Filomat 29:3 (2015), 573–584 DOI 10.2298/FIL1503573S
Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat
Some Properties of ET-Projective Tensors Obtained from Weyl Projective Tensor Mi´ca S. Stankovi´ca , Milan Lj. Zlatanovi´ca , Nenad O. Vesi´ca a Faculty
of Science and Mathematics, University of Niˇs, Serbia
Abstract. Vanishing of linearly independent curvature tensors of a non-symmetric affine connection space as functions of vanished curvature tensor of the associated space of this one are analyzed in the first part of this paper. Projective curvature tensors of a non-symmetric affine connection space are expressed as functions of the affine connection coefficients and Weyl projective tensor of the corresponding associated affine connection space in the second part of this paper.
1. Introduction Many authors have been interested in non-symmetric affine connection spaces theory research. Einstein was the first one who used non-symmetric affine connection spaces in his research area (see [2–4]). In the Unified Field Theory (UFT) instead of Riemannian space, he involves non-symmetric affine connection coefficients Γijk which are independent of metric tensor 1i j . While at Riemannian space connection coefficients are expressed in terms of 1i j , functional connections of these quantities are determined with the following equations in Einstein’s works about UFT: p
p
1 ij ;m ≡ 1i j,m − Γim 1p j − Γm j 1ip = 0. +−
(1)
After Einstein, a lot of authors have developed theory of non-symmetric affine connection spaces. Some of them are M. Prvanovi´c [17], R. S. Mishra [15], S. M. Minˇci´c [7–14], Lj. S. Velimirovi´c [1, 13] and many others. Definition 1.1. An N-dimensional manifold M endowed with affine connection coefficients Lijk = Lijk (x), x ∈ M, is an N-dimensional affine connection space. Affine connection coefficients Lijk of this space satisfy the following equation 0
0
j
0
Lij0 k0 = xii x j0 xkk0 Lijk + xii xij0 k0 ,
(2)
2010 Mathematics Subject Classification. Primary 53B05 Keywords. geodesic mapping, invariant, curvature tensor, projective curvature tensor, flatness Received: 4 September 2014; Accepted: 10 November 2014 Communicated by Ljubica Velimirovi´c Research financially supported by project 174012 of Serbian Ministry of Education, Science and Technological Development Email addresses:
[email protected] (Mi´ca S. Stankovi´c),
[email protected] (Milan Lj. Zlatanovi´c),
[email protected] (Nenad O. Vesi´c)
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0
∂xi . ∂xi Based on eventually non-symmetry of affine connection coefficients Lijk by indices j and k, the symmetric 0
where e.g. xii =
part and the non-symmetric one (a half of a torsion tensor Tijk ) of these connection coefficients respectively are defined as
Lijk
1 i 1 i 1 i i i = L + Lk j and L jk = L jk − Lk j = Tijk . 2 jk 2 2 ∨
(3)
It is evident it holds the equality 1 Lijk = Lijk + Lijk = Lijk + Tijk . 2 ∨
(4)
Definition 1.2. An N-dimensional space AN with Tijk = 0 is an affine connection space without torsion and a space GAN where it exist indices i0 , j0 , k0 such that Tij0 k , 0 is an affine connection space with torsion. 0 0
Remark 1.3. An affine connection space AN without torsion is a symmetric affine connection space. An affine connection space GAN with torsion is a non-symmetric affine connection space. 1.1. Affine connection space without torsion An affine connection on an N-dimensional manifold M is a mapping ∇ which maps any pair (X, Y) of vector fields to a vector field F = ∇X Y such that (see [6]) ∇X (Y + Z) = ∇X Y + ∇X Z; ∇X ( f Y) = f ∇X Z + (X f )Y;
(5)
∇ f X+1Y Z = f ∇X Z + 1∇Y Z, for any vector fields X, Y, Z and any differentiable functions f, 1 on M. Remark 1.4. In our case, operator ∇ is partial derivative operator. Let AN be an N-dimensional space, and let ` : [a, b] → AN be a curve on it. A vector field along ` is a function that assigns to each point of ` a tangent vector to the space at that point. That is, a vector field along ` is a smooth function v : [a, b] → RN with the property that v(t) ∈ T`(t) AN for any t ∈ [a, b]. Let ` be a curve on an affine connection space AN . We say that a vector field X is parallel along a curve ` if X satisfies the condition de f
∇t X = ∇λ X = 0,
(6) .
for any t, where λ = λ(t) = `(t). The definition below is the definition of parallel transport. Definition 1.5. [6] Let x0 = `(t0 ) and x1 = `(t1 ) be points on the given curve ` = `(t). A vector X1 from the tangent space Tx1 AN in x1 is a result of the parallel transport along ` from the point x0 to the point x1 if along `, there exists a parallel vector field X(t) for which X(t0 ) = X0 and X(t1 ) = X1 . The following definition is the one of a geodesic line.
