On Quasi-Recurrent Spaces With Ricci
Quarter-Symmetric Metric Connection
S. Aynur Uysal, E. ¨Ozkara Canfes and C. Elvan Din¸c
Technical University of Istanbul Faculty of Sciences and Letters
Department of Mathematics 34469 Maslak - Istanbul, Turkey
auysal@dogus.edu.tr canfes@itu.edu.tr elvand@khas.edu.tr
Abstract
In [3], Mishra and Pandey defined Ricci quarter-symmetric met-ric connection in Riemanian manifold. In [5],Uysal and Do˘gan de-fined D-recurrent spaces with semi-symmetric metric connection and constructed an example of these spaces. In these paper we define quasi-recurrent spaces with Ricci quarter- symmetric metric connection and establish an example of such spaces.
Mathematics Subject Classification: 53B05
Keywords: Ricci quarter-symmetric metric connection, Recurrent space,
Quasi-recurrent spaces
1
Introduction
A non-flat Riemannian space Mn (n > 2) is called a quasi-recurrent space if the curvature tensor Rkjih satisfies the condition
∇lRkjih = alRkjih + bkRljih+ cjRklih+ diRkjlh+ ehRkjil (1.1)
where a, b, c, d, e are 1-forms (non-zero simultaneously) and ∇ denotes the co-variant differentiation with respect to Levi Civita connection
h ji
. a, b, c, d, e are called the associated 1-forms of the space and n-dimensional space of this
kind is denoted by (QR)n.
Let (Mn, g) be an n-dimensional differentiable manifold of class C∞with metric tensor g and Levi Civita connection ∇. A linear connection D on (Mn, g) is said
quarter-symmetric metric to be a Ricci Quarter-Symmetric Metric Connection if the torsion tensor T of the connection D and the metric tensor g of the manifold satisfy [2], [6],
Tjkh = Rjhwk− Rkhwj, Dkgji= 0 , (1.2) where w is a 1-form associated with the torsion tensor of the connection D and Rhj = Rjagah, Rij is the Ricci tensor of the Levi Civita connection ∇.
Then we have Γhji = h ji + Rjhwi− Rjiwh, (1.3)
where Γhji are the connection coefficients of the Ricci quarter-symmetric metric connection and wh = wtgth.
The curvature tensor Lhkji of the manifold Mn is defined in [6] by, Lhkji = ∂kΓhji− ∂jΓhki+ ΓhkaΓaji− ΓhjaΓaki, ∂k = ∂ ∂xk (1.4) and the curvature tensor of the Riemannian manifold
Rhkji = ∂k h ji − ∂j h ki + h ka a ji − jah kia , (1.5) then substituting (1.3) in (1.4), we obtain the following equation for the cur-vature tensor Lhkji of (Mn, g, D) .
Lkjih = Rkjih− Rkhwji+ Rjhwki− wkhRji+ wjhRki + (∇kRjh− ∇jRkh) wi− (∇kRji− ∇jRki) wh, (1.6) where we put wji = ∇jwi− Rjtwtwi + 1 2wtw tR ji; whk = wkagah. (1.7)
The curvature tensor of the Ricci quarter-symmetric metric connection Lkjih satisfies the following properties:
(i) Lkjih =−Ljkih, (ii) Lkjih =−Lkjhi, (iii) Lkjih = Ljkhi,
Definition 1.1. An n-dimensional, (n > 2), quasi-recurrent space with Ricci
quarter-symmetric metric connection is a non-flat space satisfying the condi-tion
DlLkjih = alLkjih + bkLljih + cjLklih+ diLkjlh+ ehLkjil, (1.9)
where Lkjih is the curvature tensor of the space and a, b, c, d, e are 1-forms (non-zero simultaneously). Such spaces will be represented by ((QR)n, D) in short.
Theorem ,. Five associated 1-forms cannot be all different.
Proof˙, By using the following method which is used in [1], we obtain that
b = c and d = e.
Interchanging the indices k and j in (1.9) we obtain
DlLjkih = alLjkih+ bjLlkih+ ckLjlih + diLjklh+ ehLjkil. (1.10)
Adding (1.9) and (1.10) and using (1.8i), we get
(bk− ck)Lljih + (bj− cj)Llkih= 0 (1.11) or
AkLljih+ AjLlkih = 0 (1.12)
where Ak = (bk− ck).
We want to show that Ak = 0 (k = 1, 2, ..., n).
On the contrary assume that there exists a fixed q for which Aq = 0. If we set k = j = q in (1.12), then we find that AqLlqih+ AqLlqih = 0 which means that the curvature tensor Llqih = 0 for all l, i, h.
On the other hand, if we take k = q in (1.12), we obtain that AqLljih + AjLlqih = 0.
