THE GENERALIZED RECURRENT WEYL SPACES HAVING A DECOMPOSABLE CURVATURE TENSOR

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1 International Mathematical Forum, 1, 2006, no. 4, THE GENERALIZED RECURRENT WEYL SPACES HAVING A DECOMPOSABLE CURVATURE TENSOR Hakan Demirbüker and Fatma Öztürk Faculty of Science and Letters Davutpaşa, Istanbul, Turkey hdemirbu@yahoo.com, fatozmat@yahoo.com Leyla Zeren Akgün Mimar Sinan University Faculty of Science and Letters Beşiktaş, Istanbul, Turkey fakgun1946@yahoo.com Abstract In this paper, we have studied the generalized recurrent Weyl spaces the curvature tensor of which is decomposed in the form L i jkl = vi ϕ jkl, and proved some theorems concerning such spaces. Mathematics Subject Classification: 53A40 Keywords: Generalized Weyl space, recurrent Weyl space, recurrent tensor, decomposable curvature tensor.

2 166 Hakan Demirbüker, Fatma Öztürk and Leyla Zeren Akgün 1. INTRODUCTION A differentiable manifold W n of dimension n having a symmetric connection and a symmetric conformal metric tensor g ij preserved by is called a Weyl space. Accordingly, in local coordinates, there exists a covariant vector field T ( complementary vector field ) such that the condition k g ij 2T k g ij =0, (1.1) holds [1], [2], [3]. Writing (1.1) out in full, we have k g ij g hj Γ h ik g ihγ h jk 2T kg ij =0 (1.2) where Γ i kl are the connection coefficients of the connection are given by { } i Γ i kl = g im (g mk T l + g ml T k g kl T m ). (1.3) kl We denote such a Weyl space as W n (Γ i jk,g ij, T k ). An n dimensional manifold GW n is said to be a generalized Weyl space if it has an asymmetric connection and a symmetric conformal metric tensor g ij satisfying the compatibility condition given by the equation k g ij 2T k g ij =0 (1.4) where T k denotes a covariant vector field and k g ij denotes the usual covariant derivative [4], [5]. Writing (1.4) out in full, we have k g ij g hj L h ik g ih L h jk 2T k g ij =0 (1.5) where L i kl are the connection coefficients of the connection k are given by L i kl =Γ i kl + χ i kl (1.6)

3 THE GENERALIZED RECURRENT WEYL SPACES where Γ i kl and χi kl L i kl namely and are the symmetric and anti-symmetric parts respectively of Γ i kl = 1 2 (Li kl + L i lk) (1.7) χ i kl = 1 2 (Li kl Li lk ). (1.8) We denote such a Weyl space GW n (L i jk, g ij,t k ). The Weyl space W n (Γ i jk, g ij,t k ) is called the associate Weyl space to the generalized Weyl space GW n (L i jk, g ij,t k ). Using the relations (1.2), (1.5) and (1.6) we obtain χ jik + χ ijk =0 (1.9) where the tensor χ jik is defined by χ jik = g hj χ h ik. (1.10) Under a renormalization of the fundamental tensor of the form ğ ij = λ 2 g ij, (1.11) the complementary vector field T k is transformed by the law T k = T k + k lnλ, (1.12) where λ is a scalar function defined on GW n (L i jk,g ij, T k ). A quantity A is called a satellite of weight {p} of the tensor g ij, if it admits a transformation of the form Ă = λ p A (1.13) under the renormalization (1.11) of the metric tensor g ij of weight {p} is defined by [3], [6] k A = k A pt k A. (1.14) We note that the prolonged covariant derivative preserves the weight.

