Filomat 28:1 (2014), 107–117 DOI 10.2298/FIL1401107C
Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat
New Cartan’s Tensors and Pseudotensors in a Generalized Finsler Space Milica D. Cvetkovi´ca , Milan Lj. Zlatanovi´cb a University
of Niˇs, Faculty of Science and Mathematics, 18000 Niˇs, Serbia, and College for Applied Technical Sciences, 18000 Niˇs, Serbia, b University of Niˇs, Faculty of Science and Mathematics, 18000 Niˇs, Serbia
Abstract. In this work we defined a generalized Finsler space (GFN ) as 2N-dimensional differentiable ˙ = 0, θ = 1, 2. Based on ˙ which applies that 1ij | m (x, x) manifold with a non-symmetric basic tensor 1i j (x, x), θ
non-symmetry of basic tensor, we obtained ten Ricci type identities, comparing to two kinds of covariant derivative of a tensor in Rund’s sense. There appear two new curvature tensors and fifteen magnitudes, we called ”curvature pseudotensors” .
1. Introduction Finsler geometry is a natural and fundamental generalization of Riemann geometry. It was first suggested by Riemann as early as 1854 [15], and studied systematically by Finsler in 1918 [5]. The name ”Finsler Geometry” was first given by J. Taylor in 1927. The non-symmetric connection was interesting to many authors: K. Yano [1], A.C. Shamihoke [17], S.Minˇci´c [8]-[10], S. Manoff [6], C. K. Mishra[11] and many others: [7], [20], [21]. Finsler [5] spaces FN are N-dimensional manifolds where the infinitesimal distance between two neighboring points xi , xi + dxi is given by: ds = F(xi , dxi ), i = 1, . . . , N,
(1)
where F is required to satisfy some properties [16]: 1) F(xi , dxi ) > 0; 2) F(xi , λdxi ) = λF(x, dxi ), for any λ > 0;
(2)
∂ F (x , dx ) i j ξ ξ > 0, for all vector ξi and any (xi , dxi ). ∂dxi ∂dx j Then, the metric tensor is defined from (1) as: 2 2
i
i
3) The quadric form
1ij =
1 ∂2 F2 (xi , x˙ i ) , 2 ∂x˙ i ∂x˙ j
2010 Mathematics Subject Classification. Primary 53A45; Secondary 53B05, 53B40 Keywords. The generalized Finsler space, Ricci type identities, h-differentiation. Received: 16 May 2013; Accepted: 22 September 2013 Communicated by Ljubica Velimirovi´c Research supported by the research projects 174012 of the Serbian Ministry of Science Email addresses:
[email protected] (Milica D. Cvetkovi´c),
[email protected] (Milan Lj. Zlatanovi´c)
(3)
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108
dxi are the tangent vectors to a curve C : xi = xi (t) in the manifold space, or elements of the dt tangent space Tn (xi ) at point xi . Using the second condition in (2), 1i j are homogeneous of degree zero in the other set of variables and we can write: where x˙ i =
ds2 = 1ij (xk , dxk )dxi dx j .
(4)
Definition 1.1. The generalized Finsler space (GFN ) is a differentiable manifold with non-symmetric basic tensor ˙ , where 1i j (x, x) ˙ , 1 ji (x, x) ˙ , 1ij (x, x)
(1 = det(1ij ) , 0) .
(5)
Based on (5), it can be defined the symmetric, respectively, antisymmetric part of 1i j : 1ij =
1 (1ij + 1 ji ) , 2
1 (1ij − 1 ji ) , 2
1i j = ∨
(6)
where, following [17], it is true that: a) 1ij =
˙ 1 ∂2 F2 (x, x) , j i 2 ∂x˙ ∂x˙
∂1ij b)
∨
∂x˙ k
=0,
(7)
where F is a metric function in GFN , having the properties known from the theory of usual Finsler space (2). In the papers [8–10, 23, 24] we studied generalized Finsler space. Introducing a Cartan tensor Cijk , similar as in FN , we have: de f
˙ = Ci jk (x, x)
1 1 1 1 k = 1 k = F2x˙ i x˙ j x˙ k , 2 ij,x˙ (7b) 2 i j,x˙ 4
(8)
where ” = ” signifies ”equal based on (7b)”. We can conclude that Ci jk is symmetric in relation to each pair (7b)
of indices. Also, we have: de f
Cijk = hip Cpjk = hip C jpk = hip C jkp = Cik j .
