Filomat 31:6 (2017), 1709–1719 DOI 10.2298/FIL1706709A
Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat
Odd-Dimensional Weyl and Pseudo-Weyl spaces with Additional Tensor Structures Musa Ajetia , Marta Teofilovab , Georgi Zlatanovb a State
Universty of Tetovo, Faculty of Natural Sciences and Mathematics, dispersed in Skopje, Macedonia b University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria
Abstract. Odd-dimensional Weyl and pseudo-Weyl spaces admitting almost contact, almost paracontact and nilpotent structures are considered in this work. The results are obtained by means of the apparatus of the prolonged covariant differentiation. A linear connection with torsion is constructed. With respect to this connection the prolonged covariant derivatives of the fundamental tensors of the Weyl and pseudo-Weyl spaces are found to be zero. The curvature tensor with respect to this connection is considered.
1. Introduction Riemannian spaces with almost contact and almost paracontact structures have been studied in [1, 3, 5, 6, 13–15]. In [11, 12, 16, 17] Weyl spaces are studied, and in [19] nilpotent structures have been considered. In this paper, we study odd-dimensional Weyl and pseudo-Weyl spaces endowed with various structures: almost contact, almost paracontact and nilpotent. In our investigations we use the apparatus of the prolonged covariant differentiation which is defined in [4] and developed in [18, 21, 22]. The affinors of the considered structures are defined by means of 2n + 1 independent directional fields vα (σ, α = 1, 2, ..., 2n + 1) σ
σ
and their reciprocal covectors vα [2, 23, 24]. We pay special attention to the spaces with parallel structures with respect to the Levi-Civita connection of the metric, i.e. the so called K¨ahler-like classes. For such spaces we obtain a decomposition of three mutually orthogonal subspaces and also their line elements (fundamental forms). In the last section, we introduce a linear non-symmetric connection. With respect to this connection the prolonged covariant derivatives of the fundamental tensors of the Weyl and pseudo-Weyl spaces are proved to be zero. We study the curvature tensor corresponding to the introduced connection. 2. Preliminaries A set of quantities that differ from each other by a non-zero factor is called a pseudo-quantity. A particular quantity of this set is called a representative of the pseudo-quantity. The choice of a representative from a pseudo-quantity is called normalization. 2010 Mathematics Subject Classification. Primary 53B05, 53B99. Keywords. composition, almost (para-)contact structure, nilpotent structure, Weyl space, prolonged differentiation. Received: 27 March 2015; Accepted: 27 September 2015 Communicated by Dragan S. Djordjevi´c This work is partially supported by project NI15–FMI–004 of the Scientific Research Fund, Plovdiv University, Bulgaria. Email addresses:
[email protected] (Musa Ajeti),
[email protected] (Marta Teofilova),
[email protected] (Georgi Zlatanov)
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
1710
Let An be an n-dimensional space with an affine connection and the pseudo-quantity A ∈ An . The following definitions are given in [7, 10, 20]: Definition 2.1. By a pseudo-quantity with weight k it is meant a set of objects A admitting a transformation (renormalization) of the form A˘ = λk A, 1 2
(1) n
where λ = λ(u, u, ..., u) is a non-zero function of the point, and k ∈ R. We denote a pseudo-quantity with weight k by A{k}. Definition 2.2. A normalizer is defined as a covector (1-form) Tσ which is transformed by the rule T˘ σ = Tσ + ∂σ ln λ,
∂σ ln λ =
∂ ln λ . ∂uσ
(2)
In [4], V. Hlavat y´ introduced the notion of prolonged derivative of a pseudo-quantity A{k} by ∂•σ A = ∂σ A − kTσ A.
(3)
Because of (1) and (3) we have ∂•σ A˘ = λk ∂•σ A from which it follows that the prolonged differentiation preservers the weight of a pseudo-quantity. It is known that if A{p} and B{q}, then AB{p + q} and ∂•σ (AB) = (∂•σ A)B + A(∂•σ B). Definition 2.3. The prolonged covariant derivative of a pseudo-quantity A{k} is called the object [7, 10, 20] •
∇σ A = ∇σ A − kTσ A,
(4)
where ∇σ A is the usual covariant derivative of A. Let M2n+1 (1αβ , Tσ ) be a (2n + 1)-dimensional smooth manifold with a Weyl connection ∇, a symmetric pseudo-tensor 1αβ and an additional covector Tσ . The space M2n+1 (1αβ , Tσ ) will be denoted by W2n+1 and will be called a Weyl space. The coefficients Γσαβ of the Weyl connection are given by [7](p. 154) Γσαβ = {σαβ } − Tα δσβ + Tβ δσα − Tν 1νσ 1αβ ,
(5)
where {σαβ } are the Christoffel symbols of the tensor 1αβ . According to [7](p. 152), the fundamental tensor 1αβ admits a transformation of the form 1˘αβ = λ2 1αβ , 1 2
(6) 2n+1
where λ = λ(u, u, ..., u ), λ , 0, is an arbitrary smooth function of the point. By the renormalization (6) of 1αβ , the additional covector Tσ is transformed by formula (2) ([7], p. 152). According to [7](p. 152), the following hold ∇σ 1αβ = 2 Tσ 1αβ ,
∇σ 1αβ = −2 Tσ 1αβ ,
(7)
where 1αβ 1αν = δνβ .
