CURVATURE TENSORS IN SP-KENMOTSU MANIFOLDS WITH RESPECT TO QUARTER-SYMMETRIC METRIC CONNECTION Текст научной статьи по специальности «Математика»

CURVATURE TENSORS IN SP-KENMOTSU MANIFOLDS WITH RESPECT TO QUARTER.SYMMETRIC METRIC CONNECTION S. Sunitha Devi1, T. Satyanarayana2, K. L. Sai Prasad3,. � Department of Mathematics 1 Vignan�s Institute of Information Technology, Visakhapatnam, 530 049, INDIA 2 Pragati Engineering College, Surampalem, Near Peddapuram, Andhra Pradesh, India 3,. Gayatri Vidya Parishad College of Engineering for Women, Visakhapatnam, 530 048, INDIA [email protected] [email protected] [email protected],. Abstract A conformal curvature tensor and con-circular curvature tensor in an SP-Kenmotsu manifold are derived in this article which admits a quarter-symmetric metric connection. Conclusively, we veri.ed our results by considering a case of 3-D SP-Kenmotsu manifold. Keywords: .-Einstein manifold,SP-Kenmotsu manifold, con-circular curvature tensor, Quarter.symmetric metric connection, Ricci tensor, conformal curvature tensor. 2010 Mathematics Subject Classi.cation: 53C07, 53C25 I. Introduction A Mn (Riemannian manifold) is symmetrical locally if ..R = 0 and symmetric if R(X, Y).R = 0 where R(X, Y)Z = .X.YZ ..Y.XZ ..[X,Y]Z appears as a derivation. If R(X, Y).R = 0, then Mn is turns to be the pseudo symmetric space that is de.ned with the criteria R.R = L(g, R). A manifold Mn is conformally symmetric if ..C = 0 and if R.C = 0, it is said to be Weyl semi symmetric which are characterised by the condition R.C = LCQ(g, C). Schouten & Friedman proposed the concept of semi-symmetric linear connection on a differ.entiable manifold. Some of the semi-symmetric curvature criteria in Riemannian manifolds are given by Yano [12]. Semi symmetric metric connection plays a very signi.cant part in the geometry of Riemannian manifolds. For instance, a semi-symmetric metric is the displacement of the earth�s surface after a .xed point. A quarter-symmetric connection is a linear connection . on an n-dimensional Riemannian manifold (Mn, g) if Tis T(X, Y)= .(Y).X . .(X).Y. Sato [8] proposed concepts of almost para contact Riemannian manifold. In 1977, Matsumoto and Adati [1] characterized special para-Sasakian as well as para-Sasakian manifolds as a par.ticular type of almost contact Riemannian manifolds. Before Sato, Kenmotsu [6] characterized a type of this manifold. In 1995, Sinha and Sai Prasad [9] characterized a type of almost para contact metric manifolds mainly para-Kenmotsu and special para-Kenmotsu manifolds. For the literature, on Para-Kenmotsu manifolds one can refer to Balga [2], Srivastava and Srivastava [10], Olszak [7] . On the other hand, various geometers of Riemannian manifolds and speci.cally, SP-Sasakian manifolds were widely explored for the quarter-symmetric metric connections [3, 4, 5]. Inspired by these studies, in this work, we explore a class of special para-Kenmotsu manifolds that allow.ing the quarter-symmetric metric