We study half lightlike submanifolds M of an indefinite trans-Sasakian manifold of quasi-constant curvature subject to the condition that the 1-form θ and the vector field ζ , defined by (1.1), are identical with the 1-form θ and the vector field ζ of the indefinite trans-Sasakian structure { J , θ , ζ } of . 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-Bejancu [3] and later studied by many authors (see two books [5 , 6] ). Half lightlike submanifold M is a lightlike submanifold of codimension 2 such that rank { Rad ( TM )} = 1, where Rad ( TM ) is the radical distribution of M . It is a special case of an r -lightlike submanifold [3] such that r = 1. Its geometry is more general than that of lightlike hypersurfaces or coisotropic submanifolds which are lightlike submanifolds M of codimension 2 such that rank { Rad ( TM )} = 2. Much of its theory will be immediately generalized in a formal way to arbitrary r -lightlike submanifolds. For this reason, we study half lightlike submanifolds. B.Y. Chen and K. Yano [2] introduced the notion of a Riemannian manifold of quasi-constant curvature as a Riemannian manifold endowed with the curvature tensor satisfying the following form: for any vector fields X , Y and Z of , where ℓ and ħ are smooth functions, ζ is a smooth vector field and θ is a 1-form associated with ζ by If ħ = 0, then is a space of constant curvature ℓ . J.A. Oubina [10] introduced the notion of a trans-Sasakian manifold of type ( α , β ). We say that a trans-Sasakian manifold of type ( α , β ) is an indefinite trans-Sasakian manifold if is a semi-Riemannian manifold. Indefinite Sasakian, Kenmotsu and cosymplectic manifolds are three important kinds of trans-Sasakian manifold such that α = 1, β = 0, and α = 0, β = 1, and α = β = 0, respectively. In this paper, we study half lightlike submanifolds M of an indefinite trans-Sasakian manifold of quasi-constant curvature subject to the condition that the 1-form θ and the vector field ζ , defined by (1.1), are identical with the 1-form θ and the vector field ζ of the indefinite trans-Sasakian structure { J , ζ , θ } of . The paper contains several new results which are related to the induced structure on M . Let ( M , g ) be a half lightlike submanifold, with the radical distribution Rad ( TM ), and screen and coscreen distributions S ( TM ) and S ( TM ^⊥ ) respectively, of a semi-Riemannian manifold . We follow Duggal and Jin [4] for notations and structure equations used in this article. Denote by F ( M ) the algebra of smooth functions on M , by Г( 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. For any null section ξ of Rad ( TM ) on a coordinate neighborhood U ⊂ M , there exists a uniquely defined null vector field N ∈ Г( S ( TM ^⊥ ) ^⊥ ) satisfying Denote by ltr ( TM ) the subbundle of S ( TM ^⊥ ) ^⊥ locally spanned by N . Then we show that S ( TM ^⊥ ) ^⊥ = Rad ( TM ) ⊕ ltr ( TM ). Let tr ( TM ) = S ( TM ^⊥ )⊕ _orth ltr ( TM ). We call N , ltr ( TM ) and tr ( TM ) the lightlike transversal vector field, lightlike transversal vector bundle and transversal vector bundle of M with respect to the screen distribution S ( TM ) respectively. Let be the Levi-Civita connection of and P the projection morphism of TM on S ( TM ). Then the local Gauss and Weingarten formulas of M and S ( TM ) are given respectively by where ∇ and ∇* are induced connections on TM and S ( TM ) respectively, B and D are called the local second fundamental forms of M , C is called the local second fundamental form on S ( TM ). A_N , and A_L are called the shape operators , and τ , ρ and ϕ are 1-forms on TM . From now and in the sequel, let X , Y , Z and W be the vector fields on M , unless otherwise specified. Since the connection on is torsion-free, the induced connection ∇ on M is also torsion-free, and B and D are symmetric. The above three local second fundamental forms of M and S ( TM ) are related to their shape operators by where η is a 1-form on TM such that for any X ∈ Г( TM ). From (2.6), (2.7) and (2.8), we see that B and D satisfy and A_N are S ( TM )-valued, and is self-adjoint on TM such that Denote by , R and R * the curvature tensors of the connections , ∇ and ∇* respectively. Using the local Gauss-Weingarten formulas for M and S ( TM ), we have the Gauss equations for M and S ( TM ) such that In the case R = 0, we say that M is flat . An odd-dimensional semi-Riemannian manifold is called an indefinite trans-Sasakian manifold [10] if there