From (6) it follows that 1αβ {2} and 1αβ {−2}. Then, according to (4) and (7), we obtain •
∇σ 1αβ = 0,
•
∇σ 1αβ = 0.
(8) β
•
β
Since the identity affinor has zero weight, i.e. δα {0}, then ∇σ δα = 0.
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
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Let us introduce the notations α, β, γ, σ, τ, ν, δ = 1, 2, ..., 2n + 1; j, s, k, l, m = 1, 2, ..., n;
p, q, r, t = 1, 2, ..., 2n;
(9)
¯ l,¯ m ¯ = n + 1, n + 2, ..., 2n. j,¯ s¯, k,
Let vβ (α = 1, 2, ..., 2n + 1) be 2n + 1 independent directional fields over M2n+1 . The pseudo-vectors vβ are α
α
renormalized by the condition 1αβ vα vβ = 1.
(10)
σ σ
From (10) it follows that vβ {−1}. According to [7](p. 153) and (10), we have α
1αβ vα vβ = cos ω, σ ν
(11)
σν
where ω{0} is the angle between the pseudo-vectors vα and vβ . σ
σν
Let the following conditions hold 1αβ vα vβ = 0,
1αβ vα v
k s¯
β
p 2n+1
ν
= 0.
(12) α
The net defined by the pseudo-vectors vβ is denoted by {v}. The pseudo-covectors vβ are given by α
σ
β
β
α
β
vβ vα = δα ⇐⇒ vσ vσ = δα , σ
(13)
α
α
from which it follows that vβ {1}. We choose {v} to be the coordinate net. Then, from (10), (11), (12) and (13) we have α
vβ 1
√1 , 0, 0, ..., 0 111
, vβ 0, 2
√1 , 0, ..., 0 122
, ..., v
2n+1
β
0, 0, ..., 0,
√
1 12n+1
2n+1
;
2 √ √ 2n+1 √ vβ 111 , 0, 0, ..., 0 , vβ 0, 122 , 0, ..., 0 , ..., v β 0, 0, ..., 0, 12n+1 2n+1 . 1
(14)
In the parameters of the coordinate net {v} the matrix of the fundamental tensor 1αβ has the following block diagonal form
1ks
1αβ =
0
0
α
0 1k¯ ¯s 0
0 0 12n+1 2n+1
,
det 1αβ , 0,
1αα > 0.
(15)
eαβ whose matrix has the following form in the parameters of the Let us consider the pseudo-tensor 1 coordinate net {v}: α
1ks
e 1αβ =
0
0
0 −1k¯ ¯s 0
0 0 12n+1 2n+1
.
(16)
By (8), (15) and (16) we get eαβ = 2Tσ 1 eαβ , ∇σ 1
eαβ = −2Tσ 1 eαβ , ∇σ 1
(17)
eαβ 1 eασ = δσβ . According to (16) and 1 eαβ 1 eασ = δσβ , it follows that 1 eαβ {2} and 1 eαβ {−2}. Then, by (4) and where 1 (17) we obtain •
eαβ = 0, ∇σ 1
•
eαβ = 0. ∇σ 1
(18)
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
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e 2n+1 and will be called a pseudo-Weyl space with fundamental The space W2n+1 (e 1αβ , Tσ ) will be denoted by W e 2n+1 coincide with the eαβ and additional covector Tσ . The coefficients of the connection of W tensor 1 coefficients of the connection of the space W2n+1 . From (10) and (16) it follows eαβ vα vβ = 1, 1
eαβ vα vβ = −1, 1
eαβ v 1
k¯ k¯
k k
α
v
β
2n+1 2n+1
= 1.
(19) α
In the parameters of the coordinate net {v} it it easy to prove that 1αβ v α
2n+1
2n+1
= v
β
eαβ v and 1
2n+1
The direction fields vβ satisfy the following derivative equations [22]:
α
2n+1
= v β.