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Definition 1.6. [6] A curve ` in space AN is geodesic when its tangent vector field remains in tangent distribution of ` during parallel transport along the curve. Special diffeomorphisms are of interest for research in symmetric affine connection spaces and nonsymmetric ones. Definition 1.7. [6] Let AN and AN be symmetric affine connection spaces. A diffeomorphism f : AN → AN is said to be a geodesic mapping of AN onto AN if any geodesic curve in AN it maps onto a geodesic curve in AN . An invariant geometrical object under a geodesic mapping f : AN → AN is a Weyl projective tensor W ijmn = Rijmn +
1 N+1 R[mn]
+
N δi R N2 −1 [m jn]
+
1 δi R , N2 −1 [m n]j
(7)
where by [·, ·] it is denoted an anti-symmetrization without division. A symmetrization without division is going to be denoted by (·, ·). A magnitude p
p
Rijmn = Lijm,n − Lijn,m + L jm Lipn − L jn Lipm ,
(8) p
is Riemann-Christoffel curvature tensor and Rmn = Rmnp is Ricci tensor. Definition 1.8. [6] An N-dimensional affine connection space AN with vanished Riemann-Christoffel curvature tensor Rijmn is a flat space. Definition 1.9. [6] An N-dimensional affine connection space AN with vanished Ricci tensor Rmn is a Ricci-flat space. Definition 1.10. An N-dimensional affine connection space AN with vanished Weyl projective tensor is a projectively flat space. It is easy to conclude that a flat space M is a projectively flat one. On the contrary, Ricci-flat projective flat space M is a flat space. 1.2. Affine connection space with torsion S. Minˇci´c and M. Stankovi´c tried to generalize Weyl projective tensor in non-symmetric case. They did not succeed in that. For this reason, they have started research into equitorsion mappings. Definition 1.11. [8, 18] A mapping f : GAN → GAN is equitorsion (ET)-mapping if the torsion tensors of the spaces GAN and GAN are equal. Definition 1.12. A space AN is an associated space to a space GAN if symmetric part Lijk of connection coefficients in space GAN are used for connection coefficients in the space AN . Using non-symmetry of connection, S. Minˇci´c (see [10]) involved four types of covariant differentiation A of tensors. Let Xij11ij22...i be a tensor in a non-symmetric affine connection space GAN with affine connection ... jB coefficients Lijk . Covariant derivatives of this tensor are: A A Xij1 ij2 ...i = Xij1 ij2 ...i + ... j | k ... j ,k 1 2
B
1 2 3 4
1 2
B
A X α=1
i ...i
piα+1 ...iA
α−1 α Lipm X j11 j2 ...j B mp pm mp
−
B X α=1
A L jα m Xij11...i ...jα−1 p jα+1 ... jB
p
m jα m jα jα m
(9)
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In the corresponding Ricci type identities, there are four curvature tensors [11]: p
p
p
p
Rijmn = Lijm,n − Lijn,m + L jm Lipn − L jn Lipm , 1
Rijmn = Lim j,n − Lin j,m + Lm j Linp − Ln j Limp , 2
(10)
p p p Rijmn = Lijm,n − Lin j,m + L jm Linp − Ln j Lipm + Lnm Lip j − Lijp , 3
p
p
p
Rijmn = Lijm,n − Lin j,m + L jm Linp − Ln j Lipm + Lmn (Lip j − Lijp ). 4
In these Ricci type identities appear fifteen magnitudes Aijmn which are not tensors: θ
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
Aijmn = Lijm,n − Lijn,m + L jm Linp − L jn Limp , Aijmn = Lijm,n − Lin j,m + L jm Lipn − Ln j Lipm , 1
p
9
p
Aijmn = Lijm,n − Lijn,m + Lm j Lipn − Ln j Lipm , Aijmn = Lijm,n − Lin j,m + L jm Linp − L jn Lipm , 2
p
p
Aijmn = Lim j,n − Lin j,m + Lm j Lipn − Ln j Lipm , 3
p
p
Aijmn = Lim j,n − Lin j,m + L jm Linp − L jn Limp , 4
p
p
Aijmn = Lijm,n − Lin j,m + L jm Lipn − Ln j Limp , 5
p
p
Aijmn = Lijm,n − Lin j,m + Lm j Linp − L jn Lipm , 6
p
p
Aijmn = Lijm,n − Lijn,m + L jm Lipn − L jn Limp , 7
p
p
Aijmn = Lijm,n − Lijn,m + Lm j Lipn − L jn Lipm .
