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RICCI QUARTER-SIMETRIK METRİK KONNEKSİYONLU D-REKÜRANT UZAYLAR ÖZET Walker, reküran Riemanrı uzaylarını inceleyerek reküran Riemann uzaylanna örnek vermiştir [5]. Zayıf simetrik uzay kavramı, Tamassy ve Binh tarafından ortaya konmuştur [9]. De ve Bandyopadhyay tararından zayıf simetrik Riemann uzaylanna bir örnek gösterilmiştir [6]. Semi-simetrik metrik konneksiyonlu reküran ve semi-simetrik metrik konneksiyonlu zayıf simetrik uzaylara da örnekler Uysal ve Doğan tarafından verilmiştir [12,15]. Bu çalışmada, Ricci qurter-simetrik metrik konneksiyonlu reküran uzaylar ve Ricci quarter-simetrik metrik konneksiyonlu zayıf simetrik uzaylar incelenerek bu uzaylara örnek gösterilmiştir. Bu çalışmanın birinci bölümünde RT Riemann uzaylanna ait bazı temel kavramlara yer verilmiştir. Sıfırdan farklı herhangi bir Xı kovaryant vektör alam için VlRkjih = hRkjih (1) şartını sağlayan (Vn, g, V) uzaylarına V-Rekürant Uzayları denir. R? (n > 4) de g metriği, ds2 = `{xl) = 2ıp3(x1) diferansiyel denklemine eşdeğerdir. Bu diferansiyel denklemin çözümünden c2 keyfi bir sabit olmak üzere ıp(xx) fonksiyonu, X1 +c2 olarak bulunur. Sonuç olarak, (2) metriğine sahip semi-simetrik konneksiyonlu metrik uzay için D-rekürantlık koşulu sağlanacak şekilde ^(x1) fonksiyonu ve dolayısıyla uzayın konneksiyon katsayıları belirlenmiş olur. tp fonksiyonunu tp = öaşxax^e^ _2^(*)* formunda, aj,&,,c,,dj ve e$ kovaryant vektörleri _ / -2ıj){xl), i = 1 üi ` / 0, » ^ 1 ve b- - c - i ~^X^ İ = l °l ~ Q ~ / 0, * ^ 1 flî - e% - / o, %±/ olarak seçilirse, (2) metriğine sahip (VÇj, #, D) semi-simetrik metrik konneksiyonlu uzayın zayıf simetrik uzay olma koşulu, VIformuna indirgenir. Bu eşitlik ise a,, bi, di kovaryant vektörlerinin seçimine bağlı olarak, diferansiyel denklemine eşdeğerdir. Bu diferansiyel denklemin çözümünden '(xa)] = e*1 [1 - (n - 2)^'(rcQ)] (12) ~exl (n - 2W(xQ) = exl [1 - (n - 2)^'(arQ)] (13) diferansiyel denklem sistemi elde edilir. (12) diferansiyel denklemi her ıp'{xa) fonksiyonu için sağlanır. (13) diferansiyel denkleminin çözümü ise c% ve c-ı keyfi sabitler olmak üzere, İ>{xa) =ö + cıe* + c2 n - 2 olarak bulunur. Sonuç olarak, (2) metriğine sahip Ricci quarter-simetrik metrik konneksiyonlu uzay için jD-rekürantlık koşulu sağlanacak şekilde ıp(xa) fonksiyonu ve dolayısıyla uzayın konneksiyon katsayıları belirlenmiş olur. (p fonksiyonu ip = 8aşxax®ex formunda, o$, öj, Cj, di ve e, kovaryant vektörleri de ` _ / I' * = l,a ai-/0, i*l,a °* ^ / 0, i^l.a ve i = l,a =«={£,fcl, a vınolarak seçilirse (2) ile tanımlı metriğe sahip (Vn,g,D) Ricci Quarter-simetrik metrik konneksiyonlu uzayın zayıf simetrik uzay olma koşulu, (aa + ba + daj-klaal = DaLıaaı şekline girer. Bu eşitlik ise a*, 6,-, di kovaryant vektörlerinin seçimine bağlı olarak, i>`{xa) = ıp'(xa) î n-2 2.mertebeden sabit katsayılı lineer diferansiyel denklemine eşdeğerdir. Bu diferansiyel denklemin çözümünden c/ ve c2 keyfi sabitler olmak üzere %jj{xa) fonksiyonu, n - 2 olarak bulunur. Sonuç olarak (2) metriğine sahip Ricci Quarter-simetrik metrik konneksiyonlu uzay için {(WS)n,g,D) zayıf simetrik uzay olma bağıntısını sağlıyacak şekilde ıp{xa) fonksiyonu ve dolayısıyla uzayın konneksiyon katsayıları belirlenmiş olur. IX D-RECURRENT SPACES WITH RICCI QUARTER- SYMMETRIC CONNECTION SUMMARY The recurrent Riemannian space was introduced and studied by Walker [5] in 1950. The notion of weakly symmetric space was introduced by Tamassy and Binh [9] in 1989, then an example of weakly symmetric Riemannian spaces was constructed by De and Bandyopadhyay [6] in 1999. Examples of recurrent and also