Half Lightlike Submanifolds of an Indefinite Generalized Sasakian Space Form with a Quarter-Symmetric Metric Connection

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1 International Mathematical Forum, Vol. 10, 2015, no. 3, HIKARI Ltd, Half Lightlike Submanifolds of an Indefinite Generalized Sasakian Space Form with a Quarter-Symmetric Metric Connection Dae Ho Jin Department of Mathematics, Dongguk University Gyeongju , Republic of Korea Copyright c 2015 Dae Ho Jin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We study the geometry of half lightlike submanifolds of an indefinite generalized Sasakian space form M(f 1, f 2, f 3 ) with a quarter-symmetric metric connection. Our main result is several characterization theorems for such a half lightlike submanifold of M(f1, f 2, f 3 ) endowed with an indefinite trans-sasakian structure of type (α, β). Mathematics Subject Classification: 53C25, 53C40, 53C50 Keywords: quarter-symmetric connection, metric connection, half lightlike submanifold 1 Introduction The theory of lightlike submanifolds is an important topic of research in differential geometry due to its application in mathematical physics. The study of such notion was initiated by Duggal and Bejancu [5] and then studied by many authors [7, 8]. Half lightlike submanifold M [6] is a lightlike submanifold of codimension 2 such that rank{rad(t M)} = 1, where Rad(T M) is the radical distribution of M. It is a special case of an r-lightlike submanifold [5] such that r = 1. Much of its theory will be immediately generalized in a formal way to r-lightlike submanifolds. Its geometry is more general than that of lightlike

2 128 Dae Ho Jin hypersurfaces or coisotropic submanifolds [4] which are lightlike submanifolds M of codimension 2 such that rank{rad(t M)} = 2. A linear connection on a semi-riemannian manifold ( M, ḡ) is said to be a quarter-symmetric metric connection if it is metric, i.e., ḡ = 0 and its torsion tensor T, defined by T = X Y Y X [X, Y ], is satisfied T (X, Y ) = θ(y )JX θ(x)jy, (1.1) for any vector field X and Y on M, where J is a (1, 1)-type tensor field and θ is a 1-form associated with a smooth vector field ζ by θ(x) = ḡ(x, ζ). It have been studied by K. Yano and T. Imai [12] and later studied by many authors. Although now we have lightlike version of a large variety of Riemannian submanifolds, the theory of lightlike submanifolds of semi-riemannian manifolds with quarter-symmetric metric connections is few known. In this paper, we study the geometry of half lightlike submanifolds of an indefinite generalized Sasakian space form with a quarter-symmetric metric connection, in which the tensor field J, the 1-form θ and the vector field ζ defined by (1.1) are identical with the tensor field J, the 1-form θ and the vector field ζ of the indefinite trans-sasakian structure (J, θ, ζ, ḡ) on M. 2 Preliminaries Let (M, g) be a half lightlike submanifold of a semi-riemannian manifold ( M, ḡ) with the following objects; the radical distribution Rad(T M) = T M T M, a screen distribution S(T M), and a coscreen distribution S(T M ). We follow Duggal and Jin [6] for notations and structure equations used in this article. Then we obtain the following two decompositions: T M = Rad(T M) orth S(T M), T M = Rad(T M) orth S(T M ), where T M and T M are the tangent and normal bundles of M. Denote by F (M) the algebra of smooth functions on M, by Γ(E) the F (M) module of smooth sections of a vector bundle E over M and by (. ) i the i-th equation of (. ). We use the same notations for any others. It is known [6] that, for any null section ξ of Rad(T M) on a coordinate neighborhood U M, there exists a uniquely defined null vector field N Γ(S(T M ) ) satisfying ḡ(ξ, N) = 1, ḡ(n, N) = ḡ(n, X) = ḡ(n, L) = 0, X Γ(S(T M)). Denote by ltr(t M) the vector subbundle of S(T M ) locally spanned by N. Then we show that S(T M ) = Rad(T M) ltr(t M). Let tr(t M) = S(T M ) orth ltr(t M). We call N, ltr(t M) and tr(t M) the lightlike transversal vector field, lightlike transversal vector bundle and transversal vector bundle

