New commutation formulas for δ-differentiation in a generalized

New commutation formulas for δ-differentiation in a generalized Finsler space Svetislav M. Minˇci´c and Milan Lj. Zlatanovi´c

Abstract. In this work we consider a generalized Finsler space GFN as an N -dimensional differentiable manifold on which a non-symmetric basic tensor gij (x, x) ˙ is defined by virtue (1.3). Some basic properties of GFN are given (§1). In the work [4] we defined two kinds of covariant derivative of a tensor in the Rund’s sense and obtained 10 Ricci type idetities in a GFN . Non-symmetric basic tensor gij (x, x) ˙ gives possibility to define two more kinds of covariant derivative in the Rund’s sense too. In this work we consider new 10 Ricci type identities, and we get 3 curvature tensors and 15 curvature pseudotensors, defined in [4] and also a new curvature e. tensor K 4

M.S.C. 2000: 53A45, 53B05, 53B40. Key words: Generalized Finsler spaces; Ricci type identities; covariant derivative of the third kind; covariant derivative of the forth kind, non-symmetric connection; curvature tensor; curvature pseudotensor.

1

Introduction

The Finsler space and it’s generalizations were investigated by many authors, for example H. D., Pande and K. K., Gupta [7, 8], M. K. Gupta [3], C., Nitescu [6], A. C., Shamihoke [10]-[13], U. P., Singh [14, 15], M., Verma [16] and many others. The generalized Finsler space (GFN ) is a differentiable manifold with non-symmetric basic tensor gij (x1 , . . . , xN , x˙ 1 , . . . , x˙ N ) ≡ gij (x, x), ˙ where (1.1)

gij (x, x) ˙ 6= gji (x, x), ˙

(g = det(gij ) 6= 0, x˙ = dx/dt).

Based on (1.1), one can defined the symmetric respectively anti-symmetric part of gij (1.2)

g ij =

1 (gij + gji ), 2

g ij = ∨

1 (gij − gji ), 2

where, following [10]-[13], is (1.3)

a) g ij (x, x) ˙ =

˙ 1 ∂ 2 F 2 (x, x) , 2 ∂ x˙ i ∂ x˙ j

∂g ij b)

∨

∂ x˙ k

Differential Geometry - Dynamical Systems, Vol.12, 2010, pp. c Balkan Society of Geometers, Geometry Balkan Press 2010. °

= 0,

145-157.

146

Svetislav M. Minˇci´c and Milan Lj. Zlatanovi´c

where F (x, x) ˙ is a metric function in GFN , having the properties known from the theory of usual Finsler space (FN ) (see e.g. [1, 9]). The lowering and the raising of indices one defines by the tensors g ij and hij respectively, where hij is defined as follows g ij hjk = δik ,

(1.4)

(g = det(gij ) 6= 0).

We can define the generalized Cristoffel symbols of the 1st and the 2nd kind: 1 (gji,k − gjk,i + gik,j ) 6= γi.kj , 2 1 i = hip γp.jk = hip (gjp,k − gjk,p + gpk,j ) 6= γkj , 2

(1.5)

γi.jk =

(1.6)

i γjk

where, e.g., gji,k = ∂gji /∂xk . Then we have p γjk g ip = γs.jk hps g ip = γs.jk δis = γi.jk .

(1.7)

Theorem 1.1. In GFN the following relations are valid (1.8)

γi.jk + γj.ik = gij,k , γi.jk + γk.ji = gik,j , 1 1 1 γi.jk x˙ i x˙ j = gij,k x˙ i x˙ j = gij,k x˙ i x˙ j = Fx˙2i x˙ j xk x˙ i x˙ j , 2 2 4 1 1 1 i k i k i k γi.jk x˙ x˙ = gik,j x˙ x˙ = gik,j x˙ x˙ = Fx˙2i xj x˙ k x˙ i x˙ k , 2 2 4 1 j k j k γi.jk x˙ x˙ = (gij,k − gjk,i )x˙ x˙ , 2