(9)
(8)
In GFN the next equations are valid: Ci jk x˙ i = Ci jk x˙ j = Cijk x˙ k = 0 .
(10)
One obtains coefficients of non-symmetric affine connections in the Cartan sense [2]: Γ∗ijk = γijk − hil (C jlp Γks + Cklp Γ js − C jkp Γls )x˙ s , Γ∗ikj ,
(11)
Γ∗i.jk = Γ∗rjk 1ir = γi.jk − (Ci jp Γks + Cikp Γ js − C jkp Γis )x˙ s , Γ∗i.k j .
(12)
p
p
p
p
p
p
We defined the coefficients: p p p e Γ∗ijk = γijk − hil (Cklp Γs j + C jlp Γsk − Ck jp Γsl )x˙ s , Γ∗ikj ,
(13)
p p p e Γ∗i.k j . Γ∗i.jk = e Γ∗rjk 1ir = γi.jk − (Cikp Γsj + Cijp Γsk − Ck jp Γsi )x˙ s , e
(14)
Let us denote: ˙ = Γ∗ijk − Γ∗ikj , T∗ijk (x, x)
Γi( jk) = Γ∗ijk + Γik j ,
e ˙ =e T∗ijk (x, x) Γ∗ijk − e Γ∗ikj ,
e Γi( jk) = e Γ∗ijk + e Γik j ,
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as torsion tensors of the connections Γ∗ , e Γ∗ , respectively. Based on non-symmetry of the coefficients of the connection, it can be defined two kinds of h-covariant derivative: ∗p
Tij| m = Tij,m + T j Γ∗ipm − Tpi Γ jm − Tij,s˙ Γsrm x˙ r , p
1
Tij| m 2
(15)
∗p p ∗i Γm j − Tij,s˙ Γsmr x˙ r . = Tij,m + T j e Γmp − Tpi e
By the procedure that is similar in a Finsler space, it can be proved that covariant derivative (15) of a tensor also is a tensor. ˙ based on both kinds of derivative (15) in GFN it is valid: Theorem 1.2. For the tensor 1ij (x, x) ˙ =0, 1ij | m (x, x)
θ = 1, 2.
θ
(16)
Proof. Starting from (15), we get: ∗p
p
∗p
˙ = 1ij ,m − 1i j,p˙ Γrm x˙ r − Γim 1pj − Γ jm 1ip = 1ij | m (x, x) 1
= 1ij ,m − 2Cijp Γrm x˙ r − (Γ∗i.jm + Γ∗j.im ) = 0 . p
(17)
(8,12)
The same result one obtains for 1ij| m : 2
p ∗p ∗p ˙ = 1ij ,m − 1i j,p˙ Γmr x˙ r − e Γmi 1pj − e Γmj 1ip = 1ij | m (x, x) 2
p Γ∗i.mj ) = 0 , = 1ij ,m − 2Cijp Γmr x˙ r − (e Γ∗j.mi + e
(18)
(8,12)
and we have proved (16).
2. Ricci type identities for h-differentiation in GFN It is known that in Finsler space there is only one Ricci identity for h-differentiation, corresponding to alternated covariant derivative of the 2nd order. In the case of non-symmetric affine connection there are 10 possibilities to form the difference: ...ru ...ru art11...t − art11...t v |m|n v |m | n λ
µ
ν
(λ, µ, ν, ω = 1, 2) ,
(19)
ω
where | , | denote two kinds of covariant derivative based on (1.17, 1.18), and we can obtain ten Ricci type 1 2
identities and two tensors of curvature. Corresponding identities in GFN may be proved by total induction method. The mentioned possibilities are obtained for these combinations: (λ, µ; ν, ω) ∈ {(1, 1; 1, 1), (2, 2; 2, 2), (1, 2; 1, 2), (2, 1; 2, 1), (1, 1; 2, 2), (1, 1; 1, 2), (1, 1; 2, 1), (2, 2; 1, 2), (2, 2; 2, 1), (1, 2; 2, 1)}.