α
ν
•
α
α
α
• α
∇σ vβ = Tσ vβ ,
ν
∇σ vβ = −Tσ vβ ,
ν
(20)
ν
ν
where Tσ {0}. α
Lemma 2.4. In the parameters of the coordinate net {v} the coefficients of the derivative equations (20) have the form: α
γ
Tσ = α
√ 1γγ γ √ 1αα Γσα ,
α
γ , α,
1 ∂σ 1αα 2 1αα
Tσ = Γασα − α
(21) + Tσ
(no summation over α). ν
•
Proof. According to (4) and the first of the equalities (20), we have ∇σ vβ = ∇σ vβ + Tσ vβ = Tσ vβ , from which we get α
ν
α
α
α ν
β
Tσ vβ = ∂σ vβ + Γσν vν + Tσ vβ . α ν
α
α
(22)
α
γ
Having in mind (13), after contracting (22) with vβ , we obtain γ
γ
γ
β
γ
Tσ = ∂σ vβ vβ + Γσν vν vβ + Tσ δα . α
α
(23)
α
By (23) we get γ
γ
γ
β
Tσ = ∂σ vβ vβ + Γσν vν vβ , α
α
α
α
α
α
β
Tσ = ∂σ vβ vβ + Γσν vν vβ + Tσ
γ , α,
α
α
α
(no summation over α).
(24)
We choose {v} for the coordinate net. Then, according to (14), equalities (24) take the form (21). α
β
Let us consider the affinor field aα defined by [2, 23, 24] p
β
aα = vβ vα − v p
2n+1
β 2n+1 v α.
(25) β
β
By (13) and (25) it follows that aα aσβ = δσα . Hence, the affinor aα defines a composition X2n × X1 of the basic manifolds X2n and X1 [8]. The positions (tangent planes) of the basic manifolds X2n and X1 are denoted by P(X2n ) and P(X1 ), respectively [8]. According to [8, 9], the affinors 1β
β
p
β
aα = 12 (δα + aα ) = vβ vα , p
2β
β
β
aα = 12 (δα − aα ) = v
2n+1
β 2n+1
v
α
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
1713 1β
2β
are the projecting affinors of the composition X2n ×X1 . If vβ is an arbitrary vector, we have vβ = aα vα + aα vα = 1
2
1
2
1β
2β
V β + V β , where V β = aα vα ∈ P(X2n ) and V β = aα vα ∈ P(X1 ). Obviously, vα ∈ P(X2n ), and v p
2n+1
α
∈ P(X1 ). The
β 1β 2β
affinors aα , aα , aα have zero weights. Let Xa × Xb (a + b = 2n + 1) be an arbitrary composition in the space W2n+1 , and P(Xa ) and P(Xb ) be the positions of the differentiable manifolds Xa and Xb , respectively. According to [9], the composition Xa × Xb is of the type (c, c), i.e. (Cartesian, Cartesian), if the positions P(Xa ) and P(Xb ) are translated parallelly along any line in the space M2n+1 . f2n+1 3. Almost Contact and Almost Paracontact Structures in W2n+1 and W Let us consider the following affinor fields ! ¯ β β k β k bα = κ v vα − v vα ,
(26)
k¯
k
κ
β
where κ = 1, i (i2 = −1). According to (13) and (26), we have bα v
2n+1 σ 2n+1
κ
β
Let κ = 1. From (13) and (26) we obtain bα bσβ = δσα − v 1
paracontact structure on W2n+1 .
1
structure on W2n+1 .
= 0. β
1
β
i
β 2n+1 β κ
= 0 and bα v
v α , i.e. the affinor bα defines an almost
2n+1
β
σ 2n+1
Let κ = i. By (13) and (26) it follows that bα bσβ = −δσα + v i
α
v α , i.e. bα defines an almost contact i
2n+1
β
Theorem 3.1. The affinor bα is parallel with respect to the Weyl connection ∇, i.e. κ
β
∇σ bα = 0
(27)
κ
if and only if the coefficients of the derivative equations (20) satisfy the conditions: s¯
s
k
k¯
p
Tσ = Tσ = T
2n+1
2n+1
σ
= T p
σ
= 0.
(28)
β
Proof. Because of (4) and bα {0}, the condition (27) is equivalent to κ
•
β
∇σ bα = 0.
