The Projective Quarter Symmetric Metric Connections and Their Curvature Tensors

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1 Interntion Mtemtic Forum, 4, 009, no. 48, e Projective Qurter Symmetric Metric Connections nd eir Curvture ensors Hüy BAGDALI YILMAZ e University of Mrmr, Fcuty of Sciences nd Letters Deprtment of Mtemtics, Istnbu urkey bgdti@mrmr.edu.tr Aynur UYSAL e University of Dogus, Fcuty of Sciences nd Letters Deprtment of Mtemtics, Istnbu urkey uys@dogus.edu.tr Abstrct In tis pper, te existence of te projective qurter symmetric metric connection is proved in Riemnnin mnifods. In prticur two cses, tis connection reduces to semi-symmetric metric connection nd to projective semi-symmetric connection. Furtermore, we study scr curvture of Riemnnin mnifods wit keeping te covrint derivtive of tensor W j. Mtemtics Subject Cssifiction: 53A07, 53B5 Keywords:Projective qurter symmetric metric connection, e projective curvture tensor, -form Introduction In 970, Yno ], studied Riemnnin mnifods dmitting semi-symmetric metric connections wose curvture tensors vnis. In 980, Misr nd Prdey 6], studied qurter symmetric metric connections in Riemnnin, Keerin nd Sskin mnifods nd found some properties of curvture tensors of tem. In 98, Yno nd Imi 3], gve te most gener form of qurter symmetric metric connections nd studied its ppictions. In 008, Zo 5], investigted te properties of projective semi-symmetric metric connections of Riemnnin mnifod nd gve some interesting resuts wit respect to tis semi -symmetric connection.

2 370 H. BAGDALI YILMAZ nd A. UYSAL In te present pper, we define te projective qurter symmetric metric connections nd we so find formu for scr curvture of Riemnnin mnifods wit keeping te covrint derivtive of tensor W j. Projective Qurter Symmetric Metric Connection Let M be n-dimension Riemnnin mnifod of css C wit metric tensor g, nd be Levi- Cevit connection ssocited wit g. en, te condition Xk gx i, X j )= g ij x { } k gj { } jk gi = 0.) ods. A iner connection on M wose torsion tensor X, Y )= X Y Y X X, Y ] X, Y χm) wic is given by X, Y )=πy )tx) πx)ty ).) were π nd t re -form nd tensor of type, ), respectivey, is sid qurter symmetric connection, 6]. Aso, iner connection stisfying.) nd te condition X g ) Y,Z) = 0 were X, Y, Z χm), is ced qurter symmetric metric connection 6]. Here, χm) denote te set of differentibe vector fieds on M. If te geodesics wit respect to re wys consistent wit tose of, ten is sid projective equivent connection wit,5]. Definition If is bot te projective equivent connection wit nd te qurter symmetric metric connection, ten is ced te projective qurter symmetric metric connection. eorem Let be iner connection on M. is te projective qurter symmetric metric connection if nd ony if te coefficients Γ of te connection stisfy Γ = { } + + P.3) ) were = π k t i π i t k nd P = k δi + iδ k.

3 Projective qurter symmetric metric connections 37 Proof. Let be Xk gx i, X j )= g ij x k Γ g j Γ jk g i = 0.4) Subtrcting.4) from.), we ve Γ { }) gj + Γ jk { jk }) gi = 0.5) Now, we define tensor θ by θx, Y )= X Y X Y, X, Y χm). us, we cn write.5) in te foowing form θ g j + θ jk g i = 0.6) were te tensor θ is in te form θ =Γ { }.7) Hence, te torsion tensor ij wit te id of.6), we get cn be written s ij = θ ij θ ji.8) + θ ki) gj + jk + θ kj) gi =0 If te indices i, j, k) in.6) re cnged cycicy nd tis eqution for ec oter order is rewrote, te foowing equtions re obtined: θg j + θjkg i = 0 θji g k + θki g j = 0 θkj g i + θij g k = 0 Let us dd first two equtions nd subtrct te st eqution.us, we get ) θ + θki gj + θjk ) θ kj gi + θji ) θ ij gk =0.9) Using.8), we ve θ + θ ki ) gj + jkg i + jig k = 0.0) Since θ + θ ki =θ + ki.)

4 37 H. BAGDALI YILMAZ nd A. UYSAL e eqution.0) cn be written s θ + ki ) gj + jk g i + ji g k = 0.) If we mutipy te bove eqution by g j, we obtin θ = ki + jkg i g j + jig k g j).3) As sid before, is te projective equivent to if nd ony if te symmetric prt θ cn be written s θx, Y )+θy,x) =ΨY )X +ΨX)Y were Ψ is -form. erefore, we find X, Y χm) Contrcting for nd j in.3), we get θ + θ ki = jkg i g j + jig k g j) θ = ki + kδ i + iδk) Since te connection is projective qurter symmetric metric connection, by.), te torsion tensor is = π kt i π it k, were π is -form nd t is tensor of type, ). Hence, if we denote tensor k δi + ) i δ k by P, ten te projective qurter symmetric metric connection is given s foows wen.7) is used, Γ = { } + + P is competes te proof of eorem. Now, et us investigte some prticur cses for te connection.3).. In prticury, if P = 0 nd tensor of type, ) t i in = π kt i π it k is coiced t i = δi, ten te connection.3) is reduced to te semi -symmetric metric connection.t is to sy, Γ = { } + πk δi π iδk) tking φ k = π k nd using φ i = g im φ m. Hence,we ve, ], Γ = { } + φk δ i g φ.4)