(1.9) (1.10) (1.11)

where, for example, Fx˙2i x˙ j xk =

∂3F 2 . ∂ x˙ i ∂ x˙ j ∂xk

Proof. The equations (1.8) follow from (1.5, 1.2) and the equations (1.9-1.11) from (1.5, 1.2, 1.3). ¤ Introducing a tensor Cijk like as at FN , we have def

(1.12)

Cijk (x, x) ˙ =

1 g k 2 ij,x˙

=

(1.3b)

1 1 g k = Fx˙2i x˙ j x˙ k , 2 ij,x˙ 4

where ” = ” signifies ”equal based on (1.3b)”. We see that Cijk is symmetric in (1.3b)

relation to each pair of indices. Also, we have (1.13)

def

i Cjk = hip Cpjk = hip Cjpk = hip Cjkp . (1.12)

With help of coefficients (1.14)

p s i i i i Pjk = γjk − Cjp γsk x˙ 6= Pkj

New commutation formulas for δ-differentiation

147

one obtains coefficients of non-symmetric affine connections in the Rund’s sense [9, 12]: (1.15)

p p ∗i i ∗i Pjk = γjk − hil (Cjlp Pks + Cklp Pjs − Cjkp Plsp )x˙ s 6= Pkj ,

(1.16)

p p p ∗ ∗r ∗ Pi.jk = Pjk gir = γi.jk − (Cijp Pks + Cikp Pjs − Cjkp Pis )x˙ s 6= Pi.kj .

(1.6)

In the work [4], we defined two kinds of covariant derivative of a tensor in the GFN . ...ru For example, for a tensor art11...t (x, ξ) we have covariant derivative of the 1st and 2nd v kind (1.17) n v X X r1 ...ru ... ... p ∗rα r1 ...,rα−1 prα+1 ...ru at1 ...tv | m = a...,m + a...,p˙ ξ,m + Ppm at1 ...tv − Pt∗p ar1 ...ru . β m r1 ...rβ−1 prβ+1 ...rv 1 2

α=1

mp

β=1

mtβ

In a space GFN it is possible to define two new kinds of covariant derivative: covariant derivative of the 3rd kind and covariant derivative of the 4th kind. Generally is: (1.18) n v X X ∗p ...ru ... ... p ∗rα r1 ...,rα−1 prα+1 ...ru art11...t = a + a ξ + P a − Pmt ar1 ...ru , t ...t ...,m ..., p ˙ ,m pm | m 1 v β r1 ...rβ−1 prβ+1 ...rv v 3 4

α=1

mp

β=1

tβ m

∂a...

... where ξ(x) is an arbitrary tangent vector in the tangent space TN (x), and a... ...,p˙ = ∂ x˙ p . By the procedure that is similar to that one in a Finsler space, it can be proved that covariant derivative (1.17, 1.18) of a tensor also is a tensor.

Theorem 1.2. For the tensor g ij (x, x) ˙ based on both kinds of derivative (1.18) is valid (1.19)

g ij

| m (x, ξ)

p ∗p s = 2Cijp (ξ,m + Pms x˙ ),

θ = 3, 4,

θ

(1.20)

g ij

˙ | m (x, x) θ

= 2Cijp x˙ p| m = 2Cijp x˙ p| m , 4

θ = 3, 4.

2

Proof. Starting from (1.18), we get g ij (1.21)

| m (x, ξ) 3

∗p ∗p p = g ij ,m + gij,x˙ p ξ,m − Pmi g pj − Pmj g ip p ∗ ∗ = g ij ,m + 2Cijp ξ,m − (Pj.mi + Pi.mj ).

(1.12)

Based on (1.16) and using (1.8) one obtains (1.22)

∗ ∗ p Pj.mi + Pi.mj = gij,m − 2Cijp Pms x˙ s .

By substituting from (1.22) into (1.21) we get g ij

| m (x, ξ) 3

p p = 2Cijp (ξ,m + Pms x˙ s ).