3 Half lightlike submanifolds of an indefinite generalized Sasakian space form 129 of M with respect to the screen distribution S(T M) respectively. Let L be a unit spacelike vector field of S(T M ) without loss of generality. Let X, Y, Z and W be the vector fields on M, unless otherwise specified. As the tangent bundle T M of M is satisfied T M = T M tr(t M) = T M ltr(t M) orth S(T M ), the Gauss and Weingartan formulas of M are given respectively by X Y = X Y + B(X, Y )N + D(X, Y )L, (2.1) X N = A N X + τ(x)n + ρ(x)l, (2.2) X L = A L X + φ(x)n, (2.3) where is the linear connection on M, B and D are the local second fundamental forms of M, A N and A L are their shape operators, and τ, ρ and φ are 1-forms on T M. Let P be the projection morphism of T M on S(T M) and η a 1-form such that η(x) = ḡ(x, N). As T M = S(T M) orth Rad(T M), then the Gauss and Weingartan formulas of S(T M) are given respectively by X P Y = XP Y + C(X, P Y )ξ, (2.4) X ξ = A ξx τ(x)ξ, (2.5) where is the linear connection on S(T M), C is the local screen second fundamental form of S(T M), A ξ is its shape operator. Note that B and D are not symmetric. As ḡ( X ξ, ξ) = 0 and ḡ( X ξ, L) = ḡ(ξ, X L), from (2.1) and (2.3) we obtain B(X, ξ) = 0, D(X, ξ) = φ(x), (2.6) X ξ = A ξx τ(x)ξ φ(x)l. (2.7) The local second fundamental forms are related to their shape operators by B(X, Y ) = g(a ξx, Y ), ḡ(a ξx, N) = 0, (2.8) C(X, P Y ) = g(a N X, P Y ), ḡ(a N X, N) = 0, (2.9) D(X, Y ) = g(a L X, Y ) φ(x)η(y ), ḡ(a L X, N) = ρ(x). (2.10) 3 Indefinite trans-sasakian manifolds An odd-dimensional semi-riemannian manifold ( M, ḡ) is called an indefinite trans-sasakian manifold [11] if there exist a structure set {J, ζ, θ, ḡ} and two smooth functions α and β, where J is a (1, 1)-type tensor field, ζ is a vector field which is called the structure vector field and θ is a 1-form such that J 2 X = X + θ(x)ζ, θ(ζ) = 1, θ(x) = ɛḡ(x, ζ), (3.1) θ J = 0, ḡ(jx, JY ) = ḡ(x, Y ) ɛθ(x)θ(y ),

4 130 Dae Ho Jin ( X J)Y = α{ḡ(x, Y )ζ ɛθ(y )X} + β{ḡ(jx, Y )ζ ɛθ(y )JX}, (3.2) for any vector fields X and Y on M, where ɛ = 1 or 1 according as the vector field ζ is spacelike or timelike respectively. In this case, the set {J, ζ, θ, ḡ} is called an indefinite trans-sasakian structure of type (α, β). Note that if β = 0, then M is called an indefinite α-sasakian manifold. Indefinite Sasakian manifold is an example of indefinite α-sasakian manifold with α = 1 or 1 [9]. If α = 0, then M is called an indefinite β-kenmotsu manifold. Indefinite Kenmotsu manifold is an example of indefinite β-kenmotsu manifold such that β = 1. Indefinite cosymplectic manifold is an another important kind of indefinite trans-sasakian manifold such that α = β = 0. In this paper, we may assume that the structure vector field ζ is unit spacelike, i.e., ɛ = 1, without loss generality. From (3.1) and (3.2), we get X ζ = αjx + β(x θ(x)ζ), dθ(x, Y ) = ḡ(x, JY ). (3.3) Let M be a half lightlike submanifold of a indefinite trans-sasakian manifold M such that the structure vector field ζ is tangent to M. Cǎlin [2] proved that if ζ is tangent to M, then it belongs to S(T M) which we assumed in this paper. It is known [10] that, for any half lightlike submanifold M of an indefinite trans-sasakian manifold M, J(Rad(T M)), J(ltr(T M)) and J(S(T M )) are subbundles of S(T M), of rank 1. Thus there exists a non-degenerate almost complex distribution H o with respect to J, i.e., J(H o ) = H o, such that S(T M) = {J(Rad(T M)) J(ltr(T M))} orth J(S(T M )) orth H o. Denote by H the almost complex distribution with respect to J such that H = Rad(T M) orth J(Rad(T M)) orth H o, and denote by H the distribution on S(T M) such that H = J(ltr(T M)) orth J(S(T M )). Then the decomposition of the tangent bundle T M of M is reduced to T M = H H Consider two local null vector fields U and V, a local unit spacelike vector field W on S(T M), and their 1-forms u, v and w defined by U = JN, V = Jξ, W = JL, (3.4) u(x) = g(x, V ), v(x) = g(x, U), w(x) = g(x, W ). (3.5)