(29)
κ
According to (20) and (26), equality (29) has the form ν
k
k
ν
ν
k¯
k¯
ν
Tσ vβ vα − Tσ vβ vα − Tσ vβ vα + Tσ vβ vα = 0. k
ν
ν
k¯
k
ν
ν
k¯
If we write the sums over the index ν in a more detailed form and regroup the addends, the last equality has the form ! ! ! s¯ 2n+1 s 2n+1 s¯ s k k¯ 2n+1 2 Tσ vβ + T σ v β vα − 2 Tσ vβ + T σ v β vα + T σ vβ − T σ vβ v α = 0. (30) s¯
k
k¯
2n+1
k
k¯
s
2n+1
2n+1
s¯
2n+1
s
ν
The independence of the pseudo-covectors vα yields that (30) holds true if and only if the following equalities are valid: s¯
2n+1
2 Tσ vβ + T k
s¯
k
σ
v
2n+1
β
= 0,
s
2n+1 β σ v 2n+1 k¯
2 Tσ vβ + T k¯
s
= 0,
s¯
T
2n+1
s
σ
vβ − T s¯
2n+1
σ
vβ = 0. s
(31)
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
1714 s¯
s
Since the pseudo-vectors vβ are mutually independent, equalities (31) are valid if and only if Tσ = Tσ = 0 and s¯
s
T
2n+1
σ
= T
ν 2n+1
2n+1
2n+1
σ
= T k
= T
σ
k¯
p
σ
= 0. Because of (9) the latter conditions are equivalent to T
2n+1
k¯
k 2n+1
σ
= T p
σ
= 0. Thus we
proved that equalities (31), and hence (27), are valid if and only if conditions (28) hold. β
Corollary 3.2. If ∇σ bα = 0, in the parameters of the coordinate net {v}, the coefficients Γναβ of the Weyl connection α
κ
satisfy: ¯
p
Γkσs = Γkσ¯s = Γσ2n+1 = Γ2n+1 = 0. σp
(32)
Proof. According to (21), equalities (28) imply (32). Let the space M2n+1 be a topological product of three smooth manifolds Xa , Xb and Xc (a + b + c = 2n + 1), i.e. let M2n+1 be the space of the composition Xa × Xb × Xc . We denote by P(Xa ), P(Xb ) and P(Xc ), respectively, the positions of the manifolds Xa , Xb and Xc . Definition 3.3. The composition Xa × Xb × Xc is said to be of type (c, c, c) if the positions P(Xa ), P(Xb ) and P(Xc ) are translated parallelly along any line in W2n+1 . Definition 3.4. The composition Xa × Xb × Xc is said to be orthogonal if the positions P(Xa ), P(Xb ) and P(Xc ) are mutually orthogonal. •
β
Theorem 3.5. If ∇σ bα = 0, the space W2n+1 is a space of orthogonal compositions Xn × Xn × X1 of the type (c, c, c). κ
Proof. Let condition (29) hold true. Then, in the parameters of the net {v} the conditions (32) are valid. Let us consider α
the composition X2n × X1 defined by the affinor (25). According to [9], by (32) it follows that the composition X2n × X1 is of the type (c, c). Hence, the position P(X1 ) of the manifold X1 (which is a curve) is translated parallelly along any line in M2n+1 . Let us consider the following affinors β
k¯
k
cα = vβ vα − vβ vα − v k¯
k
2n+1
β 2n+1 v α,
β
β
k¯
k
dα = vβ vα − vβ vα + v k¯
k
β
2n+1
β 2n+1 v α. β
(33) β
By (13) and (33) it follows that cα cσβ = δσα and dα dσβ = δσα . Hence, affinors cα and dα define the compositions Xn × Yn+1 and Xn × Zn+1 , respectively, where Yn+1 and Zn+1 are smooth (n + 1)-dimensional manifolds. In the parameters of the β β β coordinate net the affinors aα , cα and dα have the form, respectively: q
δp 0
β
(aα ) =
0 −1
s δk β (cα ) = 0 0
! ,
0 −δsk¯¯ 0
0 0 −1
,
s δk β (dα ) = 0 0
0 −δsk¯¯ 0
0 0 1
.
(34)
By (34) it follows that Yn+1 = Xn ×X1 and Zn+1 = Xn ×X1 , therefore W2n+1 is a space of the composition Xn ×Xn ×X1 . We denote by P(Xn ) and P(Xn ), respectively, the positions of the manifolds Xn and Xn . According to [9], equality (32) yields that the compositions Xn × Yn+1 and Xn × Zn+1 are of the type (c, c). Hance, the positions P(Xn ) and P(Xn ) are translated parallelly along any line in W2n+1 , i.e. we proved that the composition Xn × X× X1 is of the type (c, c, c). The projecting affinors of the composition Xn × Xn × X1 are: 1β
k
cα = vβ vα , k
1
k¯
β
dα = vβ vα , k¯
2β
aα = v
2n+1
β 2n+1
v α.