connection. The current study is arranged as follows: Section 2 has certain prerequisites. In relation to the quarter symmetric metric connection in an SP-Kenmotsu manifold, we derive the equations for the Ricci tensor S& Riemannian curvature tensor Rin Section 3. The equations in relation to quarter symmetric metric connection are also derived in an SP-Kenmotsu manifold Mn for con.circular curvature tensor Zin Section 4. It is illustrated that the manifold Mn is .-Einstein given the concircular curvature tensor Zmeets either of these conditions R(., U).Z= 0, Z(., U).R= 0, Z(., U).Z = 0, Z(X, Y).S = 0. Section 5 is intended to de.ne and analyse the curvature prop.erties in the quarter-symmetric metric connection of the Weyl-conformal curvature tensor C, of form (0, 4), of SP-Kenmotsu manifold Mn. Finally, an illustration of a 3d SP-Kenmotsu manifold is considered in Section 6. II. Preliminaries Suppose Mn be an n-dimensional differentiable manifold provided with structure tensors (., ., .) such that (a) .(.)= 1 (1) (b) .2(X)= X . .(X).; X = .X. Mn is called an almost para contact manifold. Suppose that g be a Riemannian metric such that, for all vector .elds X and Y on Mn (a) g(X, .)= .(X) (b) .. = 0, .(.X)= 0, rank . = n . 1 (2) (c) g(.X, .Y)= g(X, Y) . .(X).(Y). Then it is stated that the manifold [8] Mn accepts an almost para contact structure of Riemannian (., ., ., g). Furthermore, if (., ., ., g) ful.ls the equations (a)(.X.)Y . (.Y.)X = 0; (b)(.X.Y.)Z =[.g(X, Z)+ .(X).(Z)].(Y)+[.g(X, Y)+ .(X).(Y)].(Z); (3) (c) .X. = .2X = X . .(X).; (d)(.X.)Y = .g(X, .Y). . .(Y).X; then Mn is termed a para-Kenmotsu manifold or simply a P-Kenmotsu manifold [9]. A P-Kenmotsu manifold Mn permitting a 1-form . ful.lling (a)(.X.)Y = g(X, Y) . .(X).(Y); (4) (b)(.X.)Y = .(X, Y); here . signi.es . associate, is termed a special para-Kenmotsu manifold or shortly SP-Kenmotsu manifold [9]. Suppose (Mn, g) be an n-dimensional, n � 3, differentiable manifold of class C� and let . be its connection Levi-Civita. Then curvature tensor R of class (1, 3) of the the Riemannian Christoffel is provided by: R(X, Y)Z = .X.YZ ..Y.XZ ..[X,Y] Z. (5) The (0,2)-tensor S2 and the Ricci operator S are described as follows g(SX, Y)= S(X, Y), (6) and S2(X, Y)= S(SX, Y). (7) It is known [9] that the following relationship exist in the P-Kenmotsu manifold: (a) S(X, .)= .(n . 1).(X), (b) g[R(X, Y)Z, .]= .[R(X, Y, Z)] = g(X, Z).(Y) . g(Y, Z).(X), (8) (c) R(., X)Y = g(X, Y). . .(Y)X, (d) R(X, Y). = .(Y)X . .(X)Y; when X is orthogonal to .. Almost para-contact Riemannian manifold Mn is termed to be .-Einstein and form of its Ricci tensor S(X, Y)= ag(X, Y)+ b .(X) .(Y) (9) Fields X and Y for any vector; a and b are a few scalars on Mn. In speci.c, if b = 0 thus Mn is considered to be an Einstein manifold. III. Curvature tensor A linear connection .in a Riemannian manifold Mn is called a quarter-symmetric metric con.nection [4] if their torsion tensor T(X, Y) meets T(X, Y)= .(Y) .X . .