4 168 Hakan Demirbüker, Fatma Öztürk and Leyla Zeren Akgün The components of the mixed curvature tensor and the Ricci tensor of GW n (L i jk,g ij, T k ) are respectively L i jkl = x k Li jl x l Li jk + Li hk Lh jl Li hl Lh jk (1.15) L ij = L a ija. (1.16) by On the other hand, the scalar curvature of GW n (L i jk, g ij, T k ) is defined L = g ij L ij. (1.17) It is easy to see that the curvature tensor L i jkl of GW n (L i jk, g ij, T k ) can be written as L i jkl = B i jkl + χ i jkl (1.18) where the tensors B i jkl and χi jkl are defined respectively B i jkl = x k Γi jl x l Γi jk +Γi hk Γh jl Γi hl Γh jk, (1.19) χ i jkl = kχ i jl lχ i jk + χi hl χh jk χi hk χh jl 2χi jh χh kl. (1.20) The curvature tensor of GW n (L i jk, g ij,t k ) satisfies the relations [5] L h jkl + L h jlk =0 (1.21) L j hlk + Lj hkl + Lj klh =2( k χ j lh + lχ j hk + hχ j kl +2χj lm χm hk +2χj hm χm kl +2χj km χm lh ) (1.22) m L i jkl + k L i jlm + l L i jmk =2(Li jpl χp mk + Li jpk χp lm + Li jpm χp kl ). (1.23) 2. THE GENERALIZED RECURRENT WEYL SPACES HAVING A DECOMPOSABLE CURVATURE TENSOR

5 THE GENERALIZED RECURRENT WEYL SPACES If the curvature tensor L i jkl of GW n (L i jk,g ij, T k ) satisfies the condition m L i jkl = ψ ml i jkl (2.1) where ψ m is a covariant vector field, then GW n (L i jk,g ij, T k ) is called recurrent. In this section we consider the generalized recurrent Weyl spaces denoted by RGW n (L i jk, g ij, T k ) the curvature tensor of which is decomposed in the form L i jkl = vi ϕ jkl (2.2) where v i is a contravariant vector field of weight { 1} and ϕ jkl is a covariant tensor field of weight {1}. Using the relations (1.21) and (2.2) we get ϕ jkl + ϕ jlk =0. (2.3) Taking into account the relations (1.23), (2.1) and (2.2) we obtain ψ m ϕ jkl + ψ k ϕ jlm + ψ l ϕ jmk =2(ϕ jpl χ p mk + ϕ jpkχ p lm + ϕ jpmχ p kl ). (2.4) Multiplying both hand sides of (2.4) by v m and summing for m we have ϕ jkl = α[ψ l φ jk ψ k φ jl +2(ϕ jpl v m χ p mk + ϕ jpkv m χ p lm + φ jpχ p kl )] (2.5) where α is a scalar function of weight {1}which is defined by α = 1 μ (μ = vk ψ k ) (2.6) and φ jk is a covariant tensor field of weight {0} which is defined by φ jk = v l ϕ jkl. (2.7) Using the relations (1.16), (2.2) and (2.7) we obtain L jk = φ jk. (2.8)

6 170 Hakan Demirbüker, Fatma Öztürk and Leyla Zeren Akgün Using (2.8) in (2.5) we get ϕ jkl = α[ψ l L jk ψ k L jl +2(ϕ jpl v m χ p mk + ϕ jpkv m χ p lm + L jpχ p kl )]. (2.9) Multiplying the relation (2.3) by v k v l and summing for k and l and using the relations (2.7) and (2.8) we obtain L jk v k =0 (2.10) from which it follows that Thus we obtain the det(l jk )=0. (2.11) THEOREM 2.1: If a space RGW n (L i jk, g ij, T k ) has a decomposable curvature tensor in the form L i jkl = vi ϕ jkl, then we have ϕ jkl = α[ψ l L jk ψ k L jl +2(ϕ jpl v m χ p mk + ϕ jpkv m χ p lm + L jpχ p kl )] and det(l jk )=0. If the connection of RGW n (L i jk, g ij, T k ) is symmetric that is χ p mk =0 then the relations (2.9) reduces to ϕ jkl = α(ψ l L jk ψ k )L jl. (2.12) THEOREM 2.2: If a space RGW n (L i jk, g ij, T k ) has a decomposable curvature tensor in the form L i jkl = v i ϕ jkl, then the vector field v i and the tensor field ϕ jkl are recurrent. Proof: By (1.16) and (2.1) we get m L jk = ψ m L jk. (2.13)