10
Aijmn = Lim j,n − Lijn,m + Lm j Linp − Ln j Limp , 11
Aijmn = Lim j,n − Lijn,m + Lm j Lipn − Ln j Limp , 12
Aijmn = Lim j,n − Lin j,m + Lm j Linp − Ln j Lipm , 13
(11)
Aijmn = Lim j,n − Lin j,m + L jm Linp − Ln j Limp , 14
Aijmn = Lijm,n − Lin j,m + L jm Linp − Ln j Lipm , 15
8
Derived curvature tensors of this space are: ∼
Rijmn = 12 (A + A)ijmn = 21 (A + A)ijmn , 1 ∼
Rijmn 2 ∼ Rijmn 3 ∼ Rijmn 4 ∼ Rijmn 5 ∼ Rijmn 6 ∼ Rijmn 7 ∼ Rijmn 8
=
1 1 2 (A 7
3
+
A)ijmn 13
=
1 2 (A 8
+
A)ijmn 14
=
1 6 (R 3
+A+
= (A − 1
11 A)ijmn 7
=
2 1 2 (A 9
+ A)ijmn ,
=
1 2 (A 10
+ A)ijmn ,
4
11
12 1 6 (R 3
A)ij[mn] 13
=
Aijnm 13
=
−
= (A − A)ijmn − Aijnm = 2
8
14
= (A +
A)ijmn 7
+ Aijnm =
= (A +
A)ijmn 8
+ Aijnm =
3
4
13
14
+ A + A)ij[mn] ,
12 14 i −A jmn − (A + A)ijnm , 7 11 15 i −A jmn − (A + A)ijnm , 8 12 15 i A jmn + (A − A)ijnm , 9 13 15 i A jmn + (A − A)ijnm . 10 14 15
(12)
Curvature tensors (10, 12) are generalizations of a curvature tensor Rijmn (8) of a symmetric affine connection space AN . After non-symmetric affine connection coefficients Lijk symmetrization, all of these curvature tensors become equal to Rijmn (curvature tensor of the associated space). ∼
∼
1
8
S. Minˇci´c proved in the set {Rijmn , . . . , Rijmn , Rijmn , . . . , Rijmn } of four curvature tensors and eight derived 1
4
ones of a non-symmetric affine connection space GAN it exist five linearly independent tensors in [12]. We are going to use the following ones (see [18]) in further research: Kijmn = Rijmn , 1
1
∼
∼
Kijmn = Rijmn , Kijmn = Rijmn , Kijmn = Rijmn , Kijmn = 2
1
3
3
4
3
5
1 ∼i (3R + Ri ). 4 4 jmn 1 jmn
(13)
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Definition 1.13. Space with Kijmn = 0, r ∈ {1, 2, . . . , 5}, is the r-th flat space. A space which is the r-th flat one, for all r = 1, . . . , 5, is flat space.