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
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If wβ is an arbitrary vector, we have 1
1β
β
1
2β
2
3
wβ = cα wα + dα wα + aα wα = W β + W β + W β , 1
1
2
1β
3
β
2β
where W β = cα wα ∈ P(Xn ), W β = dα wα ∈ P(Xn ) and W β = aα wα ∈ P(X1 ). Because vα ∈ P(Xn ), vα ∈ P(Xn ) and k¯
k
v
α
2n+1
∈ P(X1 ), from (15) or (16) it follows that the positions P(Xn ), P(Xn ) and P(X1 ) are mutually orthogonal. β
β
α α e 2n+1 , eαβ dudu the fundamental forms of the spaces W2n+1 and W We denote by ds2 = 1αβ dudu and de s2 = 1 respectively. •
β
Theorem 3.6. If ∇σ bα = 0, in the parameters of the coordinate net {v}, the fundamental forms of the space W2n+1 α
κ
e 2n+1 are given by and W ◦
k
s
◦
k¯
◦
k¯
s¯
◦
2n+1
◦
2n+1
2 ds2 = f 1ks dudu + f 1k¯ ¯ s dudu + f 12n+1 2n+1 d( u ) 2
1 ◦
k
s
3
s¯
(35)
2 de s 2 = f 1ks dudu − f 1k¯ ¯ s dudu + f 12n+1 2n+1 d( u ) , 2
1
3
where Ts¯ = 12 ∂s¯ ln f = 12 ∂s¯ ln f ,
Ts = 12 ∂s ln f = 21 ∂s ln f , 2
α
3
α
α
◦
◦
j
◦
T2n+1 = 12 ∂2n+1 ln f = 12 ∂2n+1 ln f ,
3
1
◦
j¯
2
1
◦
◦
2n+1
and f = f (u), f = f (u), f = f (u), 1ks = 1ks (u), 1k¯ ¯ s = 1k¯ ¯ s (u), 12n+1 2n+1 = 12n+1 2n+1 ( u ), 1
1
2
2
3
3
∂f
α
∂u
= ∂α f .
Proof. Let condition (29) hold. Then, in the parameters of the coordinate net {v}, conditions (32) will be valid. From α
(7), (17) and (32) we obtain ∂ j 1k¯ ¯ s = 2T j 1k¯ ¯s,
∂ j 12n+1 2n+1 = 2T j 12n+1 2n+1 ,
∂2n+1 1ks = 2T2n+1 1ks ,
∂ j¯1ks = 2T j¯1ks ,
∂ j¯12n+1 2n+1 = 2T j¯12n+1 2n+1 ,
∂2n+1 1k¯ ¯ s = 2T2n+1 1k¯ ¯s.
(36)
The truthfulness of the theorem follows after integration of equations (36). By (35) it follows that the positions P(Xn ) (or P(Xn ), or P(X1 )) are in conformal correspondence under parallel translation. e 2n+1 if 1 eαβ wα wβ = 0. Then, by According to (14), a direction field wα defines an isotropic direction in W (12) and (19) it is easy to prove that the direction fields vα ± vα and v α ± vα define isotropic directions in k
s¯
s¯
2n+1
e 2n+1 . the space W Let the functions f , f and f involved in (35) satisfy the condition f = f = f . Then, Tα = grad, and 2
1
3
1
2
3
e 2n+1 are Riemannian and pseudo-Riemannian, respectively, according to [7](p. 157), the spaces W2n+1 and W e which we denote by V2n+1 and V2n+1 . After renormalization of the fundamental tensor 1αβ we get [7](p. e2n+1 eαβ = 0. By (35) it follows that the line elements dS2 and de 157) ∇σ 1αβ = ∇σ 1 S 2 of the spaces V2n+1 and V have the form, respectively: j
k
s
j¯
k¯
j¯
¯
s¯
2n+1
2n+1
2 dS2 = 1ks (u)dudu + 1k¯ ¯ s (u)dudu + 12n+1 2n+1 ( u )d( u ) , j
(37)
k s k s¯ 2n+1 2n+1 2 de S 2 = 1ks (u)dudu − 1k¯ ¯ s (u)dudu + 12n+1 2n+1 ( u )d( u ) .
Equalities (37) imply that the positions P(Xn ) (or P(Xn ), or P(X1 )) are in conformal correspondence under parallel translation.
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
1716
Theorem 3.7. Condition (29) is equivalent to the following conditions: 1 •
1 •
1β
cσν ∇α cσ = 0, 1
1
1
2 •
β
dσν ∇α dσ = 0,
2β
aσν ∇α aσ = 0,
(38)
2
where cσν , dσν and aσν are the projecting affinors of the composition Xn × Xn × X1 . 1
k
1
k¯
σ 2n+1
2
Proof. Since cσν = vσ vν , dσν = vσ vν and aσν = v k¯
k
1 • 1 k¯ • s¯ β dσν ∇α dσ = vσ vν ∇α vβ vσ ,
1 • 1β k • s cσν ∇α cσ = vσ vν ∇α vβ vσ ,
k¯
s
k
v ν , we obtain
2n+1
2 •
2β
aσν ∇α aσ = v
s¯
σ 2n+1
2n+1
v
• ν
∇α
v
β 2n+1
v
2n+1
σ
. (39)
By (20) and (39) we get 1 • 1β cσν ∇α cσ
s¯
2n+1
β
= Tα v + T s¯
k
α
k
v
β
!
2n+1
1 • 1 β dσν ∇α dσ
k
vν ,
s
2n+1
β
= Tα v + T k¯
k¯
s
α
v
β
!
2n+1
k¯
vν ,
2 •
2β
p
aσν ∇α aσ = T
2n+1
α
2n+1
vβ v ν . (40) p
By (40) it follows that conditions (38) are valid if and only if conditions (28) are valid, too. Then, in accordance to Theorem 3.1, conditions (37) are equivalent to (29). f2n+1 4. A Nilpotent Structure on W2n+1 and W Let us consider the affinor β
n+i
fα = vβ v α .
