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Screen Almost Semi-Invariant Lightlike Submanifolds of Indefinite Kaehler Manifolds

In the present paper, we introduce screen almost semi-invariant (SASI) lightlike submanifolds of indefinite Keahler manifolds. We obtain the neccesary and sufficient condition for the induced connection to be a metric connection on SASI-lightlike submanifolds and construct an example for this manifold. Also we find some conditions for integrability of distributions and investigate certain characterizations.


INTRODUCTION
One of the most important issues of differential geometry is the Riemannian geometry of submanifolds [7]. Obviously, semi-Riemannian submanifolds have similar properties with Riemannian submanifolds, but the lightlike submanifolds [9] are different since (contrary to the nondegenerate cases) their normal vector bundle intersects with the tangent bundle. So, studying them becomes more difficult than studying non-degenerate submanifolds. When we think of hypersurfaces as submanifolds, we say that lightlike hypersurfaces of semi-Riemannian manifolds are important due to their physical applications in mathematical physics. Moreover, in physics, lightlike hypersurfaces are interesting in general relativity since they produce models of different types of horizons. Lightlike hypersurfaces are also studied in the theory of electromagnetism.
The geometry of lightlike submanifolds of a semi-Riemannian manifold has been presented in [9] (see also [8]) by Duggal and Bejancu. In [14], Duggal and Şahin have introduced differential geometry of lightlike submanifolds and they have studied geometry of classes of lightlike submanifolds (see [10], [11], [12], [13]). Also, the geometry of lightlike submanifolds of indefinite Kaehler manifolds has been presented in a book by Duggal and Bejancu [12]. The notion of semi-invariant lightlike submanifolds has been studied some authors. For instance, authors have introduced screen semi invariant lightlike submanifolds of semi-Riemannian product manifolds in [16] and they have given the Hence, it follows that D ′ is also invariant with respect to F . We denote the orthogonal complement to F D ⊥ in S(T M ⊥ ) by D 0 . Then, we have tr(T M) = ltr(T M)⊥F D ⊥ ⊥D 0 , where F 2 = I and g(F X, Y ) = g(X, F Y ), for any X, Y ∈ (T M).
Also, Bahadır has studied screen semi-invariant half-lightlike submanifolds of a semi-Riemannian product manifold [2]. In [21], semi-invariant lightlike submanifolds of golden semi-Riemannian manifolds have been introduced by Poyraz and Dogan. And, in [1], authors have introduced semi-invariant lightlike submanifolds of a semi-Riemannian product manifold. Finally, Gupta and friends have given geometry of semi-invariant lightlike product manifolds in [15].
On the other hand, in 1984, Bejancu and Papaghiuc [5] have given the definition of almost semi-invariant submanifolds of a Sasakian manifold as follows: Let M be a (2n + l)−dimensional almost contact metric manifold with (F, ξ, η, g) as the almost contact metric structure, where F is tensor field of type (1, 1), ξ is a vector field, η is a 1-from and g is a Riemannian metric on M . These tensor fields satisfy for all vector fields X, Y tangent to M, where I designes the identity morphism on the tangent bundle T M. It is well known that M is a Sasakian manifold if and only for all X, Y tangent to M , where ∇ is the Riemannian connection with respect to g. From For any vector bundle H on M, we denote by Γ(H) the module of all differentiable sections of H. For any X ∈ Γ(T M), we have g(F X, ξ) = 0. Then we put F X = bX + cX, (1.5) where bX ∈ Γ({ξ} ⊥ ) and cX ∈ Γ(T M ⊥ ). Thus b is an endomorphism of the tangent bundle T M and c is a normal bundle valued 1-form on M. Next, for each x ∈ M we define the following subspaces: We note that D x and D ⊥ x are two orthogonal subspaces of space T x M. In fact, using (1.1), (1.2) and (1.5), for any X Thus, the submanifold M of the Sasakian manifod M is said to be an almost semiinvariant submanifold if dim(D x ) and dim(D ⊥ x ) are constant along M and D : iii) D is neither an invariant nor an anti-invariant distribution on M, that is bX x = 0 and cX x = 0, for any x ∈ M and X x ∈ D x . Also, in [20], Papaghiuc has introduced some results on almost semi-invariant submanifolds in Sasakian manifolds. For similar studies, see [4], [6], [22] and [28].
In this paper, after giving some basic notions about lightlike submanifolds in the second Section, in Section 3 we give the definition of screen almost semi-invariant (SASI) lightlike submanifolds of indefinite Keahler manifolds and construct an example for this submanifold. After as a characteristic theorem, we obtain the neccesary and sufficient condition for the induced connection to be a metric connection on SASI-lightlike submanifolds. Also, we give some certain characterizations for SASI-lightlike submanifolds. For instance, we investigate the totally geodesic of the SASI-lightlike submanifolds.