r
1.3. Invariants of ET-mappings In a non-symmetric affine connection space GAN exist five linearly independent invariants under ETmappings caused from linearly independent curvature tensors and the corresponding torsion ones. These invariants are Eijmn 1
= Kijmn +
1 i i + NN 2 −1 δ[m K jn] + N 2 −1 δ[m K n] j 1 1 1 p q q i i L (N − 1) δ j Tqn + δn T jq (N + 1)2 (N − 1) mp 1 p q q Lnp − (N − 1) δij Tqm − δim T jq 2 (N + 1) (N − 1) 1 p q i 2 i L Nδ T + N − 1 T nm [m qn] (N + 1)2 (N − 1) jp 2 q p p i i i p L −(N − 1)δ T + δ T − δ T ; pq mn m n j jn jm (N + 1)2 1
+ + + +
1 i [mn] N+1 δ j K 1
Eijmn 2
= Kijmn +
1 i [mn] N+1 δ j K 2
Eijmn
= Kijmn +
1 i [mn] N+1 δ j K 3
3
2
+ − +
N δi K N2 −1 [m 2 jn]
+
1 δi K N2 −1 [m 2 n] j
;
1 i i + NN 2 −1 δ[m K jn] + N 2 −1 δ[m K n] j 3 3 1 q p q i i Lmp (N − 1) δ j Tqn − δn T jq + (N2 − 1)Tijn 2 (N + 1) (N − 1) 1 p i q i q 2 i (N L − − 1) δ T + δ T + (N − 1)T np qm m j jm jq (N + 1)2 (N − 1) 1 q p i L δ T (N + 1)2 (N − 1) jp [m qn] 1 p p p q Lpq −(N − 1)δij Tmn − Nδim T jn − δin T jm ; 2 (N + 1) 3
+
+
Eijmn 4
= Kijmn +
1 i [mn] N+1 δ j K 4
+
N δi K N2 −1 [m 4 jn]
+
1 δi K N2 −1 [m 4 n] j
;
Eijmn
= Kijmn +
1 i [mn] N+1 δ j K 5
+
N δi K N2 −1 [m 5 jn]
+
1 δi K N2 −1 [m 5 n] j
.
5
4
5
(14)
Theorem 1.14. [18] The magnitudes Eijmn , Eijmn , Eijmn are ET-projective tensors, and the other ones Eijmn , Eijmn are 2
ET-projective parameters.
4
5
1
3
Definition 1.15. A non-symmetric affine connection space GAN with Eijmn = 0, r ∈ {1, 2, . . . , 5}, is the r-th pror
jectively flat space. A space which is the r-th projectively flat one, for all r = 1, . . . , 5, is projectively flat space. 2. Flatness and projective flatness of affine connection spaces with torsion Moffat’s results [16] are equations of motion of test particles and linear weak field approximation in Ricci and Minkowski flat spaces. Corollary of these results are galaxy rotational velocity curves and effective gravitational constant at infinity. In this research, Moffat used flat and Ricci-flat spaces. We are going to connect flatness and projective flatness of non-symmetric affine connection spaces.
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Firstly, we consider whether the r-th flat space is the r-th projectively flat one, r = 1, 2, . . . , 5, or not. If it is not the case, we are going to analyze what is a necessary and sufficient condition which the r-th flat space have to satisfy to be the r-th projectively flat one. Next what we analyze in this paper is an expression of ET- projective tensors Eijmn , r = 1, . . . , 5, (14), of a r
space GAN as linear functions of the Weyl projective tensor W ijmn of the associated space AN . The first question is whether flat non-symmetric affine connection spaces are the projectively flat ones or not. In the following theorems the r-th flat and the r-th projective flat non-symmetric spaces, r = 1, 2, . . . , 5, are connected. Before we prove theorems which present connection of the r-th flatness and the r-th projective flatness of a space GAN , we have to prove the following propositions. Proposition 2.1. Connection coefficients Lijk satisfy the equation 1 p i p p p . L jm Linp + Lm j Lipn = 2L jm Linp + T jm Tnp 2
(15)
Proof. Based on the equation (4), the equalities 1 p 1 i 1 p 1 i p p p p ) + (Lm j + Tm j )(Lipn + Tpn ) L jm Linp + Lm j Lipn = (L jm + T jm )(Linp + Tnp 2 2 2 2 1 p i 1 p 1 p 1 p i 1 p 1 p i p p + T jm Linp + T jm Linp + Lm j Lipn + Lm j Tpn + Tm j Linp + Tm j Tpn = L jm Linp + L jm Tnp 2 2 4 2 2 4 1 p i p i = 2L jm Lnp + Tm j Tpn , 2 are valid.
Proposition 2.2. The equalities ∼ 1 p i 1 p i Rijmn = Rijmn + T jm Tnp − T jn Tmp 1 4 4 p
and
∼ 1 p i 1 p i Rijmn = Rijmn + T jm Tnp − T jn Tpm 3 4 4
(16)
p
hold, where Rijmn = Lijm,n − Lijn,m + L jm Lipn − L jn Lipm . Proof. From the equation (11), we obtain ∼
R= 1
1 1 1 p p p p (A + A)i = (Li − Lijn,m + L jm Linp − L jn Limp ) + (Lim j,n − Lin j,m + Lm j Lipn − Ln j Lipm ) 2 1 3 jmn 2 jm,n 2 1 p i 1 p i (3,15) i p p = L jm,n − Lijn,m + L jm Linp + T jm Tnp − L jn Limp + T jn Tmp 4 4 1 1 p p i i = Rijmn + T jm Tnp − T jn Tmp , 4 4
∼
1 1 p i 1 p i p p (A + A)ijmn =Lijm,n − Lin j,m + L jm Linp + T jm Tnp − L jn Lipm + T jn Tpm 3 2 10 12 4 4 1 1 p p i i = Rijmn + T jm Tnp − T jn Tpm , 4 4 which proves this proposition. R=
Curvature pseudotensors A and A satisfy the equation 12
14
1 1 p i p p − Ln j Limp . (A + A)ijmn = Lim j,n − Lijn,m + Lm j Lipn + Tm j Tpn 2 12 14 4 Let us consider the mentioned problems.
