only if, Z−sZs−tZt = 0,

(25)

Z_ss =Z_tt =Z_st = 0, (26)

Q_s−sQ_ss−tQ_st = 0, (27)

Q_t−tQ_tt−sQ_st = 0, (28)

Q_sss=Q_sst =Q_stt =Q_ttt= 0, (29)

R_s−sR_ss−tR_st = 0, (30)

Rt−tRtt−sRst = 0, (31)

R_sss =R_sst =R_stt =R_ttt= 0.

(32)

Plugging (24) into (25), we have

−(n+ 1)(Z−sZ_s−tZ_t) = (Q_s−sQ_ss−tQ_st) u−s² + (Qt−tQtt−sQst) (v−st) + (R_s−sR_ss−tR_st) (v−st) + (R_t−tR_tt−sR_st) a²−t²

.

Thus (27), (28), (30) and (31) imply (25). Finally, (26) is easy to obtain from (27)-(32). Then (25)-(32) can be reduced to (23).

3. Appendix

ω := C₄ C₂ −C₄

C₂ζu−ζs, L² :=sω+ C₄

C₂(uω+s) + 1, b₁ :=−vτ ω²−tτ ω−vζ, b₂ :=−vτ ω−tτ,

B^ija_ja_i :=tb₂+vb₁ +a²,

d₁ :=−sν b₁²−sτ ω²−tν b₁b₂−vν b₁²−vτ ω² −a²ν b₁−tν b₁−tτ ω

−ν b₁b₂−sξ−vξ−τ ω,

d2 :=−sν b1b2−tν b22−vν b1b2−a²ν b2−sτ ω−tν b2−vτ ω−ν b22−tτ

−τ + 1,

d₃ :=−sν b₁−tν b₂ −vν b₁−a²ν−tν−ν b₂+ 1, M² := (s+v)d₁+ (t+ 1)d₂+ a²+t

d₃,

e1 :=−a²σ d1d3−sν b1b2−sσ d1d2−tν b1b2−tσ d1d2−uν b12−uσ d12

−uτ ω² −vν b₁²−vσ d₁²−vσ d₁d₃−vτ ω²−a²ν b₁−sτ ω−tτ ω

−vν b₁−uξ−vξ+ 1,

e₂ :=−a²σ d₂d₃−sν b₂²−sσ d₂²−tν b₂²−tσ d₂²−uν b₁b₂−uσ d₁d₂

−vν b₁b₂−vσ d₁d₂−vσ d₂d₃ −a²ν b₂−uτ ω−vν b₂−vτ ω−sτ −tτ, e₃ :=−a²σ d₃² −sσ d₂d₃−tσ d₂d₃−uσ d₁d₃−vσ d₁d₃−vσ d₃²−sν b₂

−tν b2−uν b1−vν b1−a²ν−νv+ 1, N² := (u+v)e₁+ (s+t)e₂+ a²+v

e₃,

f₁ :=−sκ e₁²−sν b₁²−sσ d₁²−sτ ω²−tκ e₁e₃−tσ d₁d₃−tν b₁−κ e₁e₂

−ν b₁b₂−σ d₁d₂−sξ−τ ω,

f₂ :=−sκ e₁e₂−sν b₁b₂−sσ d₁d₂−tκ e₂e₃−tσ d₂d₃−sτ ω−tν b₂−κ e₂²

−ν b22−σ d22 −τ+ 1,

f₃ :=−sκ e₁e₃−sσ d₁d₃−tκ e₃²−tσ d₃²−sν b₁−κ e₂e₃−σ d₂d₃−tν

−ν b₂,

y² :=sf₁+tf₃+f₂,

C :=s²α f₁²+s²κ e₁²+s²ν b₁²+s²σ d₁²+s²τ ω²+stα f₁f₃+stκ e₁e₃ +stσ d₁d₃+stν b₁−uα f₁²−uκ e₁²−uν b₁²−uσ d₁²−uτ ω²

−vα f₁f₃−vκ e₁e₃−vσ d₁d₃+s²ξ−vν b₁−uξ+ 1,

D:=s²κ e1e2+s²ν b1b2+s²σ d1d2+stα f2f3+stκ e2e3+stσ d2d3+s²τ ω +stν b₂−uα f₁f₂−uκ e₁e₂−uν b₁b₂−uσ d₁d₂−vα f₂f₃−vκ e₂e₃

−vσ d₂d₃−uτ ω−vν b₂+α f₁f₂−s,

E :=s²α f₁f₃+s²κ e₁e₃+s²σ d₁d₃+stα f₃²+stκ e₃²+stσ d₃²+s²ν b₁

−uα f₁f₃−uκ e₁e₃ −uσ d₁d₃−vα f₃²−vκ e₃²−vσ d₃²+stν−uν b₁

−vν, G:=stα f12

+stκ e12

+stν b12

+stσ d12

+stτ ω²+t²α f1f3+t²κ e1e3

+t²σ d₁d₃−a²α f₁f₃−a²κ e₁e₃−a²σ d₁d₃ +t²ν b₁−vα f₁²−vκ e₁²

−vν b₁²−vσ d₁²−vτ ω²−a²ν b₁+stξ −vξ,

H :=stα f₁f₂+stκ e₁e₂ +stν b₁b₂+stσ d₁d₂+t²α f₂f₃+t²κ e₂e₃

+t²σ d₂d₃−a²α f₂f₃−a²κ e₂e₃−a²σ d₂d₃ +stτ ω+t²ν b₂−vα f₁f₂

−vκ e₁e₂−vν b₁b₂−vσ d₁d₂−a²ν b₂−vτ ω−t,

I :=stα f₁f₃+stκ e₁e₃ +stσ d₁d₃+t²α f₃²+t²κ e₃²+t²σ d₃²−a²α f₃²

−a²κ e32−a²σ d32

+stν b1−vα f1f3−vκ e1e3−vσ d1d3+t²ν−vν b1

−a²ν+ 1, P :=A+ BD+F H

C₀ ,