The same result one obtains for gij | m . As 4

(1.23)

p Pms x˙ s

∗p s = Pms x˙ .

148

Svetislav M. Minˇci´c and Milan Lj. Zlatanovi´c

we obtain (1.19). For ξ = x˙ one obtains g ij

∗p s = 2Cijp (x˙ p,m + Pms x˙ ),

˙ | m (x, x) 3

and the same one obtains for θ = 4. Because x˙ p do not depends from ξ, the previous equation becomes (1.20). ¤ Theorem 1.3. For the Kronecker symbol is in force δji | m = 0,

θ = 1, . . . , 4.

θ

The proof is obtained by using the definition of δ-symbols and covariant derivatives (1.17, 1.18). Theorem 1.4. For hij is force ip jq hij | m = −h h g pq

(1.24)

θ

|m,

θ = 1, . . . , 4,

θ

Proof. From hip gpq = δqi one obtains ip hip | m gpq + h gpq | m = 0. θ

θ

Composing with hjq , it results that ip jq j hip | m δp + h h gpq | m = 0,

θ = 3, 4 ⇒ (1.24). ¤

θ

θ

From (1.17, 1.18) is evidently that (1.25)

ai| m = ai| m , 3

2

1

ak| m = ak| m , 3

2

ai| m = ai| m , 4

ak| m = ak| m , 4

2

1

New Ricci type identities in GFN

If we examine the differences ...ru ...ru atr11...t − art11...t v | m| n v |m|n

(2.1)

π

ρ

σ

(π, ρ, σ, τ = 3, 4)

τ

e, K e, we get ten new Ricci type identities. In these identities the same magnitude K 1

2

e; A e, . . . , A e appear, but in different arrangement than they appear in the work [4]. K 3

1

15

e. Only in the last case appears a new curvature tensor K 4

ahij kl

Let us consider a tensor and its covariant derivative of the 3rd and the 4th kind, also their combinations, by virtue of (2.1).

New commutation formulas for δ-differentiation

149

1. Based on (1.18) we obtain hij hij hij d ∗h sij ∗i hsj ahij kl| mn = (akl| m ) | n = (akl| m ),n + akl| m,d˙ξ,n + Psn akl| m + Psn akl| m 3

3

3

3

+

3

3

∗j his Psn akl| m

−

3

∗s hij Pnk asl| m

−

3

∗s hij Pnl aks| m

3

−

∗s hij Pnm akl| s

3

i.e.

3

hij pij hpj hip ∗h ∗h pij ∗i ∗i hpj ∗j ahij kl| mn = akl,mn + Ppm,n akl + Ppm akl,n + Ppm,n akl + Ppm akl,n + Ppm,n akl 3