(X) .Y, (10) and (.Xg)(Y, Z)= 0; (11) where . iindicates a tensor .eld of the form (1, 1) and . is a 1-form. A quarter-symmetric metric connection .with torsion tensor (10) is given by .XY = .XY + .(Y) .X . .(X, Y). (12) here, . indicates Riemannian connection. Suppose manifold Mn to be an SP-Kenmotsu manifold and .(X) as .X= X. Therefore the (10) and (11) may be represented as: T(X, Y)= .(Y)X . .(X)Y (13) (.Xg)(Y, Z)= 0. (14) Let us choose the linear and Riemannian connection as .and ., respectively .XY = .XY + U(X, Y), U is a tensor of type (1, 2) (15) We have [12], for .to be a quarter symmetric metric connection in Mn, U(X, Y)= 1/2[T(X, Y)+ T . (X, Y)+ T . (Y, X)], (16) where g(T . (X, Y), Z)= g(T(Z, X), Y)]. (17) Using (13) and (17), we get . T . (X, Y)= .(X)Y . F(X, Y).; (18) here .F(X, Y)= g(X, Y), . signi.es a 1-form and . indicates the associated vector .eld. From (13) and (16), in (18), we have . U(X, Y)= .(Y)X . F(X, Y)., (19) and then (15) becomes . .XY = .XY + .(Y)X . F(X, Y).; (20) which indicates .in an SP-Kenmotsu manifold. Suppose R and R be the curvature tensors of the connections . and . correspondingly, we get R(X, Y)Z = .X.YZ ..Y.XZ ..[X, Y]Z (21) Using (20) and (5) in (21), we have R(X, Y)Z = R(X, Y)Z + g(Y, Z)X . g(X, Z)Y. (22) If we describe R(X, Y, Z, U) as g(R(X, Y)Z, U) and R(X, Y, Z, U) as g(R(X, Y)Z, U); then (22) becomes R(X, Y, Z, U)= R(X, Y, Z, U)+ g(Y, Z)g(X, U) . g(X, Z)g(Y, U). (23) The above expression (23) denotes the relation between R(X, Y)Z of Mn w.r.t. . and R(X, Y)Z w..r.t. .. Put X = U = ei in (23), where ei be an orthonormal basis of the tangent space at any point of the manifold and taking summation over i (1 � i �n), we get S(Y, Z)= S(Y, Z)+ ng(Y, Z) . .(Y).(Z); (24) here Sand S signi.es the Ricci tensors of .and .. From (24), by using Y = Z = ei, we obtain r = r + n2 . 1; (25) here r and r indicates the scalar curvatures of .and . correspondingly. Theorem 3.1: Suppose that Sbe the Ricci tensor & Rbe the curvature tensor in an SP-Kenmotsu manifold Mn w.r.t. ., then (a) R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0, (b) R(X, Y, Z, U)+ R(X, Y, U, Z)= 0, (c) R(X, Y, Z, U) . R(Z, U, X, Y)= 0, (d) R(X, Y, Z, .)= 2R(X, Y, Z, .), (e) S(X, .)= 2S(X, .). Proof: Using .rst Bianchi identity and eq.(22) w.r.t. the Riemannian connection, we obtain (a). From eq. (23), we obtain (b) & (c). By putting U = . in (23) and by using (8) we have (d). By using Y = Z = ei in equation (d) as well as summation with i, we obtain (e). Theorem 3.2: The Ricci tensor S in an SP-Kenmotsu manifold Mn w.r.t. the connection for the quarter-symmetric metric is symmetrical. Proof: The theorem-proof is based on the eq. provided in (24). IV. Concircular curvature tensor The n-dimensional Riemannian manifold Mn is provided by the concircular curvature tensor Z(X, Y) [11, 13]: r Z(X, Y)U = R(X, Y)U . [g(Y, U)X . g(X, U)Y] (26) n(n . 1) for all X,Y, U . TM. The concircular curvature tensor w.r.t. .in an SP-Kenmotsu manifold is Z. Therefore, using the equations (22) and (26), we get Z(X, Y)U = Z(X, Y)U . 