weakly symmetric spaces with semi -symmetric metric connection were given by Uysal and Doğan [12,15]. In this work, D- recurrent spaces and weakly symmetric spaces with Ricci quarter-symmetric connection are studied, then examples of D-recurrent and weakly symmetric spaces are given. This work consists of three chapters. In chapter I, the definition of Riemannian space, then some definitions and properties related to these spaces are given. After then, examples of V - recurrent and also weakly symmetric spaces are given. The space (Vn, g, V) is called V - Recurrent if there exists a covariant vector field Xi such that ^iRkjih = MRkjih- (14) We define the metric g in (R11, g, V) by the formula, ds2 = (p(dx1)2 + kaf)dxadxp + 2dxLdxn, (15) where [kap] is a symmetric non-singular matrix consisting of constant elements and (p is independent of a;`.1 For the metric (2), if we consider [kap] as 6ap and 2) is called weakly symmetric if its Riemannian curvature tensor Rkjth satisfies the condition ^iRkjih = O-lRkjih + hRljih + CjRkUh + diRkjlh + ZhRkjih (16) where a, b, c, d and e are 1-forms (non-zero simultaneously). If t = l i = l Ml and to, then R71 equipped with the metric g given in (2) is a weakly symmetric space. * = 1 «7^1 In chapter II, the definition of spaces with semi-symmetric metric connection is given, then some definitions and properties related to these spaces are given. After then, examples of the modified form of recurrent and weakly symmetric Riemannian spaces are given. For an arbitrary covariant vector field Wj, if the torsion tensor of the connection D satisfies then this connection is called semi-symmetric connection. If g is the metric of the semi-symmetric space, then the coefficients of the semi-symmetric connection are of the form T% = { *k } + 6*wk - gjkw/ Dk9ij = 0 (17) where D denotes the covariant differentiation with respect to T. If the covariant derivative of the metric tensor gji with respect to the connection given by (4) is zero, this connection is called the semi-symmetric metric connection. Ann- dimensional differentiable manifold, having the semi-symmetric metric connection D will be called the space with semi-symmetric metric connection and denoted by (Vn,g,D). xiFor a (Vn, g, D) space, if there exists a covariant vector A*, satisfying the equality, DıLkjih = ^iLkjih (18) then this space is called JD-Recurrent (Vn, g, D) space. An n- dimensional, (n > 2), weakly symmetric space with semi-symmetric metric connection is a non-flat space satisfying the condition DıLkjih = a-iLkjih + bkLijih + CjLuih + diLkjih + ZhLkju, (19) where Lkjih is the curvature tensor of the space and a, b, c, d, e are 1-forms (non-zero simultaneously). Such spaces are denoted by ((WS)n,D). The curvature tensor L/^ of Vn with semi-symmetric metric connection is Lkji = Rtji ~ &kwH + tfwki - w%9ji + wj9u, (20) where Wji = VjWi - WjWi + ^wtwtgji and R^t is the Riemannian curvature tensor of the space. If the components of the vector field w are chosen by / ^{x1), h = 1 Wh=/0, Ml then the D-recurrency condition takes the form DiWu = XiWu where A = {-Ai^{xl), 0,..., 0). For the metric given by (2), we obtain the following differential equation i>`(xl) = 2^V), where (') denotes the derivation with respect to x1. By solving this differential equation, we obtain the general solution of ^(x1) as follows ^(x1) = i-r^-. v ' x1 + oi Finally, the function ij){xx) is determined and so coefficients of the connection are determined for the D-recurrency condition. ?JC11If n-l tp = VVarV* N>(*)+Ai(t)]rf* and _ / -2ıP(xl), i = 1 °* ~ / 0, » ^ 1 0t ~ * ~ / o, * ^ i a* - e* - / 