∗p ∗p hij ∗p hij ∗p hij hij ∗j hip s akl,n − Pmk,n ahij + Ppm pl − Pmk apl,n − Pml,n akp − Pml akp,n + akl,sn ˙ ξ,m hij pij d hpj d s d ∗h ∗h pij d ∗i + ahij ξ s ξ t + ahij kl ξ,nm + akl,md˙ξ,n + Ppm,d˙akl ξ,n + Ppm akl,d˙ξ,n + Ppm,d˙akl ξ,n kl,s˙ t˙ ,m ,n ∗j ∗p ∗p hij d ∗i hpj d ∗j hip d akl,d˙ξ,n + Ppm, ahip ξ d + Ppm akl,d˙ξ,n − Pkm, ahij ξ d − Pkm apl,d˙ξ,n + Ppm d˙ kl ,n d˙ pl ,n ∗p ∗p hij s ∗h sij − Plm, ahij ξ d − Plm akp,d˙ + ahij ξ s ξ d + ahij ξ s ξ t ξ d + ahij kl ξ,md˙ + Psn akl,m kl,s˙ t˙ ,m ,d˙ ,n kl,s˙ d˙ ,m ,n d˙ kp ,n ∗h sij d ∗h ∗s pij ∗h ∗i spj ∗h ∗j sip ∗h ∗p sij akl,d˙ξ,m + Psn Ppm akl + Psn Ppm akl + Psn Ppm akl − Psn Pmk apl + Psn ∗h ∗p sij ∗i hsj ∗i hsj d ∗i ∗h psj ∗i ∗s hpj ∗i ∗j hsp − Psn Pml akp + Psn akl,m + Psn akl,d˙ξ,m + Psn Ppm akl + Psn Ppm akl + Psn Ppm akl ∗i ∗p hsj ∗i ∗p hsj ∗j his ∗j his d ∗j ∗h pis ∗j ∗i hps − Psn Pmk apl − Psn Pml akp + Psn akl,m + Psn akl,d˙ξ,m + Psn Ppm akl + Psn Ppm akl ∗j ∗s hip ∗j ∗p his ∗j ∗p his ∗s hij ∗s hij d ∗s ∗h pij + Psn Ppm akl − Psn Pmk apl − Psn Pml akp − Pnk asl,m − Pnk asl,d˙ξ,m − Pnk Ppm asl ∗s ∗i hpj ∗s ∗i hip ∗s ∗p hij ∗s ∗p hij ∗s hij ∗s hij d − Pnk Ppm asl − Pnk Ppm asl + Pnk Pms apl + Pnk Pml asp − Pnl aks,m − Pnl aks,d˙ξ,m ∗s ∗h pij ∗s ∗i hpj ∗s ∗i hip ∗s ∗p hij ∗s ∗p hij ∗s hij − Pnl Ppm aks − Pnl Ppm aks − Pnl Ppm aks + Pnl Pmk aps + Pnl Pms akp − Pnm akl| s , 3

and from here hij hij hpj hip eh ei ej ahij kl| mn − akl| nm = K pmn akl + K pmn akl + K pmn akl 3

1

3

−

1

ep K ahij 2 kmn pl

−

1

e p ahij K 2 lmn kp

∗p hij akl| p , + Tmn 3

where (2.2)

∗p ∗p ∗i ∗p ∗i ∗p s ∗i ∗i s e ijmn (x, ξ) = Pjm,n K − Pjn,m + Pjm Ppn − Pjn Ppm + Pjm, s˙ ξ,n − Pjn,s˙ ξ,m ,

(2.3)

∗p s ∗p ∗i ∗p ∗i ∗p ∗i ∗i s e ijmn (x, ξ) = Pmj,n + Pmj − Pnj,m Pnp − Pnj Pmp + Pmj, K s˙ ξ,n − Pnj,s˙ ξ,m ,

1

2

are Rund’s type curvature tensors of the 1st and the 2nd kind respectively. The torsion tensor and the double simmetric part of the connection are respectively ∗i ∗i ∗i ∗i a) Tjk = P[jk] = Pjk − Pkj ,

(2.4)

∗i ∗i ∗i b) P(jk) = Pjk + Pkj

2. Using (1.18) we get hij hij hij d ∗h sij ∗i hsj ahij kl| mn = (akl| m ) | n = (akl| m ),n + akl| m,d˙ξ,n + Pns akl| m + Pns akl| m 4

4

4

4

+

4

4

∗j his Pns akl| m 4

−

∗s hij Pkn asl| m 4

−

∗s hij Pln aks| m 4

4

−

∗s hij Pmn akl| s 4

150

Svetislav M. Minˇci´c and Milan Lj. Zlatanovi´c

hij pij hpj hip ∗h ∗h pij ∗i ∗i hpj ∗j ahij kl| mn = akl,mn + Pmp,n akl + Pmp akl,n + Pmp,n akl + Pmp akl,n + Pmp,n akl 4