1 [g(Y, U)X . g(X, U)Y], (27) n which denotes the relation between the concircular curvature tensors w.r.t. .and .. Theorem 4.1: If Z w.r.t. . in an SP-Kenmotsu manifold satis.es R(., U).Z= 0, the mani.fold is .-Einstein. Proof: Suppose R(., U).Z(X, Y). = 0, in an SP-Kenmotsu manifold. Then (R(., U).Z(X, Y).) . Z(R(., U)X, Y). . Z(X, R(., U)Y). . Z(X, Y).R(., U). = 0. (28) Also, from (8) and (22), we get R(X, Y). = 2[.(Y)X . .(X)Y] and (29) R(., X)U = 2[g(X, U). . .(U)X]. (30) Then, by using (28), (29) and (30), we get Z(X, Y)U = 0. (31) Now, using the equations (26) and (27), the equation (31) reduces to r + n . 1 R(X, Y, U)= [g(Y, U)X . g(X, U)Y]. (32) n(n . 1) We obtain with the above equation w.r.t. X, r + n . 1 S(Y, U)= [ng(Y, U)X . .(Y).(U)], (33) n(n . 1) which on further contracting, we get r = 1 . n2. (34) Using (34), the expression (33) becomes S(Y, U)= .(Y).(U) . ng(Y, U); (35) which proves .-Einstein manifold. Theorem 4.2: If Zwith respect to . in an SP-Kenmotsu manifold satis.es Z(., U).R = 0, the manifold is an .-Einstein. Proof: Suppose that Z(., U).R(X, Y). = 0, in an SP-Kenmotsu manifold. Then (Z(., U).R(X, Y).) . R(Z(., U)X, Y). . R(X, Z(., U)Y). . R(X, Y).Z(., U). = 0 (36) Also, from (8), (26) and (27), we have [][] r 1 Z(., U)Y =+ . 1 g(U, Y). . .(Y)U (37) n(n . 1) n and [][]r 1 Z(X, Y). = + . 1.(X)Y . .(Y)X. n(n . 1) n By substituting the values from (29), (30), (37) and (38) in the expression (36), we obtain (38) R(X, Y)U = g(U, Y)X . g(U, X)Y + .(U)[1 . .(X)]Y. (39) Using (22), the above eq. becomes R(X, Y)U = .(U)[1 . .(X)]Y; (40) and it proves. Theorem 4.3: If the Z w.r.t. . in an SP-Kenmotsu manifold meets Z(., U).Z= 0, the mani.fold is .-Einstein. Proof: The theorem-proof is trivial by the use of the the fact that Z(., U).Zindicates Z(., U) was acting on Zas a derivation. Theorem 4.4: If Z (concircular curvature tensor) with respect to .(quarter symmetric metric connection) in an SP-Kenmotsu manifold ful.lls Z(X, Y).S = 0, the manifold signi.es .-Einstein. Proof: Let Z(X, Y).S(U, V)= 0 in an SP-Kenmotsu manifold. Then it means S(Z(X, Y)U, V)+ S(U, Z(X, Y)V)= 0. (41) By choosing X = . in (41) and on using the equations (37) and (24), we obtain [][ r 1 + . 1 . .(U)S(Y, V) . n.(U)g(Y, V)+ 2.(U).(V).(Y) n(n . 1) n (42) ] . .(V)S(U, Y) . n.(V)g(U, Y)= 0. Again by using U = . in the eq. (42), we get S(Y, V)= .(Y).(V) . ng(Y, V); (43) which provides the required result. V. Conformal curvature tensor The Weyl conformal curvature tensor C of the type (0, 4) of a manifold Mn w.r.t. a Riemannian connection provided by [12, 13]: 1 C(X, Y, Z, U)= R(X, Y, Z, U) . [S(Y, Z)g(X, U) . S(X, Z)g(Y, U) n . 2 + g(Y, Z)S(X, U) . g(X, Z)S(Y, U)] (44) r +[g(Y, Z)g(X, U) . g(X, Z)g(Y, U)]. (n . 1)(n . 2) Analogous to this, we de.ne C i.e.Weyl conformal curvature tensor of the type (0, 4), of an SP-Kenmotsu manifold w.r.t. the quarter-symmetric metric connection as: 1 C(X, Y, Z, U)=R(X, Y, Z, U) . [S(Y, Z)g(X, U) . S(X, Z)g(Y, U) n . 