(41)
i
•
β
β
β
β
β
Obviously, fα {0} and hence ∇σ fα = ∇σ fα . By (13) and (15) we get fα fβσ = 0, i.e. the affinor fα is nilpotent. β
In the parameters of the coordinate net {v} the matrix of fα is given by α
√
f β
=
α
0
...
0 ... 0 0 ... 0
... 0 ... ... ... 0 ... 0 ... ... ... 0
0
1n+1 n+1 √ 111
0 ... 0 0 ... 0
0
...
0
... 0 ... ... ...
... ... ... 0 ... 0
0 ... √
√ 1n+2 n+2 √ 122
12n 2n √ 1nn
... ... ...
0
0
...
. 0
0
...
0
β
Theorem 4.1. The affinor fα is parallel with respect to ∇, i.e. β
∇σ fα = 0
(42)
if and only if the coefficients of the derivative equations (20) satisfy the following conditions: n+k
n+k
Tσ= T s
2n+1
2n+1 •
σ
= T s
β
σ
k
n+k
s
n+s
= Tσ − T σ = 0. β
Proof. Since ∇σ fα = ∇σ fα , and because of (20) and (41), equality (42) has the form ν
n+s
T σ vβ v s
ν
n+s
α
ν
− T σ vβ vα = 0. ν
s
(43)
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
1717
If we write the sums over the index ν in a more detailed form and regroup the addends, the last equality is equivalent to ! " # ! n+k k n+k n+k 2n+1 n+k β s β β β n+s β 2n+1 T σ v vα − (Tσ − T σ )v + T σ v + T σ v vα+ T σv v α = 0. (44) s
s
k
n+s
s
k
s
n+k
2n+1
2n+1
k
ν
Because of the independence of the pseudo-covectors vα , equality (44) is valid if and only if the following equalities hold: n+k
n+k
T σ vβ = 0, s
T
2n+1
k
σ
vβ = 0,
k
n+k
n+k
s
n+s
s
2n+1
(Tσ − T σ )vβ + T σ v β + T
k
k
s
n+k
σ
β
v
2n+1
= 0.
The independence of the pseudo-vectors vβ yields that the last equalities, and hence (42), are valid if and only if ν
conditions (43) hold true. β
Corollary 4.2. If ∇σ fα = 0, the coefficients of the Weyl connection Γναβ satisfy the following conditions in the parameters of the coordinate net {v}: α
¯ ¯ 2n+1 Γkσs = Γkσ2n+1 = Γσs = 0, √ √ 1kk 1n+k n+k n+k √ Γkσs − √ 1ss 1n+s n+s Γσn+s =
(45) k , s,
0,
Γkσk −
1 ∂σ 1kk 2 1kk
= Γn+k − σn+k
1 ∂σ 1n+k n+k 2 1n+k n+k .
(46)
Proof. According to (21), equalities (43) take the form (45) and (46). β
Proposition 4.3. If ∇σ fα = 0, in the parameters of the coordinate net {v}, the fundamental tensor 1αβ , the additional α
covector Tσ and the coefficients of the connection Γσαβ satisfy the following conditions: ◦
◦
1k¯ ¯ s = h1k¯ ¯s, Γksk + Ts = α
T j = 12 ∂ j ln h,
12n+1 2n+1 = h12n+1 2n+1 , 1 ∂s 1kk 2 1kk ,
◦
Γk2n+1 k + T2n+1 = j¯
◦
◦
1 ∂2n+1 1kk 2 1kk ,
T2n+1 = 21 ∂2n+1 ln h,
Γkjk¯ − Γn+k = ¯ jn+k
1 ∂ j¯1kk 2 1kk
−
1 ∂ j¯1n+k n+k 2 1n+k n+k ,
(47) (48)
i¯ 2n+1
◦
where h = h(u), 1k¯ ¯ s = 1k¯ ¯ s (u), 12n+1 2n+1 = 12n+1 2n+1 (u, u ). Proof. By (7) and (45) we obtain ∂ j 1k¯ ¯ s = 2T j 1k¯ ¯s,
∂2n+1 1k¯ ¯ s = 2T2n+1 1k¯ ¯s,
∂ j 12n+1 2n+1 = 2T j 12n+1 2n+1 .