(20)
For finding the general cases based on (2.1), we firstly observe the case of a tensor ai (x, ξ). Let us obtain the case when the vector field ξl is stationary comparing to the first kind of covariant derivative, i.e. ξl| h (x, ξ) = 0, 1
ξl| h (x, ξ) , 0 . 2
(21)
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And we have: ξl,h =
∂Gl (x, ξ) ∂ξl ∗l r = −Γ ξ = − = Gl,h˙ . rh ∂xh ∂x˙ h
(22)
Then, we obtained, for example, ∗p
∗p
aij| mn =(aij| m ),n + (aij| m ),s˙ ξs,n + Γ∗ipn a j| m − Γ jn aip| m − Γmn aij| p = p
1
1
1
1
1
1
∗p
∗p
=aij,mn + aij,ns˙ ξs,m + aij,s˙ ξs,mn + Γ∗ipm,n a j + Γ∗ipm a j,n − Γ jm,n aip − Γ jm aip,n + p
p
∗p
∗p
∗i a j,s˙ ξs,n − Γ jm,s˙ aip ξs,n − Γ jm aip,s˙ ξs,n + + aij,ms˙ ξs,n + aij,l˙s˙ ξl,m ξs,n + aij,l˙ξl,ms˙ ξs,n + Γ∗ipm,s˙ a j ξs,n + Γpm p
p
∗p
(23)
+ Γ∗ipn a j,m + Γ∗ipn a j,s˙ ξs,m + Γ∗ipn Γsm asj − Γ∗ipn Γ∗sjm as − p
∗p
p
∗p
p
∗p
∗p
∗p
i i − Γ jn aip,m − Γ jn aip,s˙ ξs,m − Γ jn Γ∗ism asp + Γ jn Γ∗s pm as − Γmn a j| p . 1
And similar: ∗p
∗p
aij| m| n = aij,mn + aij,ns˙ ξs,m + aij,s˙ ξs,mn + Γ∗ipm,n a j + Γ∗ipm a j,n − Γ jm,n aip − Γ jm aip,n + p
1
p
2
∗p
∗p
+ aij,ms˙ ξs,n + aij,l˙s˙ ξl,m ξs,n + aij,l˙ξl,ms˙ ξs,n + Γ∗ipm,s˙ a j ξs,n + Γ∗ipm a j,s˙ ξs,n − Γ jm,s˙ aip ξs,n − Γ jm aip,s˙ ξs,n + p
∗p
p
+e Γ∗inp a j,m + e Γ∗inp a j,s˙ ξs,m + e Γ∗inp Γsm asj − e Γ∗inp Γ∗sjm as − p
p
p
(24)
∗p ∗p ∗p ∗p i e∗p i −e Γn j aip,m − e Γn j aip,s˙ ξs,m − e Γnj Γ∗ism asp + e Γnj Γ∗s pm as − Γnm a j| p . 1
∗p
∗p
aij| m| n = aij,mn + aij,ns˙ ξs,m + aij,s˙ ξs,mn + e Γ∗imp,n a j + e Γ∗imp a j,n − e Γmj,n aip − e Γm j aip,n + p
2
p
1
∗p
∗p
+ aij,ms˙ ξs,n + aij,l˙s˙ ξl,m ξs,n + aij,l˙ξl,ms˙ ξs,n + e Γ∗imp,s˙ a j ξs,n + e Γ∗imp a j,s˙ ξs,n − e Γm j,s˙ aip ξs,n − e Γm j aip,s˙ ξs,n + p
p
p p ∗p p + Γ∗ipn a j,m + Γ∗ipn a j,s˙ ξs,m + Γ∗ipne Γms asj − Γ∗ipne Γ∗s mj as −
(25)
∗p ∗p ∗p ∗i s ∗p ∗s i ∗p − Γ jn aip,m − Γ jn aip,s˙ ξs,m − Γ jne Γms ap + Γ jne Γmp as − Γmn aij| p . 2
p p ∗p ∗p aij| mn = aij,mn + aij,ns˙ ξs,m + aij,s˙ ξs,mn + e Γ∗imp,n a j + e Γ∗imp a j,n − e Γmj,n aip − e Γm j aip,n + 2
p p ∗p ∗p + aij,ms˙ ξs,n + aij,l˙s˙ ξl,m ξs,n + aij,l˙ξl,ms˙ ξs,n + e Γ∗imp,s˙ a j ξs,n + e Γ∗imp a j,s˙ ξs,n − e Γm j,s˙ aip ξs,n − e Γm j aip,s˙ ξs,n + p p ∗p p +e Γ∗inp a j,m + e Γ∗inp a j,s˙ ξs,m + e Γ∗inpe Γ∗inpe Γms asj − e Γ∗s mj as −
(26)
∗p ∗p ∗p ∗i s ∗p ∗s i ∗p −e Γn j aip,m − e Γnj aip,s˙ ξs,m − e Γnje Γms ap + e Γnje Γmp as − e Γnm aij| p . 2
Also, we have ∂ ( ∂Gl ) ∂Gs ∂ ( ∂Gs ) + aij,l˙ s = n m ∂x ∂x˙ ∂x˙ ∂x˙ m ∂x˙ n ( ∂Γ∗s ( ) ) ∂Γ∗s rm r rm r ∗s ˙ ˙ = −aij,s˙ x − Γ + x Gl,n˙ = lm l ∂xn ˙ ∂ x ( ) ∗s ∗l ∗l l r = −aij,s˙ Γ∗s rm,n − Γlm Γrn − Γrm,l˙G,n˙ x˙ .