Since Aq = 0, then the curvature tensor Lljih = 0 for all l, j, i, h, which contradicts with the hypothesis. Therefore, Ak = 0, which means that bk = ck
for all k.
Interchanging the indices i and h in (1.9) we obtain
DlLkjhi = alLkjhi+ bkLljhi + cjLklhi+ dhLkjli+ eiLkjhl. (1.13) If we use the same method for indices i and h, we find that di = ei.
Furthermore the condition (1.9) can be expressed in
DlLkjih = alLkjih+ bkLljih+ bjLklih+ diLkjlh+ dhLkjil. (1.14)
So a non-flat space with Ricci quarter-symmetric metric connection is called quasi-recurrent if its curvature tensor Lkjih satisfies the condition (1.14).
If, in particular, bi = 0 and di = 0 in (1.9 ), then we have,
Definition 1.2. The space (M, g, D) is called D − recurrent, if there
exists a covariant vector field al = 0 such that
DlLhkji = alLhkji. (1.15)
Moreover, if Rhj = δjh in (1.2),then we obtain the results in [5].
2
An Example of Quasi-Recurrent Spaces With
Ricci Quarter-Symmetric Metric Connection
We define the metric g in the coordinate space Rn (n≥ 4) by the for-mula,1 [4]
ds2 = ϕ(dx1)2+ kαβdxαdxβ + 2dx1dxn, (2.1) where [kαβ] is a symmetric non - singular matrix consisting of constants and ϕ is independent of xn. The only non-zero components of the Christoffel’ s symbols
i jk
, the Riemannian curvature tensor Rkjih and the Ricci tensor Rji are [4] λ 11 =−1 2k λβϕ .β , n 11 = 1 2ϕ.1 , n 1α = 1 2ϕ.α (2.2)
1In this section, let each Latin indices run over 1, 2, ..., n and each Greek indices run over
and R1αβ1 = 1 2ϕ.αβ, R11= 1 2k αβϕ .αβ, (2.3)
respectively, where (.) denotes the partial differentiation with respect to coor-dinates and kαβ is the inverse matrix of kαβ.
For the metric (2.1), we consider kαβ = δαβ and ϕ = 1 2sin(kαβ π 2)x αxβex1 . (2.4)
Considering the metric, the only non-zero components of the curvature tensor Rkjih and the Ricci tensor Rji are
R1αα1 = e
x1
2 R11 = (n− 2)
ex1
2 . (2.5)
Now, differentiating covariantly Rkjih and Rji with respect to all compo-nents (l = 1, 2, ..., n), we get the non-zero compocompo-nents of ∇lRkjih and ∇lRji as ∇1R1αα1 = e x1 2 ∇1R11 = (n− 2) ex1 2 . (2.6)
For a chosen α, let
wh =
ψ(xα), h = α
0, otherwise (2.7)
where ψ(xα) is a continuous functions of xα on the interval I = [a, b ]. We obtain the non-zero components of Γhji and wji according to the our assumption (2.7) as follows Γα11 = α 11 + Rα1w1 − R11wα = α 11 − R11wα = α 11 − (n − 2)ex 1 2 ψ(x α) Γn11 = n 11 + Rn1w1− R11wn= n 11 Γn1α = n 1α + Rn1wα− R1αwn= n 1α + R11wα = n 1α + (n− 2)e x1 2 ψ(x α) Γnα1 = n α1 + Rnαw1− Rα1wn = n α1 (2.8)
w11=∇1w1− R1twtw1+ 2wtw tR 11 = 1 ∂x1 − wt t 11 − R1tw tw 1+ 2wαw αR 11 =−ψ(xα) α 11 + 1 4(n− 2)e x1 gαβwαwβ (2.9) wαα =∇αwα− Rαtwtwα+1 2wtw tR αα = ∂w∂xαα − wt t αα − Rαtwtwα+12wαwαRαα = ψ(xα) (2.10) Using (1.6), (2.5), (2.6), (2.7), (2.9) and (2.10), we get the only non-zero component of the curvature tensor Lkjih,
L1αα1 = R1αα1− R11wαα+ Rα1w1α− w11Rαα+ wα1R1α + (∇1Rα1− ∇αR11)wα− (∇1Rαα− ∇αR1α)w1 = R1αα1− R11wαα = 1 2[1 − (n − 2) ψ (xα) ] ex1 . (2.11)
Using the properties of the curve Lkjih, we have other non-zero components of the curvature tensor Lkjih