(49)
After integration of equations (49), we get (47). According to (45) and (49), equalities (46) imply (48). 5. Transformations of Connections Let us consider the connections 1 ν Γαβ
= Γναβ + Sναβ ,
1eν Γαβ
= Γναβ + e Sναβ ,
(50)
where Sναβ =
√ 1 12n+1
e Sναβ = √
2n+1
1
e2n+1 1
2n+1
P2n+1 τ Pn P2n δ ν − vν vδ , τ=1 vα 1βδ s¯=n+1 v v k=1 k s¯ k s¯ P2n+1 τ Pn P2n δ ν ν δ e v 1 v v − v v α βδ τ=1 s¯=n+1 k=1 k
s¯
k
(51)
s¯
Obviously, Sναβ {0}, e Sναβ {0}. We denote by 1 ∇ (1 e ∇) and 1 R (1 e R) the covariant derivative and the curvature tensor corresponding to ν eν the connection Γ (Γ ), respectively. αβ
αβ
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
1718
e 2n+1 , respectively, satisfy eαβ of the spaces W2n+1 and W Theorem 5.1. The fundamental tensors 1αβ and 1 1
• 1e eαβ ∇σ 1
•
∇σ 1αβ = 0,
= 0.
(52)
Proof. By (4) and (50) we obtain •
1
• 1e e ∇σ 1αβ
•
∇σ 1αβ = ∇σ 1αβ − Sνσα 1νβ − Sνσβ 1αν ,
•
eαβ − e eνβ − e eαν . = ∇σ 1 Sνσα 1 Sνσβ 1
(53)
Let us consider the tensors defined by e eνβ Tσαβ = e Sνσα 1
Tσαβ = Sνσα 1νβ ,
(54)
Obviously, Tσαβ {2}, e Tσαβ {2}. According to (50) and (54), we have Tσαβ =
√
1 12n+1
e Tσαβ = √
2n+1
1 e2n+1 1
2n+1
Pn P2n δ ν P2n+1 τ ν δ v 1 v v − v v 1νβ , σ αδ τ=1 s¯=n+1 k=1 k s¯ k s¯ P2n+1 τ P P δ ν ν δ e eαδ nk=1 2n νβ . τ=1 vσ 1 s¯=n+1 v v − v v 1 k
s¯
k
(55)
s¯
In the parameters of the coordinate net {v} we obtain α
Tσαβ =
√
1 12n+1
e Tσαβ = √
2n+1
1 e2n+1 1
2n+1
P2n+1 τ Pn P2n τ=1 vσ s¯=n+1 k=1
1αk 1β¯s −1α¯s 1βk √ √ 1kk 1s¯s¯ ,
(56)
P2n+1 τ Pn P2n eβ¯s −e eβk eαk 1 1α¯s 1 1 , τ=1 vσ s¯=n+1 √ √ k=1 ekk 1
es¯s¯ 1
from which it follows that Tσ(αβ) = 0,
e Tσ(αβ) = 0.
(57)
Then, (8), (18), (53) and (57) imply (52). In the parameters of the coordinate net {v}, by (14) and (51) we obtain the following non-zero components α
of the tensors Sναβ and e Sναβ : √ P2n 1ll j j Slm¯ = −e Slm¯ = − √1 j j √12n+1 s¯=n+1 2n+1
P j j S2n+1m¯ = −e S2n+1m¯ = − √11 j j 2n s¯=n+1 j¯
j¯
e = Slm ¯ = Slm ¯
√ 1l¯l¯ √ √ 1 j¯j¯ 12n+1
Pn k=1
2n+1
√ P2n 1l¯l¯ j j Sl¯m¯ = −e Sl¯m¯ = − √1 j j √12n+1 s¯=n+1 2n+1
1¯ √m¯s , 1s¯s¯
j¯
1¯ √m¯s , 1s¯s¯
j¯
Slm = e Slm = j¯
1 √mk , 1kk
√
√ 1ll √ 1 j¯j¯ 12n+1
j¯
S2n+1 m = e S2n+1 m =
Pn 2n+1
√1 1 j¯j¯
k=1
Pn k=1
1 √mk , 1kk
1¯ √m¯s , 1s¯s¯
(58)
1 √mk . 1kk
In the parameters of the net {v}, by (32), (50) and (58) we obtain the following non-zero coefficients of α
the connections 1 Γναβ and 1e Γναβ : 1 j Γlm
=1 e Γlm = Γlm ,
1 j¯ Γlm
=1 e Γlm = Slm ,
1 j Γ2n+1 m¯
j
j
1 j Γlm¯
= −1e Γlm¯ = Slm¯ ,
j¯
j¯
1 j¯ Γl¯m¯
=1 e Γl¯m¯ = Γl¯m¯ ,
j
j
j
= −1e Γ2n+1 m¯ = S2n+1 m¯ ,
j¯
1 j Γl¯m¯ 1 j¯ Γlm ¯
j
j¯
1 j¯ Γ2n+1 m
j¯
j
j
= −1e Γl¯m¯ = Sl¯m¯ , j¯
j¯
=1 e Γlm ¯ = Slm ¯ , j¯
=1 e Γ2n+1 m = S2n+1 m .