aij,s˙ ξs,mn + aij,l˙ξl,ms˙ ξs,n = −aij,s˙
(27)
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111
1. We considered the difference: ∗p
∗p
aij| mn − aij| nm = aij,mn + aij,ns˙ ξs,m + aij,s˙ ξs,mn + Γ∗ipm,n a j + Γ∗ipm a j,n − Γ jm,n aip − Γ jm aip,n + p
1
p
1
∗p
∗p
+ aij,ms˙ ξs,n + aij,l˙s˙ ξl,m ξs,n + aij,l˙ξl,ms˙ ξs,n + Γ∗ipm,s˙ a j ξs,n + Γ∗ipm a j,s˙ ξs,n − Γ jm,s˙ aip ξs,n − Γ jm aip,s˙ ξs,n + p
∗p
p
+ Γ∗ipn a j,m + Γ∗ipn a j,s˙ ξs,m + Γ∗ipn Γsm asj − Γ∗ipn Γ∗sjm as − p
p
∗p
∗p
∗p
∗p
∗p
p
i i − Γ jn aip,m − Γ jn aip,s˙ ξs,m − Γ jn Γ∗ism asp + Γ jn Γ∗s pm as − Γmn a j| p − 1
∗p
∗p
− aij,nm − aij,ms˙ ξs,n − aij,s˙ ξs,nm − Γ∗ipn,m a j − Γ∗ipn a j,m + Γ jn,m aip + Γ jn aip,m − p
p
(28)
∗p
∗p
− aij,ns˙ ξs,m − aij,l˙s˙ ξl,n ξs,m − aij,l˙ξl,ns˙ ξs,m − Γ∗ipn,s˙ a j ξs,m − Γ∗ipn a j,s˙ ξs,m + Γ jn,s˙ aip ξs,m + Γ jn aip,s˙ ξs,m − p
∗p
p
− Γ∗ipm a j,n − Γ∗ipm a j,s˙ ξs,n − Γ∗ipm Γsn asj + Γ∗ipm Γ∗sjn as + p
p
∗p
∗p
∗p
p
∗p
∗p
i i + Γ jm aip,n + Γ jm aip,s˙ ξs,n + Γ jm Γ∗isn asp − Γ jm Γ∗s pn as + Γnm a j| p = 1
p
p aip 1 jmn
= Ki pmn a j − K 1
∗p
− aij,s˙ Ksrmn x˙ r − Tmn aij| p , 1
1
where ∗i ∗s ∗i ∗i s ∗i s Ki pmn = Γ∗ipm,n − Γ∗ipn,m + Γ∗s pm Γsn − Γpn Γsm − Γpm,s˙ G,n˙ + Γpn,s˙ G,m˙ .
(29)
1
Theorem 2.1. In the generalized Finsler space GFN , for h-differentiation, the first Ricci type identity is expressed: ( ) ( ) u v ∑ p ... ∑ p tβ ... ∗p ... rα ... ... s r a...| mn − a...| nm = K pmn a... − K tβ mn a... − a... (30) ..., s˙ K rmn x˙ − Tmn a...| p , 1 1 1 rα p 1 1 1 α=1
β=1
where K given by (29) and 1
( ) p ... r ...r pr ...r a = at11...tvα−1 α+1 u , rα ...
( ) tβ ... ...ru a = art11...t . β−1 prβ+1 ...tv p ...