exists a structure set , where J is a tensor field of type (1, 1), ζ is a vector field which is called the structure vector field of and θ is a 1-form such that for any vector fields X and Y on , where ϵ = 1 or −1 according as the vector field ζ is spacelike or timelike respectively. In this case, the set is called an indefinite trans-Sasakian structure of type ( α , β ). In the entire discussion of this paper, we may assume that ζ is unit spacelike, i , e ., ϵ = 1, without loss generality. From (3.1) and (3.2), we get Let M be a half lightlike submanifold of an indefinite trans-Sasakian manifold such that the structure vector field ζ of is tangent to M . Călin [1] proved that if ζ is tangent to M , then it belongs to S ( TM ) which assume in this paper. It is known [8] that, for any half lightlike submanifold M of an indefinite trans-Sasakian manifold , J ( Rad ( TM )), J ( ltr ( TM )) and J ( S ( TM ^⊥ )) are subbundles of S ( TM ), 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(TM) = { J(Rad(TM)) ⊕ J(ltr(TM)) } ⊕_orth J(S(TM^⊥)) ⊕_orth H_o. Denote by H the almost complex distribution with respect to J such that H = Rad(TM) ⊕_orth J(Rad(TM)) ⊕_orth H_o, TM = H ⊕ J(ltr(TM)) ⊕_orth J(S(TM^⊥)). Consider two local null vector fields U and V , a local unit spacelike vector field W on S ( TM ), and their 1-forms u , v and w defined by Let S be the projection morphism of TM on H and F the tensor field of type (1, 1) globally defined on M by F = J ○ S . Then JX is expressed as Applying J to (3.6) and using (3.1) and (3.4), we have Applying 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 Substituting (3.6) into (3.3) and using (2.1), we see that Applying to and using (3.1) and (3.3), we have Let M be a half lightlike submanifold of an indefinite trans-Sasakian manifold of quasi-constant curvature. Comparing the tangential, lightlike transversal and co-screen components of the two equations (2.11) and (4.1), we get Taking the scalar product with N to (2.12), we have Substituting (4.1) into the last equation, we see that Theorem 4.1. Let M be a half lightlike submanifold of an indefinite trans-Sasakian manifold of quasi-constant curvature. Then α is a constant, and β = 0, ℓ = α², ħ = 0. Proof. Applying ∇ _Y to (3.16), we obtain Using this equation, (2.3) ₂ , (3.12) and (3.16), we have Replacing Z by ζ to (2.11) and then, taking the scalar product with ζ and using (3.17) and the fact that , ḡ ( ( 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)}. Taking the scalar product with ζ to (4.5) and using (3.17), we have β(α − 1)ḡ(X,JY) = 0. Taking X = U and Y = ξ to this equation, we obtain β ( α − 1) = 0. Applying ∇ _X to (3.8) ₁ : B ( Y , U ) = C ( Y , V ), we have Using (3.8), (3.9), (3.10), (3.17) and (3.18), the last equation is reduced to Substituting this equation into (4.2) such that Z = U and using (3.8), we get Comparing this equation with (4.4) such that PZ = V , we obtain (ℓ − α² + β²){u(Y)η(X) − u(X)η(Y)} = 2αβ{u(Y)v(X) − u(X)v(Y)}. Taking X = ξ and Y = U , and then, X = V and Y = U to this, we have ℓ = α ² − β ² and αβ = 0. From the facts that αβ = 0 and β ( α − 1) = 0, we obtain β = 0, i . e ., ℓ = α² − β², β = 0. Applying ∇ _Y to (3.17) ₁ and using (3.13) and (3.16) ~ (3.18), we have Substituting this into (4.2) such that Z = ζ , we have (Xα)u(Y) = (Yα)u(X). Replacing Y by U to this equation, we obtain Applying to and using (2.1) and (2.2) we have (∇_Xη)(Y) = − g(A_NX, Y) + τ(X)η(Y). Applying ∇ _Y to (3.18) and using (3.14), (3.16) and (3.18), we have Substituting this into (4.4) such that PZ = ζ and using (4.5), we get ħ{θ(X)η(Y) − θ(Y)η(X)} = (Xα)v(Y) − (Yα)v(X). Taking X = ξ and Y = ζ , and then, X = U and Y = V to this, we obtain ħ = 0, Uα = 0. As Uα = 0, from (4.6), we see that α is a constant. ☐ Corollary 1. Let be an indefinite trans-Sasakian manifold, of type (α, β), of quasi-constant curvature with a half lightlike submanifold. Then is an indefinite α-Sasakian manifold of constant positive curvature α². Theorem 4.2. Let M be a half lightlike submanifolds of an indefinite trans-Sasakian manifold of quasi-constant curvature. If one of the followings ; is satisfied, then is a flat manifold with an indefinite cosymplectic structure. In case (1), M is also flat. Proof. Denote λ , μ , 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 ). (1) If F is parallel, then, as β = 0, from (3.12) we have Replacing Y by ξ and using (2.9) and (3.5), we obtain ϕ ( X ) W = 0. From this