5 Half lightlike submanifolds of an indefinite generalized Sasakian space form 131 Let S be the projection morphism of T M on H and F the tensor field of type (1, 1) globally defined on M by F = J S. Then JX is expressed as JX = F X + u(x)n + w(x)l. (3.6) Applying J to (3.6) and using (3.1) and (3.4), we have F 2 X = X + u(x)u + w(x)w + θ(x)ζ. (3.7) Applying X to (3.4) (3.6) by turns and using (2.1), (2.2), (2.3), (2.6) (2.8), (2.10) and (3.4) (3.6), we have B(X, U) = C(X, V ), B(X, W ) = D(X, V ), C(X, W ) = D(X, U), (3.8) X U = F (A N X) + τ(x)u + ρ(x)w {αη(x) + βv(x)}ζ, (3.9) X V = F (A ξx) τ(x)v φ(x)w βu(x)ζ, (3.10) X W = F (A L X) + φ(x)u βw(x)ζ, (3.11) ( X F )(Y ) = u(y )A N X + w(y )A L X B(X, Y )U D(X, Y )W (3.12) + α{g(x, Y )ζ θ(y )X} + β{ḡ(jx, Y )ζ θ(y )F X}, ( X u)(y ) = u(y )τ(x) w(y )φ(x) βθ(y )u(x) B(X, F Y ), (3.13) ( X v)(y ) = v(y )τ(x) + w(y )ρ(x) θ(y ){αη(x) + βv(x)} (3.14) g(a N X, F Y ), ( X w)(y ) = u(y )ρ(x) βθ(y )w(x) D(X, F Y ). (3.15) Denote by R, R and R the curvature tensors of the connections, and respectively. Using the local Gauss-Weingarten formulas for M and S(T M), we have the Gauss equations for M and S(T M) such that R(X, Y )Z = R(X, Y )Z + B(X, Z)A N Y B(Y, Z)A N X (3.16) + D(X, Z)A L Y D(Y, Z)A L X + {( X B)(Y, Z) ( Y B)(X, Z) + τ(x)b(y, Z) τ(y )B(X, Z) + φ(x)d(y, Z) φ(y )D(X, Z) θ(x)b(f Y, Z) + θ(y )B(F X, Z)}N, + {( X D)(Y, Z) ( Y D)(X, Z) + ρ(x)b(y, Z) ρ(y )B(X, Z) θ(x)d(f Y, Z) + θ(y )D(F X, Z)}L, R(X, Y )P Z = R (X, Y )P Z + C(X, P Z)A ξy C(Y, P Z)A ξ X (3.17) + {( X C)(Y, P Z) ( Y C)(X, P Z) τ(x)c(y, P Z) + τ(y )C(X, P Z) In the case R = 0, we say that M is flat. θ(x)c(f Y, P Z) + θ(y )C(F X, P Z)}ξ.

6 132 Dae Ho Jin 4 Quarter-symmetric metric connection Theorem 4.1. Let M be a half lightlike submanifold of an indefinite trans- Sasakian manifold M admitting a quarter-symmetric metric connection. Then the second fundamental forms B and D are never symmetric and the functions α and β, defined by (3.2), are satisfied β(α 1) = 0. Proof. Substituting (3.6) into (3.3) 1 and using (2.1), we see that X ζ = αf X + β(x θ(x)ζ), (4.1) B(X, ζ) = αu(x), D(X, ζ) = αw(x). (4.2) Applying X to ḡ(ζ, N) = 0 and using (2.2), (3.1) (3.3) 1 and (3.5), we have C(X, ζ) = αv(x) + βη(x). (4.3) Substituting (2.1) and (3.5) into (1.1) and comparing the tangent, co-screen and lightlike transversal components of the resulting equation, we get T (X, Y ) = θ(y )F X θ(x)f Y, (4.4) B(X, Y ) B(Y, X) = θ(y )u(x) θ(x)u(y ), (4.5) D(X, Y ) D(Y, X) = θ(y )w(x) θ(x)w(y ), (4.6) where T is the torsion tensor with respect to the induced connection on M. If B is symmetric, then, from (4.5), we have θ(x)u(y ) = θ(y )u(x). Taking X = ζ and Y = U, we have 1 = 0. It is a contradiction. Thus B is never symmetric. Similarly, we see that D is also never symmetric. Applying Y to (4.1), we obtain X Y ζ = (Xα)F Y α( X F )Y αf ( X Y ) + (Xβ)Y + β X Y + αβθ(y )F X β 2 θ(y )X {(Xβ)θ(Y ) + βx(θ(y )) β 2 θ(x)θ(y )}ζ. Using this, (3.3) 2, (3.7), (3.12) and (4.4) (4.6), we have R(X, Y )ζ = (Xα)F Y + (Y α)f X + (Xβ)Y (Y β)x (4.7) + α{u(x)a N Y u(y )A N X + w(x)a L Y w(y )A L X} + (α 2 + α β 2 ){θ(y )X θ(x)y } + β(1 + 2α){θ(Y )F X θ(x)f Y } {(Xβ)θ(Y ) (Y β)θ(x) + 2β(1 α)dθ(x, Y )}ζ. Replacing Z by ζ to (3.16) and then, taking the scalar product with ζ and using (4.2) and the fact that ḡ( R(X, Y )ζ, ζ) = 0, we have g(r(x, Y )ζ, ζ) = α{u(x)g(a N Y, ζ) u(y )g(a N X, ζ)}.