(59)
M. Ajeti et al. / Filomat 31:6 (2017), 1709–1719
1
1719
By (59) and straightforward computations we get the following components of the curvature tensors Rαβσν and 1 e Rαβσν : 1
j
1e j Rskm 1
j
¯
1
j
j
¯
1 e j¯ Rs¯k¯ m¯
= Rskm − 2S[s|l¯Slk]m ,
j
j¯
j
Rskm = Rskm + 2S[s|l¯Slk]m ,
j
j
j
¯
j
j¯
j¯
j¯
, = Rs¯k¯ m¯ − 2S[¯s|l Slk] ¯m ¯
R2n+1 s¯k¯ = 2∂[2n+1 Ss¯]k¯ + S2n+1l¯Γls¯k¯ ,
1e j R2n+1 s¯k¯
j¯
, Rs¯k¯ m¯ = Rs¯k¯ m¯ + 2S[¯s|l Slk] ¯m ¯
¯
= −2∂[2n+1 Ss¯]k¯ − S2n+1l¯Γls¯k¯ ,
1
j¯
j¯
j¯
R2n+1 sk = 2∂[2n+1 Ss]k + S2n+1 l Γlsk , 1 e j¯ R2n+1 sk
j¯
j¯
= 2∂[2n+1 Ss]k + S2n+1 l Γlsk ,
where Rαβσν is the curvature tensor of the space W2n+1 . References [1] T. Adati, T. Miyazawa,On paracontact Riemannian manifolds, TRU Math. 13(2) (1977), 27–39. [2] M. Ajeti, M. Teofilova, G. Zlatanov, Triads of compositions in an even–dimensional space with a symmetric affine connection, Tensor, N.S. 73(3) (2011), 171–187. [3] D. Blair, Riemannian geometry of contact and symplectic manifolds, Prog. in Math. 203, Birkh¨auser Boston, 2002. ´ Les courbes de la vari´et´e Wn , Memor. Sci. Math., Paris, 1934. [4] V. Hlavat y, [5] S. Kaneyuki, F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99 (1985), 173–187. [6] S. Kaneyuki, M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math. 8(1) (1985), 81–98. [7] A. P. Norden, Affinely connected spaces, Izd. GRFML Moscow (1976) (in Russian). [8] A. P. Norden, Spaces of Cartesian composition, Izv. Vyssh. Uchebn. Zaved., Math. 4 (1963), 117–128 (in Russian). [9] A. P. Norden, G. N. Timofeev, Invariant criteria for special compositions of multidimensional spaces, Izv. Vyssh. Uchebn. Zaved., Math. 8 (1972), 81–89 (in Russian). [10] A. P. Norden, A. Sh. Yafarov, Theory of the nongeodesic vector field in two-dimensional affinely connected spaces, Izv. Vyssh. Uchebn. Zaved., Math. 12 (1974), 29–34 (in Russian) ¨ [11] A. Ozde ger, ˇ Conformal and generalized concircular mappings on Einstein-Weyl manifolds, Acta Math. Sci. 30B(5) (2010), 1739– 1745. ¨ [12] A. Ozde ger, Generalized Einstein tensor for a Weyl manifold and its applications, Acta Math. Sin. (Engl. Ser.) 29(2) (2013), ˇ 373–382. [13] S. Sasaki, On paracontact Riemannian manifolds, TRU Math. 16(2) (1980), 75-86. [14] S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure I, Tohoku Math. J.(2) 12(3) (1960), 459-476. [15] I. Sato, On a structure similar to the almost contact structure, Tensor, N.S. 30(3) (1976), 219–224. [16] G. N. Timofeev, Invariant characteristics of special compositions in Weyl spaces, Izv. Vyssh. Uchebn. Zaved., Math. 1 (1976), 87–99 (in Russian). [17] G. N. Timofeev, Orthogonal compositions in Weyl spaces, Izv. Vyssh. Uchebn. Zaved., Math. 3 (1976), 73–85 (in Russian). [18] B. Tsareva, G. Zlatanov, Proportional nets in Weyl spaces, J. Geom. 71 (2001), 192–196. [19] V. V. Vishnevskii, Manifolds over plural numbers and semi tangent structures, Itogi Sci. Tech., Geom. 20 (1988), 35–47 (in Russian). [20] G. Zlatanov, A. P. Norden, Orthogonal trajectories of a geodesic field, Izv. Vyssh. Uchebn. Zaved., Math. 7 (1975), 42–46 (in Russian). [21] G. Zlatanov, Nets in an n-dimensional space of Weyl, C. R. Acad. Bulg. Sci. 41 (1988), 29–32 (in Russian). [22] G. Zlatanov, Special networks in the n-dimensional space of Weyl Wn , Tensor, N.S. 48 (1989), 234–240. [23] G. Zlatanov, Compositions generated by special nets in affinely connected spaces, Serdica Math. J. 28 (2002), 1001–1012. [24] G. Zlatanov, Special compositions in affinely connected spaces without a torsion, Serdica, Math. J. 37 (2011), 211–220.