7 THE GENERALIZED RECURRENT WEYL SPACES On the other hand, from (2.2) and (2.9) we can write the curvature tensor L i jkl of RGW n (L i jk,g ij, T k ) in the form L i jkl = αv i [ψ l L jk ψ k L jl +2(ϕ jpl v m χ p mk + ϕ jpkv m χ p lm + L jpχ p kl )]. (2.14) Taking the prolonged covariant derivative both hand sides of (2.14) with respect to the coordinate u m and using the relations (2.1) and (2.13) we obtain L a jkl m v i = L i jkl m v a. (2.15) If we use the relation (2.2) then the relation (2.15) reduces to v a m v i = v i m v a (2.16) from which it follows that m v i = λ m v i (2.17) where λ m is a covariant vector field of weight {0}. From the relation (2.16) we have the vector field v i is recurrent. On the other hand taking the prolonged covariant derivative of (2.2) with respect to the coordinate u m and using the relations (2.1) and (2.17) we get m ϕ jkl =(ψ m λ m )ϕ jkl. (2.18) From the relation (2.18) we have the tensor field ϕ jkl is recurrent. Now we consider the RGW n (L i jk, g ij, T k ) spaces having a decomposable curvature tensor in the form L i jkl = vi ψ j ϕ kl, ϕ jkl = ψ j ϕ kl (2.19) where ϕ kl is a covariant tensor field of weight {1} and concerning these spaces we prove the following theorem: THEOREM 2.3: The tensor field ϕ jkl may be decomposed in the form ϕ jkl = ψ j ϕ kl if and only if the condition m ψ j +(λ m α m μ)ψ j =0

8 172 Hakan Demirbüker, Fatma Öztürk and Leyla Zeren Akgün holds true. Proof: We first prove the necessity of the condition. Suppose that the tensor ϕ jkl is decomposed in the form (2.19). Under this condition the equation (2.18) transforms into (ψ m λ m )ψ j ϕ kl =( m ψ j )ϕ kl +( m ϕ kl )ψ j. (2.20) Taking the prolonged covariant derivative of the equation μ = v i ψ i with respect to u m and using (2.17) we obtain v i ( m ψ i )= μλ m m μ. (2.21) Multiplying both sides of (2.20) by v j and summing for j and taking (2.21) into account, we get m ϕ kl =(ψ m α m μ)ϕ kl. (2.22) Taking the prolonged covariant derivative of both sides of the equation L i jkl = vi ψ j ϕ kl with respect to u m and using (2.1), (2.17) and (2.22) we obtain ψ m L i jkl = ψ ml i jkl +( m ψ j + λ m ψ j αψ j m μ)ϕ kl v i (2.23) from which it follows that m ψ j +(λ m α m μ)ψ j =0. (2.24) Conversely, the condition (2.24) is sufficient. To see this, take the prolonged covariant derivative of both sides of (2.24) with respect to u p we get p m ψ j + p [(λ m α m μ)ψ j ]=0. (2.25) Interchanging the indices m and p in (2.25) and subtracting the equation so obtained from (2.25) we obtain ϕ jmp = α[ m (λ p α p μ) p (λ m α m μ)+2χ a mp (α a μ λ a )]ψ j (2.26) where we have used the relations (2.2), (2.24) and the Ricci identity [7].

9 THE GENERALIZED RECURRENT WEYL SPACES From (2.26) it follows that the tensor ϕ jmp may be written in the form ϕ jmp = ψ j ϕ mp where ϕ mp is given by ϕ mp = α[ m (λ p α p μ) p (λ m α m μ)+2χ a mp(α a μ λ a )]. (2.27) The proof of the theorem is completed.

10 174 Hakan Demirbüker, Fatma Öztürk and Leyla Zeren Akgün REFERENCES [1] G. ZLATANOV, Nets in the n-dimensional space of Weyl, C. R. Acad. Bulgare Sci. 41, (1988), [2] B. TSAREVA AND G. ZLATANOV, On the geometry of the nets in the n-dimensional space of Weyl, J. Geom. 38, (1990), [3] A. NORDEN, Affinely connected spaces, GRMFL Moscow, (1976). [4] V. MURGESCU, Espaces de Weyl generalises, Bul. Inst. Pol. de Jassy., (1970). [5] L. ZEREN AKGÜN, On generalized Weyl spaces Bull. Col. Math. Soc., 91, (4), (1999). [6] A. NORDEN AND S. YAFAROV, Theory of non-geodesic vector fields on two dimensional affinely connected spaces, Izv. Vuzov, Math. 12, (1974), [7] EISENHART, Non-Riemannian Geometry, Newyork, Published by the American mathematical society. Received: March 3, 2005