7 Half lightlike submanifolds of an indefinite generalized Sasakian space form 133 Taking the scalar product with ζ to (4.7) and using (4.2), we have β(α 1)ḡ(X, JY ) = 0. Taking X = U and Y = ξ to this equation, we obtain β(α 1) = 0. Corollary. There exist no indefinite β-kenmotsu manifold with a quartersymmetric metric connection admits half lightlike submanifolds. Proof. Let M be an indefinite β-kenmotsu manifold with a quarter-symmetric metric connection admits half lightlike submanifolds. Then β(α 1) = 0 and α = 0. It follows that β = 0. It is a contradiction to β 0. Theorem 4.2. Let M be a half lightlike submanifold of an indefinite trans- Sasakian manifold M with a quarter-symmetric metric connection. If F holds ( X F )Y = ( Y F )X, X, Y Γ(T M), then α = β = 0. Therefore, M is an indefinite cosymplectic manifold. Proof. As ( X F )Y ( Y F )X = 0, from (3.12) we obtain 2βḡ(X, JY )ζ = u(y )A N X u(x)a N Y + w(y )A L X w(x)a L Y {B(X, Y ) B(Y, X)}U {D(X, Y ) D(Y, X)}W α{θ(y )X θ(x)y } β{θ(y )F X θ(x)f Y }. Taking the scalar product with ζ and using (4.2) and (4.3), we have 2βḡ(X, JY ) = α{u(x)v(y ) u(y )v(x)} β{u(x)η(y ) u(y )η(x)}. Taking X = U and Y = ξ to this equation, we get β = 0. Taking X = U and Y = V to the last equation such that β = 0, we obtain α = 0. In the following, denote λ, µ, ν, σ and δ by the 1-forms such that λ(x) = B(X, U) = C(X, V ), σ(x) = D(X, W ), µ(x) = B(X, W ) = D(X, V ), δ(x) = B(X, V ), ν(x) = C(X, W ) = D(X, U). Theorem 4.3. Let M be a half lightlike submanifold of an indefinite trans- Sasakian manifold M with a quarter-symmetric metric connection. If F is parallel with respect to the induced connection, then α = β = 0, φ = ρ = 0, H and H are parallel distributions on M and M is locally a product manifold M 1 M 2, where M 1 and M 2 are leaves of H and H respectively. Proof. Assume that F is parallel with respect to. First of all, we have A ξx = λ(x)v, A L X = σ(x)w, A N X = λ(x)u. (4.8)

8 134 Dae Ho Jin In fact, since X F = 0, from Theorem 4.2 we have α = β = 0. Therefore, u(y )A N X + w(y )A L X B(X, Y )U D(X, Y )W = 0. (4.9) Replacing Y by ξ to (4.9), we get φ = 0. Taking the scalar product with N, U, V and W to (4.9) by turns, we have w(y )ρ(x) = 0, u(y )C(X, U) + w(y )ν(x) = 0, u(y )λ(x) + w(y )µ(x) = B(X, Y ), u(y )ν(x) + w(y )σ(x) = D(X, Y ). From the first and second equations, we have ρ = 0, C(X, U) = 0 and ν = 0. Replacing Y by V to the fourth equation, we have µ(x) = B(X, W ) = D(X, V ) = 0. As ρ = 0, from (2.8) we see that A L X belongs to S(T M). As A ξx and A L X belong to S(T M) and S(T M) is non-degenerate, we have A ξx = λ(x)v, A L X = σ(x)w. Taking Y = U to (4.9) and using the fact that ν(x) = D(X, U) = 0, we have A N X = λ(x)u. Taking Y Γ(H) to (4.9), we have B(X, Y )U + D(X, Y )W = 0. Thus B(X, Y ) = 0, D(X, Y ) = 0, X Γ(T M), Y Γ(H). (4.10) Taking the scalar product with Z Γ(H o ) to (4.9), we get u(y )C(X, Z) + w(y )D(X, Z) = 0 for all X, Y Γ(T M). Taking Y = U to this, we have C(X, Y ) = 0, X Γ(T M), Y Γ(H o ). (4.11) By using (2.1), (3.6), (3.10), (4.10) and the fact that φ = ρ = 0, we derive g( X ξ, V ) = g(ξ, X V ) = B(X, V ) = 0, g( X V, V ) = 0, g( X Y, V ) = g(y, X V ) = g(a ξx, JY ) = B(X, F Y ) = 0, g( X ξ, W ) = D(X, V ) = 0, g( X V, W ) = φ(x) = 0, g( X Y, W ) = g(y, X W ) = D(X, F Y ) + u(y )ρ(x) = 0, for all X Γ(T M) and Y Γ(H o ), or equivalently, we get X Y Γ(H), X Γ(T M), Y Γ(H). This result implies that H is a parallel distribution on M.