5 Projective qurter symmetric metric connections 373. If P 0 nd we coice t i = δ i in te projective qurter symmetric metric connection, ten te connection.3) is reduced to projective semi -symmetric connection. = π kt i π it k = = π kδi π iδk.5) nd since P = k δi + i k) δ P n) = π i δk using ψ i = n) π i in.6), we get, 5] n) + π k δi.6) Γ = { } + πk δ i π i δ k + ψ i δ k + ψ k δ i.7) 3 e Riemnnin Mnifods wit Keeping te Covrint Derivtive of ensor W j We denote by R j R j = Γ x j Γ ij x k +Γ Γ j Γ ij Γ k te curvture tensor of te connection, 4]. We compute te curvture tensor using.3), we find R j = R j + k ij j + kp ij jp + 4 ] ij k j + P ij Pk PP j ] + ] ij Pk P j 3.) were Rj re te curvture tensor corresponding to. It is esiy sown tt 3.) stisfies te foowing properties: i) R mj = R mijk ii) R mk = 0 Contrcting 3.) wit respect to index nd j, we find : 3.)

6 374 H. BAGDALI YILMAZ nd A. UYSAL R = R + k j ij j j + kp j ij j P j + 4 ij j k j ] j + P ij P j k P P j j ] + ij P j k P j ] j 3.3) were R nd R re te Ricci tensor corresponding to nd, respectivey. Using 3.) nd 3.3) in te formu of te projective curvture tensor wit repect to wic is given by te formu ], we ve W j = R j R δj n R ijδk) 3.4) W j = Rj + kij j + kpij jp ij k ] j + P ij Pk P P ] j n n ) 4n ) n ) n ) ij Pk P ] j R δj R ijδk) k i n ) + ji ] ij k Pi P + jpi ] Pij i k ij + i ] j P i P k P P P ijp + P ip j ] i Pk P ijp + ip j ] 3.5) Furtermore, it cn be esiy obtined tt te projective curvture tensor 3.5) stisfies te foowing property W k = ) R R n ) Let us denote te covrint derivtive of nd by nd, respectivey. Using.3) one cn obtin:

7 Projective qurter symmetric metric connections 375 W j m = W j x m +Γ rm W r j Γr im W rkj Γr km W irj Γr jm W r ] ] = W j,m + W r j rm + P rm W rkj im r + P im r ] ] W irj km r + Pkm r W r jm r + Pjm r us, we ve were W j m = W j,m + W jm 3.6) ] ] ] ] W jm W r j rm + P rm W rkj im r + P im r W irj km r + P km r W r jm r + P jm r 3.7) us, we ve : eorem W j m = W j,m if nd ony if W jm =0. eorem 3 Let M be mnifod dmitting te connection.4). If W j m = W j,m, tere exists te foowing formu for -form π π i kj π i jk = ] R π j + n ) n +) nr n 4) kj R jk )π i + n +) R ijπ k Proof. From eorem 3 nd 3.7), we write W r j ] ] ] ] rm + Prm W rkj im r + Pim r W irj km r + Pkm r W r jm r + Pjm r =0 3.8) Moreover, substituting t i = δi in.), we obtin = π kδ i π iδ k 3.9) After contrcting 3.8) wit respect to index nd m, substituting 3.9) in 3.8), nd since P = 0, we obtin te foowing eqution n +)W m jπ m + W m mkjπ i + W m imjπ k + W m mπ j = 0 3.0)

8 376 H. BAGDALI YILMAZ nd A. UYSAL Wit te ep of 3.4), 3.0) cn be written in te form R m jπ m = n ) R ] π j + nrkj R jk )π i +n 4)R ij π k n + )n ) Since R m j π m = π i kj π i jk, we get π i kj π i jk = n ) R π j + 3.) { } ] nrkj R jk )π i +n 4)R ij π k 3.) n +) Concusion 3. Let M be mnifod dmitting te connection.4).ifw j m = W j,m nd π i kj = π i jk, ten scr curvture R of M is te foowing form R = nrkj R n +)π jk )π k π j +n 4)R ij π i π j] eorem 4 Let M be mnifod dmitting te connection.7). If W j m = W j,m, tere exists te foowing formu for -form π { } ] π i kj π i jk = R π j + nnrkj R n ) n jk )π i +n 4)R ij π k n + 4)n ) Proof. e proof is s bove te proof. Concusion 3. Let M be mnifod dmitting te connection.7).ifw j m = W j,m nd π i kj = π i jk, ten scr curvture R of M is te foowing form References R = nnrkj R n n +4)π jk )π k π j +n 4)R ij π i π j] ] K. Yno, On Semi-Symetric Metric Connection, Rev., Roum., Mt., Pureset App., 5 970), ] K. Yno, Some Remrks On ensor Fied nd Curvture, Anns of Mtemtics No:, Vo. 5595), ] K. Yno nd. Imi, Qurter Symmetric Metric Connections nd eir Curvture ensors, ensor N.S., Vo. 3898). 4] N. J. Hicks, Notes on Differenti Geometry,Princeton, H.J., Vn Nostrnd965). 5] P. Zo, Some Properties of Projective Semi-Symmetric Connections, Interntion Mtemtic Forum, 3, No: 7, 008), ] R.S. Misr nd S. N. Prdey, On Qurter Symmetric Metric F-Connection, ensor N.S., Vo ) Received: My, 009