HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE TRANS-SASAKIAN MANIFOLD OF QUASI-CONSTANT CURVATURE

Abstract

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 .

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-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.

 

2. HALF LIGHTLIKE SUBMANIFOLD

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). AN, and AL 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 AN 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.

 

3. INDEFINITE TRANS-SASAKIAN MANIFOLDS

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 Ho with respect to J, i.e., J(Ho) = Ho, such that

S(TM) = { J(Rad(TM)) ⊕ J(ltr(TM)) } ⊕orth J(S(TM⊥)) ⊕orth Ho.

Denote by H the almost complex distribution with respect to J such that

H = Rad(TM) ⊕orth J(Rad(TM)) ⊕orth Ho, 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

 

4. MANIFOLD OF QUASI-CONSTANT CURVATURE

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(AN Y, ζ) − u(Y)g(AN 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 = α2 − β2 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(ANX, 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 ; (1) F is parallel with respect to the connection ∇, (2) U is parallel with respect to the connection ∇, (3) V is parallel with respect to the connection ∇, and (4) W is parallel with respect to the connection ∇ 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(ANX) + w(Y )v(ALX) − 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 ALX belongs to S(TM). As and ALX 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(ANX) − u(ANX)N − w(ANX)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(ALX) − u(ALX)N − w(ALX)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 ALX = −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(AN Y, F(ALX)).

Taking PZ = W to (4.4) and using the last two equations, we obtain g(ANX, F(ALY )) − g(AN Y, F(ALX)) = 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(ALW) = 0 and ALξ = 0, we obtain ℓ = 0. As ℓ = 0, we see that α = 0 and is a flat manifold with an indefinite cosymplectic structure. ☐

 

5. RECURRENT HALF LIGHTLIKE SUBMANIFOLDS

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 LXF 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. ☐