∗p ∗p hij ∗p hij ∗p hij hij ∗j hip s + Pmp akl,n − Pkm,n ahij pl − Pkm apl,n − Plm,n akp − Plm akp,n + akl,sn ˙ ξ,m hij pij d hpj d s d ∗h ∗h pij d ∗i + ahij ξ s ξ t + ahij kl ξ,mn + akl,md˙ξ,n + Pmp,d˙akl ξ,n + Pmp akl,d˙ξ,n + Pmp,d˙akl ξ,n kl,s˙ t˙ ,m ,n ∗p hij d ∗j ∗p ∗j hip d ∗i hpj d apl,d˙ξ,n akl,d˙ξ,n − Pkm, ahij ξ d − Pkm ahip ξ d + Pmp + Pmp akl,d˙ξ,n + Pmp, d˙ pl ,n d˙ kl ,n ∗p hij ∗p s ∗h sij akp,d˙ + ahij ξ s ξ d + ahij − Plm, ahij ξ d − Plm ξ s ξ t ξ d + ahij kl ξ,md˙ + Pns akl,m kl,s˙ t˙ ,m ,d˙ ,n kl,s˙ d˙ ,m ,n d˙ kp ,n ∗h sij d ∗h ∗s pij ∗h ∗i spj ∗h ∗j sip ∗h ∗p sij + Pns akl,d˙ξ,m + Pns Pmp akl + Pns Pmp akl + Pns Pmp akl − Pns Pkm apl ∗h ∗p sij ∗i hsj ∗i hsj d ∗i ∗h psj ∗i ∗s hpj ∗i ∗j hsp Plm akp + Pns − Pns akl,m + Pns akl,d˙ξ,m + Pns Pmp akl + Pns Pmp akl + Pns Pmp akl ∗i ∗p hsj ∗i ∗p hsj ∗j his ∗j his d ∗j ∗h pis ∗j ∗i hps − Pns Pkm apl − Pns Plm akp + Pns akl,m + Pns akl,d˙ξ,m + Pns Pmp akl + Pns Pmp akl ∗j ∗s hip ∗j ∗p his ∗j ∗p his ∗s hij ∗s hij d ∗s ∗h pij + Pns Pmp akl − Pns Pkm apl − Pns Plm akp − Pkn asl,m − Pkn asl,d˙ξ,m − Pkn Pmp asl ∗s ∗i hpj ∗s ∗i hip ∗s ∗p hij ∗s ∗p hij ∗s hij ∗s hij d Pmp asl − Pkn Pmp asl + Pkn Pms apl + Pkn Plm asp − Pln aks,m − Pnl aks,d˙ξ,m − Pkn ∗s ∗h pij ∗s ∗i hpj ∗s ∗i hip ∗s ∗p hij ∗s ∗p hij ∗s hij − Pln Pmp aks − Pln Pmp aks − Pln Pmp aks + Pln Pkm aps + Pln Pms akp − Pmn akl| s , 4

from where one obtains hij hij hpj hip eh ei ej ahij kl| mn − akl| nm = K pmn akl + K pmn akl + K pmn akl 4

2

4

−

2

ep K ahij 1 kmn pl

−

2

e p ahij K 1 lmn kp

∗p hij akl| p , − Tmn 4

Generally, we can state by total induction method (as like in [4]): Theorem 2.1. In a generalized Finsler space GFN the 1st and 2nd new Ricci type identity are ...ru ...ru art11...t − art11...t = v | mn v | nm

(2.5)

θ

θ

u X

µ α e rpmn K

1

α=1

β=1

2

θ

− (−1)

¶ µ ¶ v X p tβ ... p e a... − K a t mn 2 β rα ... p ... 1

∗p r1 ...ru Tmn at1 ...tv | p

, θ = 3, 4,

θ

e , θ = 1, 2 are given in (2.2, 2.3), and where K θ

µ (2.6)

¶ p r ...r pr ...r a... = at11...tvα−1 α+1 u , rα ...

µ ¶ tβ ... ...ru a = art11...t . β−1 prβ+1 ...tv p ...