9 Half lightlike submanifolds of an indefinite generalized Sasakian space form 135 For all X Γ(T M) and Y Γ(H o ), using (3.9) and (4.11), we derive g( X U, N) = v(a N X) = 0, g( X U, U) = g(a N X, N) = 0, g( X U, Y ) = g(f (A N X), Y ) = g(a N X, JY ) = C(X, F Y ) = 0, g( X W, N) = v(a L X) = 0, g( X W, U) = ρ(x) = 0, g( X W, Y ) = g(a L X, JY ) = D(X, F Y ) u(y )ρ(x) = 0, X Z Γ(H ), X Γ(T M), Z Γ(H ). Thus H is also a parallel distribution of M. As T M = H H, and H and H are parallel distributions, by the decomposition theorem of de Rham [3], M is locally a product manifold M 1 M 2, where M 1 and M 2 are leaves of H and H respectively. Theorem 4.4. Let M be a half lightlike submanifold of an indefinite trans- Sasakian manifold M with a quarter-symmetric metric connection. If U is parallel with respect to the induced connection, then α = β = 0, i.e., M is an indefinite cosymplectic manifold, τ = ρ = 0, and i.e., A N X = λ(x)u + ν(x)w. (4.12) Proof. If U is parallel with respect to, then, from (3.6) and (3.9), we have J(A N X) u(a N X)N w(a N X)L + τ(x)u + ρ(x)w {αη(x) + βv(x)}ζ = 0, X Γ(T M). Taking the scalar product with ζ, V and W to this equation by turns, we get αη(x) + βv(x) = 0, τ = 0 and ρ = 0 respectively. From the first result, we get α = β = 0. Thus M is an indefinite cosymplectic manifold. Applying J to the first equation and using (3.1), (3.4) and (4.3), we obtain (4.12). Theorem 4.5. Let M be a half lightlike submanifold of an indefinite trans- Sasakian manifold M with a quarter-symmetric metric connection. If V is parallel with respect to, then α = 1 and β = 0, i.e., M is an indefinite Sasakian manifold, τ = φ = 0, and A ξx = u(x)ζ + δ(x)u + µ(x)w. (4.13) Proof. If V is parallel with respect, then, from (3.6) and (3.10), we have J(A ξx) u(a ξx)n w(a ξx)l τ(x)v φ(x)w βu(x)ζ = 0. Taking the scalar product with ζ, U and W by turns, we get β = 0, τ = 0 and φ = 0 respectively. Applying J to the first equation, we have A ξx = αu(x)ζ + δ(x)u + µ(x)w.

10 136 Dae Ho Jin Taking the scalar product with U to this equation, we get B(X, U) = 0. Replacing Y by U to (4.5) and using the result B(X, U) = 0, we have B(U, X) = θ(x). Taking X = U to (4.2) 1 and using the last equation, we get α = αu(u) = B(U, ζ) = θ(ζ) = 1. As α = 1 and β = 0, M is an indefinite Sasakian manifold. We have (4.13). Theorem 4.6. Let M be a half lightlike submanifold of an indefinite trans- Sasakian manifold M. If W is parallel with respect to, then M is an indefinite α-sasakian manifold, i.e., β = 0, φ = ρ = 0 and A L X = αw(x)ζ + µ(x)u + σ(x)w. (4.14) Proof. If W is parallel, then, from (3.6) and (3.11) we get J(A L X) u(a L X)N w(a L X)L + φ(x)u βw(x)ζ = 0. Taking the scalar product with ζ and V by turns, we have β = 0 and φ = 0 respectively. As β = 0, M is an indefinite α-sasakian manifold. Applying J to the first equation and using (3.1), (4.2) and the fact that φ = β = 0, we have (4.14). Taking the scalar product with N to (4.14), we obtain ρ = 0. 5 Indefinite generalized Sasakian space forms Definition. An indefinite trans-sasakian manifold ( M, J, ζ, θ, ḡ) is called an indefinite generalized Sasakian space form [1, 10], denote it by M(f 1, f 2, f 3 ), if there exist three smooth functions f 1, f 2 and f 3 on M such that R(X, Y )Z = f 1 {ḡ(y, Z)X ḡ(x, Z)Y } (5.1) + f 2 {ḡ(x, JZ)JY ḡ(y, JZ)JX + 2ḡ(X, JY )JZ} + f 3 {θ(x)θ(z)y θ(y )θ(z)x for any vector fields X, Y and Z on M. + ḡ(x, Z)θ(Y )ζ ḡ(y, Z)θ(X)ζ}, Example. Indefinite Sasakian, Kenmotsu and cosymplectic space forms are important kinds of indefinite generalized Sasakian space forms such that f 1 = c+3 4, f 2 = f 3 = c 1 4 ; f 1 = c 3 4, f 2 = f 3 = c+1 4 ; f 1 = f 2 = f 3 = c 4