2 + g(Y, Z)S(X, U) . g(X, Z)S(Y, U)] (45) r +[g(Y, Z)g(X, U) . g(X, Z)g(Y, U)]. (n . 1)(n . 2) Then, using the equations (23), (24), (25), (44) and (45), we get C(X, Y, Z, U)= C(X, Y, Z, U), (46) which implies the following statement: Theorem 5.1: The conformal curvature tensors of .and . are equal in an SP-Kenmotsu mani.fold. Suppose that R= 0. Then S= 0 and r = 0. From (45) we get that C= 0 and hence using (46), we get C = 0. Therefore, we provide the following theorem. Theorem 5.2: The manifold is conformally .at in an SP-Kenmotsu manifold if the conformal curvature tensor Cof .vanishes. Let S= 0. Then r = 0. Hence from (24) and (25), we get S(Y, Z)= .(Y).(Z) . ng(Y, Z) (47) and r = 1 . n2. (48) Then by using (23), (44), (47) and (48), we obtain R(X, Y, Z, U)= C(X, Y, Z, U). (49) From (49), we state that Theorem 5.3: Conformal curvature tensor C of the manifold is identical in an SP-Kenmotsu manifold if S(Ricci tensor) of . i.e quarter-symmetric metric connection vanishes, then Ri.e. curvature tensor of .. Using theorem (5.2) and (5.3), we state that Theorem 5.4: If S of . in an SP-Kenmotsu manifold disappears, then the manifold is con.formally .at if Rof .vanishes. VI. Example of a 3d SP-Kenmotsu manifold admitting the quarter-symmetric metric connection Example 6.1: Suppose that 3d manifold M = {(x, y, u) . R3}, where (x, y, u) indicates "standard coordinates" in R3. Considering e1, e2& e3 be .elds of vector in M as . .. .u .u e1 = e , e2 = e , e3 = . (50) .x .y .u for each point of M are linearly independent vectors and constitute a basis of .(M). Riemannian metric g(X, Y) is . .1, if i = j g(ei, ej)= .0, if i .= j; i, j = 1, 2, 3, 4, 5. Let.(Z)= g(Z, e3), f or any Z . .(M) Let . be a 1-form & (1, 1)-tensor .eld on M expressed by . de.ned as .2(e1)= e1, .2(e2)= e2, .2(e3)= 0. The g(X, Y) and linearity of . yields that .(e3)= 1, .2(X)= X . .(X)e3; and g(.X, .Y)= g(X, Y) . .(X).(Y) for all vector .elds X, Y . .(M). Thus for e3 = ., (., ., ., g) describes an almost para-contact structure in M. Let . be a Riemannian connection in regard to the Riemannian metric g. [] e1, e2 = 0, [e1, e3]= e1, [e2, e3]= e2. The formula of Koszul�s is 2g(.XY, Z)= Xg(Y, Z)+ Yg(Z, X) . Zg(X, Y) (51) . g(X, [Y, Z]) . g(Y, [X, Z]) + g(Z, [X, Y]). By taking e3 = . in (51), one can get .e1 e1 = .e3, .e1 e2 = 0, .e1 e3 = e1; .e2 e1 = 0, .e2 e2 = .e3, .e2 e3 = e2; .e3 e1 = 0, .e3 e2 = 0, .e3 e3 = 0. Therefore manifold under consideration satis.es .X. = .2X = X . .(X)., .(.) = 1 and the expression (3)d. The above expressions satisfy all the properties of SP-Kenmotsu manifold with (., ., ., g) . Thus M(., ., ., g) is a 3-dimensional manifold. Further from (20), we get .e1 e1 = .2e3, .e1 e2 = 0, .e1 e3 = 2e1; .e2 e1 = 0, .e2 e2 = .2e3, .e2 e3 = 2e2; .e3 e1 = 0, .e3 e2 = 0, .e3 e3 = 0; Therefore T(X, Y) of .can be expressed as: T(ei, ei)= 0, for i = 1, 2, 3; and T(e1, e2)= 0, T(e1, e3)= e1, T(e2, e3)= e2. 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