(17)
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Theorem 2.3. A first-flat space GAN is the first-projectively flat one if and only if it satisfies the condition p q q p q q p i Lmp (N − 1)δij Tqn + δin T jq + L jp Nδi[m Tqn] + (N2 − 1)Tnm + (N − 1)δim Lpq T jn p q q p p q =Lnp (N − 1)δij Tqm + δim T jq + (N − 1)Lpq (N − 1)δij Tmn + δin T jm .
(18)
p
Proof. For a first-flat space the equation Kijmn = 0 is valid, which implies Kmn = Kmnp = 0. Then, it holds 1
Kijmn + 1
1
1
N 1 1 δij K[mn] + 2 δi[m K jn] + 2 δi Kn]j = 0. r N+1 N −1 N − 1 [m r
(19)
From the equation (19) and the definition of Eijmn (14), we get 1
1 p q q Lmp (N − 1) δij Tqn + δin T jq 2 1 (N + 1) (N − 1) 1 q q p + Lnp − (N − 1) δij Tqm − δim T jq 2 (N + 1) (N − 1) 1 p q i 2 i + Nδ T L + N − 1 T nm [m qn] (N + 1)2 (N − 1) jp N−1 q i p i p i p + δ T − δ T + L −(N − 1)δ T pq mn m n j jn jm (N + 1)2 (N − 1) which is equal zero if and only if the equality (18) is satisfied.
Eijmn =
Theorem 2.4. A second-flat space GAN is the second-projectively flat one. Proof. For a second-flat space we have it holds Kijmn = 0, i.e. 2
Kmn = 0.
(20)
2
The second equation in (14), immediately causes 1 N 1 δi K[mn] + 2 δi K jn] + 2 δi Kn]j N+1 j2 N − 1 [m 2 N − 1 [m 2 which proves this theorem. Eijmn = Kijmn + 2
2
(20)
= 0,
Theorem 2.5. A third-flat space GAN is the third-projectively flat one if and only if it satisfies the condition p q p q Lmp (N − 1)δij Tqn + (N2 − 1)Tijn + Lnp δim T jq + (N2 − 1)Tijm p
q
p
q
q
p
= δin Lmp T jq + (N − 1)δij Lnp Tqm + δi[m Tqn] L jp q p p p + (N − 1)Lpq (N − 1)δij Tmn + Nδim T jn + δin T jm . Proof. Analogously as in the proof of the Theorem 2.1, we have it holds 1 p q q Lmp (N − 1) δij Tqn − δin T jq + (N2 − 1)Tijn 2 (N + 1) (N − 1) 1 p i q i q 2 i (N + − − 1) δ T + δ T + (N − 1)T L np qm m jm j jq (N + 1)2 (N − 1) 1 p q − L δi T (N + 1)2 (N − 1) jp [m qn] N−1 q p p p + Lpq −(N − 1)δij Tmn − Nδim T jn − δin T jm 2 (N + 1) (N − 1)
Eijmn = 3
(21)
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which is equal zero if and only if it holds the equality (21).