(31)
Using (2.6) it is easy to prove that: Ki pmn x˙ r = 1
s s ∂2 Gi ∂2 Gi i ∂G i ∂G i s i s − + G − G = Gi,nm˙ − Gi,nm sm sn ˙ + Gsm G,n˙ − Gsn G,m˙ , ∂xn ∂x˙ m ∂xm ∂x˙ n ∂x˙ n ∂x˙ m
where Gimn =
(32)
∂2 Gi = Ginm . ∂x˙ n ∂x˙ m
We considered: p
p aip 1 jmn
aij| mn − aij| nm = Ki pmn a j − K 1
1
1
∗p
− aij,s˙ Ksrmn x˙ r − Tmn aij| p = 1
p
1
p 1 jmn
= (Ki pmn + Aips Ksrmn lr )a j − (K 1
1
∗p
p
+ A js Ksrmn lr )aip − aij |s Ksrmn lr − Tmn aij| p , 1
1
(33)
1
and finally, we get: p
∗p
p
aij| mn − aij| nm = Ri pmn a j − Rp jmn a j − aij |s Ksrmn lr − Tmn aij| p , 1
1
1
1
1
1
(34)
where we denoted the third tensor of Cartan, our the first ”new” curvature tensor (see Rund): Ri pmn = Ki pmn + Cips Ksrmn x˙ r = 1
1
=
Γ∗ipm,n
1
−
Γ∗ipn,m
( ) (35) q q ∗i ∗s ∗i ∗i s ∗i s i s s i s + Γ∗s pm Γsn − Γpn Γsm − Γpm,s˙ G,n˙ + Γpn,s˙ G,m˙ + Cps G,nm˙ − G,nm ˙ + Gqm G,n˙ − Gqn G,m˙ ,
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and also it is valid: Ri pmn x˙ p = Ki pmn x˙ p . 1
(36)
1
2. For the next difference, we got: p
∗p
+ aij,s˙ Ksrmn x˙ r + e Tmn aij| p =
p aip 2 jmn
aij| mn − aij| nm = Ki pmn a j − K 2
2
2
= (K pmn + i
2
1
p Aips Ksrmn lr )a j 1
−
2
p (K jmn 2
+
p A js Ksrmn lr )aip 1
∗p
Tmn aij| p = − aij |s Ksrmn lr + e 1
(37)
2
∗p p p Tmn aij| p , = Ri pmn a j − Rp jmn a j − aij |s Ksrmn lr + e 2
2
1
2
where ∗p ∗i ∗p ∗i ∗p ∗p Γmp − e Γmj,s˙ Gs,n˙ + e Γnj,s˙ Gs,m˙ , Γnp − e Γnje Ki jmn = e Γ∗im j,n − e Γ∗inj,m + e Γm je
(38)
2
and the second curvature tensor is given with: Ri pmn =Ki pmn + Cips Ksrmn x˙ r = 2
2
1
( ) ∗p ∗i ∗p ∗i ∗p ∗p q q i s =e Γ∗imj,n − e Γ∗inj,m + e Γmje Γnp − e Γnje Γmp − e Γmj,s˙ Gs,n˙ + e Γnj,s˙ Gs,m˙ + Cips Gs,nm˙ − Gs,nm ˙ + Gqm G,n˙ − Gqn G,m˙ .