3. Combining the 3rd and 4th kind of derivation on the base of (1.18) we have: hij hij hij d ∗h sij ∗i hsj ahij kl| m| n = (akl| m ) | n = (akl| m ),n + akl| m,d˙ξ,n + Pns akl| m + Pns akl| m 3

4

3

4

3

+

3

3

∗j his Pns akl| m 3

−

∗s hij Pkn asl| m 3

−

∗s hij Pln aks| m 3

3

−

∗s hij Pmn akl| s 3

i.e.

New commutation formulas for δ-differentiation

151

hij pij hpj hip ∗h ∗h pij ∗i ∗i hpj ∗j ahij kl| m| n = akl,mn + Ppm,n akl + Ppm akl,n + Ppm,n akl + Ppm akl,n + Ppm,n akl 3

4

∗p ∗p hij ∗p hij ∗p hij hij ∗j hip s + Ppm akl,n − Pmk,n ahij pl − Pmk apl,n − Pml,n akp − Pml akp,n + akl,sn ˙ ξ,m hij pij d hpj d s d ∗h ∗h pij d ∗i + ahij ξ s ξ t + ahij kl ξ,mn + akl,md˙ξ,n + Ppm,d˙akl ξ,n + Ppm akl,d˙ξ,n + Ppm,d˙akl ξ,n kl,s˙ t˙ ,m ,n ∗p hij d ∗j ∗p ∗j hip d ∗i hpj d apl,d˙ξ,n akl,d˙ξ,n − Pkm, ahij ξ d − Pkm ahip ξ d + Ppm + Ppm akl,d˙ξ,n + Ppm, d˙ pl ,n d˙ kl ,n ∗p hij ∗p s ∗h sij akp,d˙ + ahij ξ s ξ d + ahij − Plm, ahij ξ d − Plm ξ s ξ t ξ d + ahij kl ξ,md˙ + Pns akl,m kl,s˙ t˙ ,m ,d˙ ,n kl,s˙ d˙ ,m ,n d˙ kp ,n ∗h sij d ∗h ∗s pij ∗h ∗i spj ∗h ∗j sip ∗h ∗p sij + Pns akl,d˙ξ,m + Pns Ppm akl + Pns Ppm akl + Pns Ppm akl − Pns Pmk apl ∗h ∗p sij ∗i hsj ∗i hsj d ∗i ∗h psj ∗i ∗s hpj ∗i ∗j hsp Pml akp + Pns − Pns akl,m + Pns akl,d˙ξ,m + Pns Ppm akl + Pns Ppm akl + Pns Ppm akl ∗i ∗p hsj ∗i ∗p hsj ∗j his ∗j his d ∗j ∗h pis ∗j ∗i hps − Pns Pmk apl − Pns Pml akp + Pns akl,m + Pns akl,d˙ξ,m + Pns Ppm akl + Pns Ppm akl ∗j ∗s hip ∗j ∗p his ∗j ∗p his ∗s hij ∗s hij d ∗s ∗h pij + Pns Ppm akl − Pns Pmk apl − Pns Pml akp − Pkn asl,m − Pkn asl,d˙ξ,m − Pkn Ppm asl ∗s ∗i hpj ∗s ∗i hip ∗s ∗p hij ∗s ∗p hij ∗s hij ∗s hij d Ppm asl − Pkn Ppm asl + Pkn Pms apl + Pkn Pml asp − Pln aks,m − Pnl aks,d˙ξ,m − Pkn ∗s ∗h pij ∗s ∗i hpj ∗s ∗i hip ∗s ∗p hij ∗s ∗p hij ∗s hij − Pln Ppm aks − Pln Ppm aks − Pln Ppm aks + Pln Pmk aps + Pln Pms akp − Pmn akl| s , 3

So, one gets hij pij hpj hip hij eh ei ej ep ahij kl| m| n − akl| n| m = A pmn akl + A pmn akl + A pmn akl − A knm apl 3

(2.7)