Following the procedures used into the last three proofs, it is easy to prove next two theorems. Theorem 2.6. A fourth-flat space GAN is the fourth-projectively flat one. Theorem 2.7. A fifth-flat space GAN is the fifth-projectively flat one. Based on the theorems presented above, we directly conclude the following corollary is valid. Corollary 2.8. A flat space GAN which satisfies the conditions (18) and (21) is a projectively flat one. 3. Functionally connected projective curvature tensors and parameters Invariants of different geometrical mappings and their exact presentations are some of research subjects which results are used in different applications (for example, see[5]). Expressions of projective curvature tensors Eijmn , Eq. (14), of a non-symmetric affine connection space GAN as functions of Weyl projective k
tensor W ijmn , Eq. (7), of the corresponding associated space AN are the main purposes of this part of our research. Theorem 3.1. A Weyl projective tensor W ijmn of an affine connection space AN and parameter Eijmn of the generalized 1
one GAN satisfy the equation 1 ei Eijmn = W ijmn + V , 1 4 1 jmn
(22)
where 4 p p i e i = 2Ti V L Ti jmn j[m;n] + T j[m Tpn] − N + 1 jp mn 1 4 2 p p p q p q δij 2Tmn;p − T[mp;n] + Tmn Tpq + + δi L T N+1 (N + 1)2 j [mp qn] 2 1 p q p q p p q p Nδi[m T jp;n] + δi[m Tqn] T jp + 2δi[m T jn];p + δi[m T jn] Tpq − δi[m Tqn] T jp − 2 N+1 N −1 4 p q p q i i + Nδ[m L jp Tqn] − δ[m Ln]p T jq . (N + 1)2 (N − 1) Proof. The equality 1 1 N δij K[mn] + 2 δi[m K jn] + 2 δi Kn]j 1 1 N+1 1 N −1 N − 1 [m 1 1 p p i q i q i q i q (N (N + L − 1) δ T + δ T + L − − 1) δ T − δ T np n jq m jq j qn j qm (N + 1)2 (N − 1) mp 1 q p i + L jp Nδi[m Tqn] + N2 − 1 Tnm 2 (N + 1) (N − 1) 1 q i p 2 i p i p + T − (N − 1)δ T L −(N − 1) δ T + (N − 1)δ mn pq m jn n jm j (N + 1)2 (N − 1)
Eijmn = Kijmn + 1
holds. After the transformation of the definition of Kijmn , we obtain the following equation holds: 1
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1 1 1 p 1 p i Kijmn = Rijmn + Tijm;n − Tijn;m + T jm Tijn − T jn Tpm 1 2 2 4 4 Based on the previous equation, we conclude Ricci tensor of the first type becomes 1 p 1 p q 1 p q 1 p Kmn = Rmn + Tmn;p − Tmp;n + Tmn Tpq − Tmq Tqn 1 2 2 4 4 which induces the succeeding equation: 1 p 1 p q 1 p p K[mn] = R[mn] + Tmn;p − Tmp;n + Tnp;m + Tmn Tpq . 1 2 2 2 Further, N 1 N 1 δi K jn] + 2 δi Kn] j = 2 δi R jn] + 2 δi Rn]j N2 − 1 [m 1 N − 1 [m 1 N − 1 [m N − 1 [m 1 p N 1 p 1 p q 1 p q δim T jn;p − T jp;n + T jn TTpq − T jq Tpn + 2 2 2 4 4 N −1 1 p 1 1 p 1 p q 1 p q i δm Tn j;p − Tnp; j + Tn j Tpq − Tnq Tp j + 2 2 2 4 4 N −1 N 1 p 1 p q 1 p q i 1 p − 2 δn T jm;p − T jp;m + T jm Tpq − T jq Tpm 2 2 4 4 N −1 1 1 1 1 1 p q p p p q − 2 δin Tm j;p − Tmp; j + Tm j Tpq − Tmq Tp j 2 2 4 4 N −1 1 1 1 p N 1 p q = 2 δi[m R jn] + 2 δi[m Rn] j + δi[m δrn] T jr;p + T jr Tpq N+1 2 4 N −1 N −1 1 1 N p q p p i i i δ T T − δ T − δ T . − 4(N + 1) [m jq pn] 2(N2 − 1) [m jp;n] 2(N2 − 1) [m n]p; j Because of that, it holds 1 ei Eijmn = W ijmn + V , 1 4 1 jmn which proves this theorem.