11 Half lightlike submanifolds of an indefinite generalized Sasakian space form 137 respectively, where c is a constant J-sectional curvature of each space forms. Theorem 5.1. Let M be a half lightlike submanifold of an indefinite generalized Sasakian space form M(f 1, f 2, f 3 ) with a quarter-symmetric metric connection. Then (1) α is a constant and (2) β = 0, and f 2 f 1 = α 2, f 1 f 3 = α(α + 1). Proof. Comparing the tangential, lightlike transversal and co-screen components of the two equations (3.16) and (5.1), and using (3.6), we get R(X, Y )Z = f 1 {g(y, Z)X g(x, Z)Y } (5.2) + f 2 {ḡ(x, JZ)F Y ḡ(y, JZ)F X + 2ḡ(X, JY )F Z} + f 3 {θ(x)θ(z)y θ(y )θ(z)x + ḡ(x, Z)θ(Y )ζ ḡ(y, Z)θ(X)ζ} + B(Y, Z)A N X B(X, Z)A N Y + D(Y, Z)A L X D(X, Z)A L Y, ( X B)(Y, Z) ( Y B)(X, Z) + τ(x)b(y, Z) τ(y )B(X, Z) (5.3) + φ(x)d(y, Z) φ(y )D(X, Z) θ(x)b(f Y, Z) + θ(y )B(F X, Z) = f 2 {u(y )ḡ(x, JZ) u(x)ḡ(y, JZ) + 2u(Z)ḡ(X, JY )}, ( X D)(Y, Z) ( Y D)(X, Z) + ρ(x)b(y, Z) ρ(y )B(X, Z) (5.4) θ(x)d(f Y, Z) + θ(y )D(F X, Z) = f 2 {w(y )ḡ(x, JZ) w(x)ḡ(y, JZ) + 2w(Z)ḡ(X, JY )}. Taking the scalar product with N to (3.17), we have g(r(x, Y )P Z, N) = ( X C)(Y, P Z) ( Y C)(X, P Z) τ(x)c(y, P Z) + τ(y )C(X, P Z) θ(x)c(f Y, P Z) + θ(y )C(F X, P Z). Substituting (5.2) into the last equation and using (2.10) 2, we obtain Applying Y ( X C)(Y, P Z) ( Y C)(X, P Z) τ(x)c(y, P Z) (5.5) + τ(y )C(X, P Z) ρ(x)d(y, P Z) + ρ(y )D(X, P Z) θ(x)c(f Y, P Z) + θ(y )C(F X, P Z) = f 1 {g(y, P Z)η(X) g(x, P Z)η(Y )} + f 2 {v(y )ḡ(x, JP Z) v(x)ḡ(y, JP Z) + 2v(P Z)ḡ(X, JY )} + f 3 {θ(x)η(y ) θ(y )η(x)}θ(p Z). ( X B)(Y, U) to (3.8) 1 and using (3.9), (3.10), (4.2) 1 and (4.3), we have = ( X C)(Y, V ) 2τ(X)C(Y, V ) φ(x)c(y, W ) ρ(x)b(y, W ) α 2 u(y )η(x) β 2 u(x)η(y ) + αβ{u(x)v(y ) u(y )v(x)} g(a ξx, F (A N Y )) g(a ξy, F (A N X)).

12 138 Dae Ho Jin Substituting this equation into (5.4) such that Z = U, we get ( X C)(Y, V ) ( Y C)(X, V ) τ(x)c(y, V ) + τ(y )C(X, V ) φ(x)c(y, W ) + φ(y )C(X, W ) ρ(x)b(y, W ) + ρ(y )B(X, W ) + (α 2 β 2 ){u(x)η(y ) u(y )η(x)} + 2αβ{u(X)v(Y ) u(y )v(x)} = f 2 {u(y )η(x) u(x)η(y ) + 2ḡ(X, JY )}. Comparing this with (5.5) such that P Z = V and using (3.8), we obtain {f 1 f 2 (α 2 β 2 )}[u(y )η(x) u(x)η(y )] = 2αβ{u(Y )v(x) u(x)v(y )}. Taking X = V and Y = U, and X = ξ and Y = U by turns, we have αβ = 0 and f 1 f 2 = α 2 β 2. As αβ = 0, by Theorem 4.1, we have β = 0. Thus f 1 f 2 = α 2, β = 0. Applying X to η(y ) = ḡ(y, N) and using (2.1) and (2.2) we have ( X η)(y ) = g(a N X, Y ) + τ(x)η(y ). Applying Y to (4.3) and using (3.14), (4.1) and (4.3), we have ( X C)(Y, ζ) = (Xα)v(Y ) α{τ(x)v(y ) + ρ(x)w(y )} + α 2 θ(y )η(x) + α{g(a N X, F Y ) + g(a N Y, F X)}. Substituting this, (4.2) 2 and (4.3) into (4.5) such that P Z = ζ, we get A{θ(X)η(Y ) θ(y )η(x)} = (Xα)v(Y ) (Y α)v(x), where A = f 1 f 3 α(α + 1). Taking X = ξ and Y = ζ, and then, taking X = U and Y = V to this equation, we obtain f 1 f 3 = α(α + 1), Uα = 0. Applying Y to (4.2) 1 and using (3.13) and (4.1), we have ( X B)(Y, ζ) = (Xα)u(Y ) + α{u(y )τ(x) + w(y )φ(x) + B(X, F Y ) + B(Y, F X)}. Substituting this into (5.3) such that Z = ζ and using (4.2), we have (Xα)u(Y ) = (Y α)u(x).