4

1

3 4

−

1

ep ahij A 4 lnm kp

+

1

ahij

+

4

ahij 6[kl]>

−

∗p hij Tmn akl| s . 3

The denotations, introduced in (2.7) [4] are (2.8)

∗p ∗p ∗i ∗p ∗i ∗p s s ∗i ∗i eijmn = Pjm,n + Pjm − Pjn,m Pnp − Pjn Pmp + Pjm, A s˙ ξ,n − Pjn,s˙ ξ,m ,

(2.9)

∗p s ∗p ∗i ∗p ∗i ∗p s ∗i ∗i eijmn = Pmj,n A − Pnj,m + Pjm Pnp − Pjn Pmp + Pmj, s˙ ξ,n − Pnj,s˙ ξ,m ,

1

4

pij pij s hpj hpj s ∗h ∗i ahij kl = Tpm (akl,n + akl,s˙ ξ,n ) + Tpm (akl,n + akl,s˙ ξ,n ) hip s ∗p hij hij s ∗j + Tpm (ahip kl,n + akl,s˙ ξ,n ) − Tmk (apl,n + apl,s˙ ξ,n ) ∗p hij s − Tml (akp,n + ahij kp,s˙ ξ,n ), psj ∗h pis ∗h ∗j hps ∗i ∗j pij ∗h ∗i ∗s ahij kl6mn> = akl P[pm Pns] + akl P[pm Pns] + akl P[pm Pns] − asl P[pm Pkn] `

`

−

∗h ∗s apij ks P[pm Pln]

−

−

`

`

∗j ∗s ahip ks P[pm Pln]

`

+

`

∗h ∗i P[pm Pns ] `

`

`

−

`

∗i ∗s ahpj ks P[pm Pln]

`

−

`

`

`

∗j ∗s ahip sl P[pm Pkn] `

`

∗p ∗s ahij ps P[mk Pln] ,

`

`

`

`

∗i ∗s ahpj sl P[pm Pkn]

=

`

`

∗h ∗i Ppm Pns

∗h ∗i − Pmp Psn .

4. Differentiating in inverse order then in preceding case, we obtain hij hij hij d ∗h sij ∗i hsj ahij kl| m| n = (akl| m ) | n = (akl| m ),n + akl| m,d˙ξ,n + Psn akl| m + Psn akl| m 4

3

4

3

4

+

4

4

∗j his Psn akl| m 4

−

∗s hij Pnk asl| m 4

−

∗s hij Pnl aks| m 4

4

−

∗s hij Pnm akl| s 4

i.e.

152

Svetislav M. Minˇci´c and Milan Lj. Zlatanovi´c

hij pij hpj hip ∗h ∗h pij ∗i ∗i hpj ∗j ahij kl| m| n = akl,mn + Pmp,n akl + Pmp akl,n + Pmp,n akl + Pmp akl,n + Pmp,n akl 4