(39)
It is easy to prove that: Ri pmn x˙ p = Ki pmn x˙ p . 2
(40)
2
Theorem 2.2. In the generalized Finsler space GFN , for h-differentiation, the second Ricci type identity is expressed: ( ) ( ) u v ∑ p ... ∑ p tβ ... rα ... r s e∗p ... a... − a = K a − K a... − a... (41) pmn ... ..., s˙ K rmn x˙ + Tmn a...| p , ...| mn ...| nm tβ mn 2 2 1 r p α 2 2 2 α=1
β=1
where K given by (38). 2
3. For the difference, we can get: p p ∗p Tmn aij| p = aij| m| n − aij| n| m = Aipmn a j − A jmn aip + 2aij + 2aij6mn> − ai,s˙ Ksrmn x˙ r + e 1
2
2
1
1 2
=
(Aipmn 1
+
+
2aij
=
p Bipmn a j 1
∨
∨
p Aips Ksrmn lr )a j 1
+
2aij6mn>
−
p p B jmn a j 2
∨
−
−
−
ai,s˙ Ksrmn x˙ r 1
aij |s Ksrmn lr 1
1
∨
p (A jmn 2
+
p A js Ksrmn lr )aip 1
1
− aij |s Ksrmn lr + 1
(42)
∗p +e Tmn aij| p = 1
+
2aij ∨
∗p + 2aij6mn> − ai,s˙ Ksrmn x˙ r + e Tmn aij| p , ∨
1
1
where ∗s e∗i ∗i s ∗i s e∗i Aipmn = Γ∗ipm,n − Γ∗ipn,m + Γ∗s pm Γns − Γpn Γms − Γpm,s˙ G,n˙ + Γpn,s˙ G,m˙ ,
(43)
∗i ∗i s ∗i s e∗s ∗i Aipmn = Γ∗ipm,n − Γ∗ipn,m + e Γ∗s mp Γsn − Γnp Γsm − Γpm,s˙ G,n˙ + Γpn,s˙ G,m˙ ,
(44)
Bipmn = Aipmn + Cips Ksrmn x˙ r ,
(45)
1
2
1
1
1
Bipmn = Aipmn + Cips Ksrmn x˙ r , 2
∗p
2
aij = M∗ipm (a j,n + a j,s˙ ξs,n ) − M jm (aip,n + aip,s˙ ξs,n ) , p
p
1
(46)
M. Cvetkovi´c, M. Zlatanovi´c / Filomat 28:1 (2014), 107–117 p aij6mn> = (e Γ∗imp Γ∗sjn − Γ∗ipme Γ∗s nj )as .
113 (47)
It is easy to prove that: Bipmn x˙ p = Aipmn x˙ p , 1
Bipmn x˙ p = Aipmn x˙ p . 2
1
(48)
2
Theorem 2.3. In GFN , for h-differentiation, the 3rd Ricci type identity is expressed by: ( ) ( ) u v ∑ p ... ∑ p tβ ... rα ... ... s r a...| m| n − a...| n| m = A pmn a... − A tβ mn a − a... ..., s˙ K rmn x˙ + 2 1 1 r p ... α 1 2 1 2 α=1
+
β=1
2a... ... ∨
+
2a... ...6mn> ∨
(49)
∗p +e Tmn a... ...| p , 1
where A and A are given by equations (43, 44) and 2
1
a... ... =
u ∑
∗rα Mpm
α=1
...ru art11...t = v 6mn>
+
( ) ( ) u ∑ p ... ∗p tβ s s (a...,n + a... ξ ) − M (a... + a... ...,s˙ ,n ...,s˙ ξ,n ) , tβ m p ...,n rα
( )( ) ( )( ) u−1 ∑ u u v ∑ s ... ∑ ∑ ∗rα ∗s p tβ ... ∗rα ∗rβ p Γ[pm Γns] a... − Γ[pm Γntβ ] a + rα rβ rα s ... ` ` ` ` β=2 α=1 (α − Tmn a...| p , ∨
where A and A are given by (53, 54). 3
4
∨
2
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114
5. We considered the difference: ∗p
∗p
Γmn aij| p , aij| mn − aij| nm = Aipmn a j − A jmn aip + 2aij + 2aij0mn1 − ai,s˙ Ksrmn x˙ r − Γmn aij| p + e p
2
1
p
5
6
1
(56)
2
1
where ∗i ∗i s e∗s e∗i e∗i s Aipmn = Γ∗ipm,n − e Γ∗inp,m + Γ∗s pm Γsn − Γnp Γms − Γpm,s˙ G,n˙ + Γnp,s˙ G,m˙ ,
(57)
∗s ∗i ∗i s e∗i e∗i s Aipmn = Γ∗ipm,n − e Γ∗inp,m + e Γ∗s mp Γns − Γpn Γsm − Γpm,s˙ G,n˙ + Γnp,s˙ G,m˙ .
(58)
5
6
Theorem 2.5. In GFN , for h-differentiation, there is the 5th Ricci type identity: ( ) ( ) u v ∑ p ... ∑ p tβ ... rα ... ... s r a...| mn − a...| nm = A pmn A tβ mn a... − a − a... ..., s˙ K rmn x˙ + 5 6 1 r p ... α 2 1 α=1
+
β=1
2a... ...
+
2a... ...0mn1
−
∗p Γmn a... ...| p 1
(59)
∗p +e Γmn a... ...| p , 2
where A and A are given by (57, 58). 5
6
...ru art11...t v 0mn1
=
u−1 ∑ u ∑ α=1 β=2 (α