13 Half lightlike submanifolds of an indefinite generalized Sasakian space form 139 Taking Y = U and using the fact that Uα = 0, we have Xα = 0. Thus α is a constant. This completes the proof of the theorem. Definition. A screen distribution S(T M) is called totally umbilical [5] in M if there exists a smooth function γ such that A N X = γp X, or equivalently, C(X, P Y ) = γg(x, Y ). In case γ = 0, we say that S(T M) is totally geodesic in M. Theorem 5.2. Let M be a half lightlike submanifold of M(f1, f 2, f 3 ) with a quarter-symmetric metric connection. If one of the following three conditions (1) F is parallel with respect to, (2) U is parallel with respect to, (3) S(T M) is totally umbilical in M is satisfied, then M(f 1, f 2, f 3 ) is a flat manifold with an indefinite cosymplectic structure. Moreover, in case (1), M is also flat and in case (3), S(T M) is totally geodesic in M and the curvature tensor R of M is given by R(X, Y )Z = D(Y, Z)A L X D(X, Z)A L Y. (5.6) Proof. (1) Assume that F is parallel with respect to. By Theorem 4.3, we get (4.8) and the facts α = β = 0 and φ = ρ = 0. As α = 0, by Theorem 5.1, we have f 1 = f 2 = f 3. Taking the scalar product with U to (4.8) 3, we get C(X, U) = 0. Applying X to C(Y, U) = 0 and using (3.9), (4.8) 3 and F U = 0, we get ( X C)(Y, U) = 0. Substituting the last two equations into (5.5) with P Z = U, we have (f 1 + f 2 ){v(y )η(x) v(x)η(y )} = 0. Taking X = V and Y = ξ to this result, we obtain f 1 + f 2 = 0. Therefore, we see that f 1 = f 2 = f 3 = 0. Thus M(f 1, f 2, f 3 ) is flat. As f 1 = f 2 = f 3 = 0, (5.2) is reduced to R(X, Y )Z = B(Y, Z)A N X B(X, Z)A N Y + D(Y, Z)A L X D(X, Z)A L Y. Substituting (4.8) into the last equation using the fact that φ = 0, we get R(X, Y )Z = {λ(y )λ(x) λ(x)λ(y )}u(z)u + {σ(y )σ(x) σ(x)σ(y )}w(z)w = 0, for all X, Y, Z Γ(T M). Therefore R = 0 and M is also flat.

14 140 Dae Ho Jin (2) Assume that U is parallel with respect to. By Theorem 4.4, we get (4.12) and the facts α = β = 0 and τ = ρ = 0. As α = 0, by Theorem 5.1, we see that f 1 = f 2 = f 3. Taking the scalar product with U to (4.12), we get C(X, U) = 0. Applying X to C(Y, U) = 0 and using (3.9) and (4.12), we obtain ( X C)(Y, U) = 0. Substituting the last two equations into (5.5) with P Z = U, we have (f 1 + f 2 ){v(y )η(x) v(x)η(y )} = 0. Taking X = V and Y = ξ to this equation, we obtain f 1 + f 2 = 0. Therefore, f 1 = f 2 = f 3 = 0 and M(f 1, f 2, f 3 ) is flat. (3) Assume that S(T M) is totally umbilical. Then (4.3) is reduced to γθ(x) = αv(x), due to β = 0. Taking X = ζ and X = V to this by turns, we have γ = 0 and α = 0 respectively. As γ = 0, S(T M) is totally geodesic in M. As α = β = 0, M is an indefinite cosymplectic manifold and f 1 = f 2 = f 3 by Theorem 5.1. As C = 0, (5.5) is reduced to ρ(x)d(y, P Z) + ρ(y )D(X, P Z) = f 1 {g(y, P Z)η(X) g(x, P Z)η(Y )} + f 2 {v(y )ḡ(x, JP Z) v(x)ḡ(y, JP Z) + 2v(P Z)ḡ(X, JY )} + f 3 {θ(x)η(y ) θ(y )η(x)}θ(p Z). Taking P Z = U and using the fact that D(X, U) = C(X, W ) = 0, we get (f 1 + f 2 ){v(y )η(x) v(x)η(y )} = 0. Taking X = ξ and Y = V, we get f 1 + f 2 = 0. Thus f 1 = f 2 = f 3 = 0 and M(f 1, f 2, f 3 ) is a flat manifold with an indefinite cosymplectic structure. From (5.2) and the facts that f 1 = f 2 = f 3 = 0 and A N = 0, we see that (5.6). Theorem 5.3. Let M be a half lightlike submanifold of an indefinite generalized Sasakian space form M(f 1, f 2, f 3 ) with a quart-symmetric metric connection. If V is parallel with respect to, then M(f 1, f 2, f 3 ) is a space form with an indefinite Sasakian structure of the curvature functions f 1 = f 3 = 4 3, f 2 = 1 3.