Lα1α1 =−L1αα1=−e x1 2 [1− (n − 2)ψ (xα)] L1α1α =−L1αα1=−e x1 2 [1− (n − 2)ψ (xα)] Lα11α = L1αα1= e x1 2 [1− (n − 2)ψ (xα)]. (2.12) Hence, we get the non-zero components of DlLkjih
D1L1αα1 = ∂L1αα1 ∂x1 − (Lsαα1+ L1ααs) Γ s 11− (L1sα1+ L1αs1) Γs1α = e x1 2 [1− (n − 2)ψ (xα)] (2.13) and DαL1αα1 = ∂L1αα1 ∂xα − (Lsαα1+ L1ααs) Γ s α1− (L1sα1+ L1αs1) Γsαα =−e x1 2 (n− 2)ψ (xα). (2.14)
We want to show that (Mn, g, D) is ((QR)n, D). Let us consider the 1-forms ai = ⎧ ⎨ ⎩ 1 3, i = 1 1, i = α 0, otherwise bi = ⎧ ⎨ ⎩ 1 3, i = 1 tanh(xα), i = α 0, otherwise and di = 1 3, i = 1 0, otherwise (2.15) In order to verify the condition (1.14), it is sufficient to check the following relations: (A) D1L1αα1 = a1L1αα1+ b1L1αα1+ bαL11α1+ dαL1α11+ d1L1αα1 (B) DαL11α1 = aαL11α1+ b1Lα1α1+ b1L1αα1+ dαL11α1+ d1L11αα (C) DαL1α11 = aαL1α11+ b1Lαα11+ bαL1α11+ d1L1αα1+ d1L1α1α (D) DαL1αα1 = aαL1αα1+ b1Lααα1+ bαL1αα1+ dαL1αα1+ d1L1ααα (E) D1Lααα1 = a1Lααα1+ bαL1αα1+ bαLα1α1+ dαLαα11+ d1Lααα1 (F ) D1L1ααα= a1L1ααα+ b1L1ααα+ bαL11αα+ dαL1α1α+ dαL1αα1
For the other cases, (1.14) holds trivially.
From (1.8 iv), (2.13) and (2.15), we get the following relation for the right-hand side (r.h.s) and the left-right-hand side (l.h.s) of (A):
r.h.s of (A)= (a1 + b1+ d1)L1αα1 = 1
2[1 − (n − 2) ψ
(xα) ] ex1
= D1L1αα1=l.h.s of (A)
where () denotes the derivative with respect to xα.
Using the relations of (1.8 i) and (1.8 iv), we obtain that r.h.s. of (B) = b1(Lα1α1 + L1αα1) = 0.
By using the similar argument in (B), it is easy to see that the relation (C) is also true.
From (1.8 iv), (2.14) and (2.15), we get the following relation for the right-hand side (r.h.s) and the left-right-hand side (l.h.s) of (D):
r.h.s of (D) = (aα+ bα+ dα)L1αα1 = 1 2[1 + tanh(x α)] [1 − (n − 2) ψ(xα) ] ex1 =−e x1 2 (n− 2)ψ (xα) = DαL1αα1 = l.h.s of (D)
From the above equation, we have the second order differential equation ψ(xα) = [1 + tanh(xα)] [ψ(xα) − 1
n− 2] (2.16)
which has the solution
ψ(xα) = ( 1 n− 2 + c1 2 ) x α + c1 4 e 2 xα + c2 (2.17)
with arbitrary constant c1 and c2.
Again using the relations (1.8 i) and (1.8 iv), we obtain that r.h.s. of (E) = bα(L1αα1 + Lα1α1) = 0.
By using the similar argument in (E), it can be shown that the relation (F) is also true.
Finally we have determined the functions ψ(xα), therefore the coefficients of the connection are determined. Hence, Rn with the metric (2.1) is a ((QR)n, D).
REFERENCES
[1] U.C.De and S.Bandyopadyay, On Weakly-symmetric Riemannian Spaces, P ubl. M ath.Debrecen,54,(1999), 377-381.
[2] D. Kamilya and U. C. De, Some properties of a Ricci Quarter-symmetric Metric Connection in a Riemannian Manifold, Indian J. P ure Appl. M ath. , 26(1), (1995)29-34.
[3] R. S. Mishra and S. N. Pandey, On Quartersymmetric Metric F -, Connections-, T ensor-, N. S.-, 34-, (1980)-, 1-7.
[4] W. Roter, On Conformally Related Conformally Recurrent Metrics I., Some General Results, Colloquium M athematicum, 47, (1982), 39-46. [5]S. A. Uysal and R. ¨O. Do˘gan, On Recurrent Spaces with Semi-symmetric
Singh Connection, Indian J. P ure Appl. M ath., 34(7), (2003), 1061-1065. [6] K. Yano and T. Imai, Quarter-symmetric Metric Connections and their
Curvature Tensors, T ensors, N. S., 38, (1982), 13-18.