3

∗p ∗p hij ∗p hij ∗p hij hij ∗j hip s + Pmp akl,n − Pkm,n ahij pl − Pkm apl,n − Plm,n akp − Plm akp,n + akl,sn ˙ ξ,m hij pij d hpj d s d ∗h ∗h pij d ∗i + ahij ξ s ξ t + ahij kl ξ,mn + akl,md˙ξ,n + Pmp,d˙akl ξ,n + Pmp akl,d˙ξ,n + Pmp,d˙akl ξ,n kl,s˙ t˙ ,m ,n ∗p hij d ∗j ∗p ∗j hip d ∗i hpj d apl,d˙ξ,n akl,d˙ξ,n − Pmk, ahij ξ d − Pmk ahip ξ d + Pmp + Pmp akl,d˙ξ,n + Pmp, d˙ pl ,n d˙ kl ,n ∗p hij ∗p s ∗h sij akp,d˙ + ahij ξ s ξ d + ahij − Pml, ahij ξ d − Pml ξ s ξ t ξ d + ahij kl ξ,md˙ + Psn akl,m kl,s˙ t˙ ,m ,d˙ ,n kl,s˙ d˙ ,m ,n d˙ kp ,n ∗h sij d ∗h ∗s pij ∗h ∗i spj ∗h ∗j sip ∗h ∗p sij + Psn akl,d˙ξ,m + Psn Pmp akl + Psn Pmp akl + Psn Pmp akl − Psn Pkm apl ∗h ∗p sij ∗i hsj ∗i hsj d ∗i ∗h psj ∗i ∗s hpj ∗i ∗j hsp Plm akp + Psn − Psn akl,m + Psn akl,d˙ξ,m + Psn Pmp akl + Psn Pmp akl + Psn Pmp akl ∗i ∗p hsj ∗i ∗p hsj ∗j his ∗j his d ∗j ∗h pis ∗j ∗i hps − Psn Pkm apl − Psn Plm akp + Psn akl,m + Psn akl,d˙ξ,m + Psn Pmp akl + Psn Pmp akl ∗j ∗s hip ∗j ∗p his ∗j ∗p his ∗s hij ∗s hij d ∗s ∗h pij + Psn Pmp akl − Psn Pkm apl − Psn Plm akp − Pnk asl,m − Pnk asl,d˙ξ,m − Pnk Pmp asl ∗s ∗i hpj ∗s ∗i hip ∗s ∗p hij ∗s ∗p hij ∗s hij ∗s hij d Pmp asl − Pnk Pmp asl + Pnk Psm apl + Pnk Plm asp − Pnl aks,m − Pnl aks,d˙ξ,m − Pnk ∗s ∗h pij ∗s ∗i hpj ∗s ∗i hip ∗s ∗p hij ∗s ∗p hij ∗s hij − Pnl Pmp aks − Pnl Pmp aks − Pnl Pmp aks + Pnl Pkm aps + Pnl Psm akp − Pnm akl| s 4

from where hij pij hpj hip hij eh ei ej ep ahij kl| m| n − akl| n| m = A pmn akl + A pmn akl + A pmn akl − A knm apl 4

(2.10)

3

3

4 3

−

3

ep ahij A 2 lnm kp

−

3

ahij

−

ahij 6[kl]>

2

+

∗p hij Tmn akl| s , 3

where we have denoted [4] (2.11)

∗p s ∗p ∗i ∗p ∗i ∗p s ∗i ∗i ei A jmn = Pmj,n − Pnj,m + Pmj Ppn − Pnj Ppm + Pmj,s˙ ξ,n − Pnj,s˙ ξ,m ,

(2.12)

∗p ∗p ∗i ∗p ∗i ∗p s s ∗i ∗i eijmn = Pjm,n A − Pjn,m + Pmj Ppn − Pnj Ppm + Pjm, s˙ ξ,n − Pjn,s˙ ξ,m ,

3 2

The next two theorems are valid. Theorem 2.2. In GFN the 3rd new Ricci type identity is in force: ...ru ...ru art11...t − atr11...t = v | m| n v | n| m

(2.13)

3

4

3 4

u X α=1

+

µ α erpmn A

1

¶ µ ¶ v X p tβ ... p e a... − A a t mn 4 β rα ... p ...

...ru atr11...t v

+

β=1 r1 ...ru at1 ...tv 6[mn]>

∗p r1 ...ru − Tmn at1 ...tv | p ,

e, A e are given at (2.8, 2.9) and where A 1

4

...ru art11...t v

¶ p ... s = (a... ...,n + a...,s˙ ξ,n ) r α α=1 µ ¶ u X tβ ∗p ... s − Tmt (a... ...,n + a...,s˙ ξ,n ) , β p u X

β=1

µ

∗rα Tpm

3

New commutation formulas for δ-differentiation

...ru art11...t = v 6mn>

u−1 X

u X

µ

u X v X

`

+ µ

p rα

v X

`

µ `

`

∗p P[mt P ∗s α tβ n] `

α=1 β=1 (α