15 Half lightlike submanifolds of an indefinite generalized Sasakian space form 141 Proof. If V is parallel with respect to, then we have (4.13) and the facts α = 1, β = 0 and τ = φ = 0. As α = 1, by Theorem 5.1, we see that f 2 f 1 = 1 and f 1 = f 3. Taking the scalar product with U to (4.13), we get B(X, U) = 0. (5.7) Applying Y to (5.7) and using (3.9), we have ( X B)(Y, U) = g(a ξy, F (A N X)) ρ(x)b(y, W ) u(y )η(x). Substituting the last two equations into (5.2) with Z = U, we obtain g(a ξx, F (A N Y )) g(a ξy, F (A N X)) + u(x)η(y ) (5.8) u(y )η(x) + ρ(y )B(X, W ) ρ(x)b(y, W ) = f 2 {u(y )η(x) u(x)η(y ) + 2ḡ(X, JY )}. Replacing Y by ξ and U to (4.5) by turns and using (2.6) 1 and (5.7), we have B(ξ, X) = 0, A ξξ = 0, B(U, X) = θ(x), A ξu = ζ. (5.9) due to (2.8). Taking X = ξ and Y = U to (5.8) and using (5.9), we obtain 3f 2 = 1. Therefore f 1 = f 3 = 4 by Theorem Theorem 5.4. Let M be a half lightlike submanifold of an indefinite generalized Sasakian space form M(f 1, f 2, f 3 ) with a quart-symmetric metric connection. If W is parallel with respect to, then M(f 1, f 2, f 3 ) is a space form with an indefinite α-sasakian structure of the curvature functions f 1 = 2α 2, f 2 = α 2, f 3 = α(3α + 1). Proof. If W is parallel with respect to, then we have (4.14) and the facts β = 0 and ρ = φ = 0. As φ = 0, we have D(X, ξ) = 0. Taking Y = ξ to (4.6), we obtain D(ξ, X) = 0. Taking the scalar product with U to (4.14), we have D(X, U) = 0. Applying X to D(Y, U) = 0 and using (3.9), we have ( X D)(Y, U) = D(Y, F (A N X)) α 2 η(x)w(y ). Substituting the last two equations into (5.4) with Z = U, we obtain D(X, F (A N Y )) D(Y, F (A N X)) + α 2 {w(x)η(y ) w(y )η(x)} = f 2 {w(y )η(x) w(x)η(y )}. Taking X = ξ and Y = W to this equation and using (2.10), we obtain g(a L W, F (A N ξ)) α 2 = f 2. Substituting (4.14) into the left term of the last result and using the facts that g(ζ, F (A N ξ)) = g(u, F (A N ξ)) = g(w, F (A N ξ)) = 0, we have f 2 = α 2. Therefore, f 1 = 2α 2, f 2 = α 2, f 3 = α(3α + 1).

16 142 Dae Ho Jin References [1] P. Alegre, D. E. Blair and A. Carriazo, Generalized Sasakian space form, Israel J. Math. 141, 2004, [2] C. Cǎlin, Contributions to geometry of CR-submanifold, Thesis, University of Iasi (Romania, 1998). [3] G. de Rham, Sur la réductibilité d un espace de Riemannian, Comm. Math. Helv. 26, 1952, [4] K.L. Duggal and A. Bejancu, Lightlike Submanifolds of codimension 2, Math. J. Toyama Univ. 15, 1992, [5] K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Acad. Publishers, Dordrecht, [6] K. L. Duggal and D. H. Jin, Half-lightlike submanifolds of codimension 2, Math. J. Toyama Univ., 22, 1999, [7] K. L. Duggal and D. H. Jin, Null curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, [8] K. L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds, Frontiers in Mathematics, Birkhäuser, [9] D. H. Jin, Geometry of lightlike hypersurfaces of an indefinite Sasakian manifold, Indian J. of Pure and Applied Math. 41(4), 2010, [10] D. H. Jin, Half lightlike submanifolds of an indefinite trans- Sasakian manifold, Bull. Korean Math. Soc. 51(4), 2014, [11] J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen 32, 1985, [12] K. Yano and T. Imai, Quarter-symmetric metric connection and their curvature tensors, Tensor, N.S., 38, Received: February 3, 2015; Published: February 25, 2015