result, we see that ϕ = 0. Taking the scalar product with U to (4.7), we get u(Y )v(A_NX) + w(Y )v(A_LX) − αθ(Y )v(X) = 0. Taking Y = W and Y = ζ to this equation by turns, we get From (4.8) ₂ , we get α = 0. By Theorem 4.1, ℓ = 0 and is flat manifold with an indefinite cosymplectic structure. Taking Y = U to (4.7), we have due to (4.8) ₁ . Taking the scalar product with N , V and W to (4.7) by turns and using (2.7), (2.8), (3.8) and (4.8) ₁ , we have Taking Y = W to the first equation, we obtain ρ = 0. As ρ = 0, from (2.8) we see that A_LX belongs to S ( TM ). As and A_LX belong to S ( TM ) and S ( TM ) is non-degenerate, from the last two equations, we have Taking the scalar product with V to the second equation, we see that μ(X) = B(X, W) = D(X, V ) = 0, As ℓ = ħ = 0, substituting (4.9) and (4.10) into (4.1), we get Thus M is also flat. (2) If U is parallel with respect to ∇, then, from (3.6) and (3.9), we have J(A_NX) − u(A_NX)N − w(A_NX)L + τ (X)U + ρ(X)W − αη(X)ζ = 0. Taking the scalar product with ζ , V and W by turns, we get αη(X) = 0, τ = 0, ρ = 0, respectively. Taking X = ξ to the first result, we have α = 0. As α = 0, we see that ℓ = 0 and is a flat manifold with an indefinite cosymplectic structure. (3) If V is parallel with respect to ∇, then, from (3.6) and (3.10), we have Taking the scalar product with U and W by turns, we get τ = 0 and ϕ = 0, respectively. Applying J to the last equation and using (3.1) and (3.17) ₁ , we have Taking the scalar product with U to this equation, we get Replacing X by ζ to this equation and using (3.17) ₁ , we get α = αu(U) = − B(U, ζ) = 0. Thus ℓ = 0 and is a flat manifold with an indefinite cosymplectic structure. (4) If W is parallel with respect to ∇, then, from (3.6) and (3.11), we get J(A_LX) − u(A_LX)N − w(A_LX)L + ϕ(X)U = 0. Taking the scalar product with V and U by turns, we have ϕ = 0, ρ = 0. respectively. Applying J to the last equation and using (3.1), (3.17) ₂ , we have A_LX = −αw(X)ζ + μ(X)U + σ(X)W. Taking the scalar product with U to this, we have D ( X , U ) = 0 and C(X,W) = 0. Applying ∇ _X to C ( Y , W ) = 0 and using (3.10) and ϕ = β = 0, we have (∇_XC)(Y, W) = − g(A_N Y, F(A_LX)). Taking PZ = W to (4.4) and using the last two equations, we obtain g(A_NX, F(A_LY )) − g(A_N Y, F(A_LX)) = ℓ{w(Y )η(X) − w(X)η(Y )} as ρ = 0. Taking X = ξ and Y = W to this and using the facts that F ( A_LW ) = 0 and A_Lξ = 0, we obtain ℓ = 0. As ℓ = 0, we see that α = 0 and is a flat manifold with an indefinite cosymplectic structure. ☐ Definition. The structure tensor field F on M is said to be recurrent [9] if there exists a 1-form on M such that Theorem 5.1. Let M be a half lightlike submanifold of an indefinite trans-Sasakian manifold of quasi-constant curvature. If F is recurrent, then it is parallel, and M are flat, and the transversal connection of M is flat. Proof. As F is recurrent, from (3.12) and the fact that β = 0, we get Replacing Y by ξ to this and using (2.10), (3.1), (3.4), (3.5) and the fact that Fξ = − V , we get . Taking the scalar product with U , we get . Thus F is parallel with respect to ∇. From Theorem 4.3, we see that and M are flat, and the transversal connection of M is flat. ☐ Definition. The structure tensor field F of M is said to be Lie recurrent [9] if there exists a 1-form 𝜗 on M such that where L_XF denotes the Lie derivative on M of F with respect to X , that is, The structure tensor field F is called Lie parallel if 𝜗 = 0. Theorem 5.2. Let M be a half lightlike submanifold of an indefinite trans-Sasakian manifold of quasi-constant curvature. If F is Lie recurrent, then it is Lie parallel, and is a flat manifold with indefinite cosymplectic structure. Proof. As F is Lie recurrent, from (3.12), (5.1) and (5.2) we get Replacing Y by ξ to (5.3) and using (2.6), (3.4) and Fξ = − V , we have Taking the scalar product with V , W and ζ to this by turns, we get On the other hand, taking Y = V to (5.3) and using (3.4), we have 𝜗(X)ξ = − B(X, V )U − D(X, V )W − ∇_ξX + F∇_V X + αu(X)ζ. Applying F and using (3.7), (5.5) and FU = FW = Fζ = 0, we get 𝜗(X)V = ∇_V X + F∇_ξX + ϕ(X)W. Comparing this with (5.4), we get 𝜗 = 0. Therefore F is Lie parallel. Taking X = U to (5.3) and using (3.7), (3.8), (3.9) and (3.18), we get Taking the scalar product with ζ to this equation and using (3.18), we get αv ( Y ) = 0. Taking Y = V to this result, we have α = 0. Therefore, ℓ = 0 and is a flat manifold with indefinite cosymplectic structure. ☐