0, « ^ 1 then the condition of being weakly symmetric of the (Vn, g, D) takes the form Aip{xx)wn - -DxWii. Hence from this relation, the following differential equation fix1) = 2t/>V) is obtained, where (') denotes the derivation with respect to xl. Integrating with respect to x1, the function X1 + c2 is obtained where Oi is an arbitrary constant. Therefore, the coefficients of the connection are obtained for the space ((WS)n,D). In chapter III, the definition of spaces with Ricci quarter-symmetric metric connection is given, the definitions of recurrent and weakly symmetric spaces with Ricci quarter-symmetric metric connection are given. Then, examples of the modified form of recurrent and weakly symmetric spaces are constructed. For an arbitrary covariant vector Wj, if the torsion tensor of the connection D satisfies ahik = RhjWk-RhkWj, then this connection is called Ricci quarter-symmetric connection, where the Rji is Ricci tensor of the Riemannian space and R% = Rjigth An n- dimensional differentiate manifold with metric g, having the Ricci quarter-symmetric connection D will be called the space with Ricci quarter-symmetric connection. -xmIn general ease, the coefficients of the Ricci quarter-symmetric connection are of the form rjfc = { * } + Rfwk - Rjkwh, Dk9ij = 0 (21) where D denotes the covariant differentiation with respect to T. If the covariant derivative of the metric tensor ## with respect to the connection given by (8) is zero, this connection is called the Ricci quarter-symmetric metric connection and the space Vn equipped with this connection will be denoted by (Vn,g,D). For a (Vn, g, D) space, if there exists a covariant vector field Xi, satisfying the equality, DiLkjih - hLkjih (22) then this space is called D- Recurrent (Vnig,D) space. An n- dimensional, (n>2), weakly symmetric space with Ricci quarter- symmetric metric connection is a non-flat space satisfying the condition DiLkjih = Q-iLkjih + bkLijih + CjLktih + diL^jih + ^hJ^kjih (23) where L^jih is the curvature tensor of the space and a, 6, c, d, e are 1-forms (non-zero simultaneously). Such spaces are represented by ((WS)n, D) in short. The curvature tensor L^ of Vn with Ricci quarter-symmetric metric connection is Lkji = Rhkii ~ RkWn + R)^ki - w^Rji + w*}Rki + (V*i§ - VjRfrwi - (VkRji - VjRki)wh (24) where Wji = VjWi - Rjtiv^i + ^w^Rji and J?j^ is the Riemannian curvature tensor of the space. For the metric given in (2), If f ib(xa), h = a Wh=/ 0, h*a and ra-1 ip = J2xaxaexl (25) a=2 XIVthen the £>-recurrency condition takes the following forms DlLiaal = AiLiaai (26) DaLiaai = XaLıaaı, (27) by choosing A = (1, 0, 0,..., 1, 0,..., 0) (xa ^ 0). Hence we obtain the following the system of differential equations exl [1 - (n - 2)i>'(xa)] = e*1 [1 - (n - 2)^'(a;a)] (28) -e*1 (n - 2)iP`{xa) = e31' [1 - (n - 2)tP'(xa)}, (29) where (') denotes the derivation with respect to xa. By solving these differential equations, the general solution of tp(xa) is found as follows, H*a) = -^ + cxexa + c2. n - I Hence, the coefficients of the connection are determined for the space (Vn, g, D). For the metric given by (2), if ip = Saşxax^ex and a.-f h » = 1»a a*-/0, i*l,a d. = e.-lh i = 1>a then the condition of being weakly symmetric of the (Vn, g, D) takes the form (aa + ba + da)Lıaaı = £>a-klaal- From this relation, i/>`(xa) = i>'(xa) - 1 n-2 is obtained where (') denotes the derivation with respect to xa. Integrating with respect to xa, xa,j;(xa) = ^- + clexa+C2 n - 2 is found, where C/, c^ are an arbitrary constants. So the coefficients of the connection, for the space ((WS)m D), are determined. XV 54 |