Theorem 3.2. A Weyl projective tensor W ijmn of an affine connection space AN and parameter Eijmn of the generalized 2
one GAN satisfy the equation 1 ei , Eijmn = W ijmn + V 2 4 2 jmn where
e i = −Tp Ti − V jmn j[m pn] 2
2 1 1 p q p q q p δi T T − δi T T + δi T T . N + 1 j mn pq N + 1 [m jn] pq N − 1 [m pn] jq
(23)
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Proof. Let we start with Eijmn . It holds 2
Eijmn 2
=
Kijmn + 2
1 1 N δi[m K jn] + δij K[mn] + 2 δi Kn]j . 2 N+1 2 N + 1 [m 2 N −1
(24)
Directly from the equations (10, 13), we have it is valid the following equation: 1 p i 1 p i + T jn Tpm . Kijmn = Rijmn − T jm Tpn 2 4 4 This equation gives us 1 p q 1 p q Kmn = Rmn − Tmn Tpq + Tmq Tpn 2 4 4
and
1 p q K[mn] = R[mn] − Tmn Tpq . 2 2
After submission of these results in (24), we obtain it is valid 1 N 1 δij R[mn] + 2 δi[m R jn] + 2 δi Rn] j N+1 N −1 N − 1 [m 1 1 p i 1 p i 1 q p p q − T jm Tpn + T jn Tpm − δi T T − δi T T 4 4 2(N + 1) j mn pq 4(N + 1) m jn pq 1 1 1 p p q p q q + δin T jm Tpq + δim T jq Tpn − δi T T , 4(N + 1) 4(N − 1) 4(N − 1) n jq pm
Eijmn = Rijmn + 2
which proves this theorem.
Theorem 3.3. A Weyl projective tensor W ijmn of an affine connection space AN and parameter Eijmn of the generalized 3
one GAN satisfy the equation 1 ei , Eijmn = W ijmn + V 3 4 3 jmn where 4 p p p i i e i = 2Ti V L Ti jmn j(m;n) − T j[m Tpn] − 2Tmn Tp j + N + 1 (mp jn) 3 2 4 p p p q p q q p i + δij 2Tmn;p + T[mp;n] − Tmn Tpq + δ L T − L T pq mn N+1 (N + 1)2 j [mp qn] 1 2 p p p p q p q + 2δi[m T jn];p − δi[m T jn] Tpq − δi[m Tn]q Tp j + 2 Nδi[m T jp;n] + δi[m Tn]p; j N+1 N −1 4 4 p q p q q p p i i + δ[m Ln]p T jq + δ[m L jp Tn]q − Lpq Nδim T jn + δin T jm . 2 2 (N + 1) (N − 1) (N + 1) Proof. It holds 1 1 1 p i 1 p i 1 p Kijmn = Rijmn + Tijm;n + Tijn;m − T jm Tpn + T jn Tpm − Tmn Tpi j 3 2 2 4 4 2 Now, we get it is valid
(25)
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1 p 1 p 1 p q p K[mn] = R[mn] + Tmn;p + Tmp;n − Tnp;m − Tmn Tpq . 3 2 2 2 Based on the previous equations, we obtain it holds 1 N 1 N δi[m K jn] + 2 δi[m Kn] j = 2 δi[m R jn] + 2 δi Rn]j 3 3 −1 N −1 N −1 N − 1 [m 1 1 p 1 p q 1 p q 1 p q + δim T jn;p − T jn Tpq + T jq Tpn − Tnq Tp j N+1 2 4 4 2 1 1 p q 1 p q 1 p q i 1 p − δn T jm;p − T jm Tpq + T jq Tpm − Tmq Tp j N+1 2 4 4 2 N 1 N 1 1 p p i i i p i p δ T + δ T − δ T − δ T , + m m n n jp;n np; j jp;m mp; j 2 N2 − 1 N2 − 1 N2 − 1 N2 − 1
N2
which combined with the definition of Eijmn , proves this theorem.
3
We are proving the following two theorems with using of procedures similarly to the ones applied in the last three proves. Theorem 3.4. A Weyl projective tensor W ijmn of an affine connection space AN and parameter Eijmn of the generalized 4
one GAN satisfy the equation 1 ei Eijmn = W ijmn + V , 4 4 4 jmn
(26)
where e i = −Tp Ti − V jmn j(m pn) 4
1 1 2 p q p q p q δij Tmn Tpq − δi[m T jn] Tpq + δi T T . N+1 N+1 N − 1 [m qn] jp
Theorem 3.5. A Weyl projective tensor W ijmn of an affine connection space AN and parameter Eijmn of the generalized 5
one GAN satisfy the equation 1 ei , Eijmn = W ijmn + V 5 8 5 jmn
(27)
where e i = −Tp Ti − V mn p j jmn 5
1 q p δi T T . N − 1 [m n]q p j
Corollary 3.6. Magnitudes
Vijmn , r r
= 1, 3 defined in Theorems 3.1 and 3.3 are parameters. Other ones
Vijmn , r r
=
2, 4, 5 are tensors. Proof. Based on the equations (22-27), we have 1 Eijmn = W ijmn + Vijmn , r = 1, . . . , 4, r 4 r
1 and Eijmn = W ijmn + Vijmn . 5 8 5
The proof of this corollary holds directly from the Theorem 1.14 proved in [18], i.e. it holds directly from the tensor characters of Eijmn and the facts that difference of tensor and parameter is parameter and r
difference of two tensors is a tensor.
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4. Conclusion Flatness and projective flatness of spaces AN and GAN were analyzed in this paper. In the second section it is proved that non-symmetric flat spaces GAN are projectively flat if they satisfy conditions (18) and (21). In the third section, ET-projective tensors in space GAN are presented as linear functions of Weyl projective tensor in an associated space AN . Acknowledgement Authors thank professor Josef Mikeˇs for the idea realized in this paper. References ´ c, M. Lj. Zlatanovi´c, M. S. Stankovi´c, Lj. S. Velimirovi´c, On geodesic mappings of equidistant generalized Riemannian [1] M. S. Ciri´ spaces, Applied Mathematics and Computation 218, 12 (2012), 6648–6655. [2] A. Einstein, A Generalization of the Relativistic Theory of Gravitation, Annals of Mathematics, Princeton, 46, (1945), 576-584. [3] A. Einstein, Bianchi Identities in the Generalized Theory of Gravitation, Canadian Journal of Mathematics, 2, (1950), 120–128. [4] A. Einstein, Relativistic Theory of the Non-symmetric field, Appendix II in the Book: The Meaning of Relativity, (5th edition), Princeton, 49, 1955. [5] F. Fu, X. Yang, P. Zhao, Geometrical and physical characteristics of a class of conformal mappings, Journal of Geometry and Physics, 62 (2012), 1467–1479. [6] J. Mikeˇs, A. Vanˇzurov´a, I. Hinterleitner, Geodesic Mappings and Some Generalizations, 2009. [7] S. M. Minˇci´c, M. S. Stankovi´c, On geodesic mapping of general affine connection spaces and of generalized Riemannian spaces, Mat.vesnik, 49 (1997), 27–33. [8] S. M. Minˇci´c, M. S. Stankovi´c, Equitorsion geodesic mappings of generalized Rimannian spaces, Publications de L’Institut Math´ematique, 61 (75) (1997) 97–104. [9] S. M. Minˇci´c, Independent curvature tensors and pseudotensors of spaces with non-summetric affine connexion, Colloquia Mathematica Societatis J´anos Bolayai, 31, Differential Geometry, Budapest (Hungary), (1979), 445–460. [10] S. M. Minˇci´c, Ricci identities in the space of non-symmetric affine connection, Matematiˇcki Vesinik, 10(25), Vol. 2, (1973), 161–172. [11] S. M. Minˇci´c, Curvature tensors of the space of non-symmetric affine connexion, obtained from the curvature pseudotensors, Matematiˇcki Vesnik, 13, (28), (1976), 421–435. [12] S. M. Minˇci´c, Independent curvature tensors and pseudotensors of spaces with non-symmetric affine connexion, Colloquia Mathematica Societatis Janos Bolayai, 31, Differential Geometry, Budapest (Hungary), (1979), 445–460. [13] S. M. Minˇci´c, Lj. S. Velimirovi´c, M. S. Stankovi´c, On spaces with non-symmetric affine connection, containing subspaces without torsion, Applied Mathematics and Computation 219, 9 (2013), 4346–4353. [14] S. M. Minˇci´c, M. S. Stankovi´c, Lj. S. Velimirovi´c, Generalized Riemannian Spaces and Spaces of Non-symmetric Affine Connection, Faculty of Science and Mathematics Niˇs, 2013. [15] R. S. Mishra, Subspaces of a generalized Riemannian space, Bull. Acad. Roy. Belgique, (1954), 1058–1071. [16] J. W. Moffat, Gravitational Theory, Galaxy Rotation Curves and Cosmology without Dark Matter, arXiv: astro-ph/0412195v3 5 May 2005. [17] M. Prvanovi´c, Four curvature tensors of non-symmetric affine connexion (in Russian), Proceedings of the conference 150 years of Lobachevski geometry, Kazan 1976, Moscow 1997, 199–205. [18] M. Lj. Zlatanovi´c, New projective tensors for equitorsion geodesic mappings, Applied Mathematics Letters, 25, (2012), 890–897.