LIGHTLIKE SUBMANIFOLDS
Let (N p+q , ρ) be a semi-Riemannian manifold and let N p be an immersed submanifold in N . If N is a lightlike manifold with respect to the metric ρ induced from ρ and the radical distribution Rad(T N) is of ranks, where 1 ≤ s ≤ p, then N is called a lightlike submanifold [9]. According to this definition, we recall basic notions about lightlike submanifolds as following. A semi-Riemannian complementary distribution of Rad(T N) in T N is called screen distribution and it is denoted by S(T N), that is T N = Rad(T N)⊥S(T N ). Now, we consider a screen transversal vector bundle S(T N ⊥ ). This vector bundle is a semi-Riemannian complementary vector bundle of Rad(T N) in For a lightlike submanifold ( N, ρ, S(T N), S(T N ⊥ )), there are the following four cases: . Now, let ∇ be the Levi-Civita connection on N . According to (2.3), for ∀X, Y ∈ Γ(T N ) and ∀V ∈ Γ(tr(T N)), the Gauss and Weingarten formula of N are given by and where {∇ X Y, A V X} and {h(X, Y ), ∇ t X V } are belong to Γ(T N) and Γ(tr(T N)), respectively. Also, ∇ and ∇ t are linear connections on N and on the vector bundle tr(T N), respectively. The second fundamental form h is a symmetric F ( N)-bilinear form on Γ(T N) with values in Γ(tr(T N)) and the shape operator A V is a linear endomorphism of Γ(T N). Then, for ∀X, Y ∈ Γ(T N), N ∈ Γ(ltr(T N)) and Z ∈ Γ(S(T N ⊥ )), we get and where ∀X, Y ∈ Γ(T N), ξ ∈ Γ(Rad(T N)) and ∇ * is induced connection on S(T N) which is a metric connection and ∇ * t is induced connection on Rad(T N). Using (2.9), (2.10), (2.11) and (2.12), we get Also, the induced connection on lightlike submanifold N is torsion-free, but it is not metric connection and satisfies the following condition For more details and studies about lightlike submanifolds, one can see [9], [10], [17], [18], [19], [23], [24], [25], [26], [27].
On the other hand, an indefinite almost Hermitian manifold (N , ρ, J) is a 2m-dimensional semi-Riemannian manifold N with a semi-Riemannian metric ρ of the constant index q, 0 < q < 2m and a (1, 1) tensor field J on N such that for ∀X, Y ∈ Γ(T N ) the following conditions are satisfied If J is parallel with respect to ∇, that is, then an indefinite almost Hermitian manifold (N, ρ, J) is called an indefinite Kaehler manifold [3], where ∀X, Y ∈ Γ(T N) and ∇ is the Levi-Civita connection with respect to ρ.

SCREEN ALMOST SEMI-INVARIANT LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KAEHLER MANIFOLDS
In this section, we introduce screen almost semi-invariant (SASI) lightlike submanifolds, give an example and obtain some characterizations. We give the necessary and sufficient condition for the induced connection which is not metric connection in general. Finally, we investigate the notion of mixed geodesic for (SASI) lightlike submanifolds. Also, we have Jltr(T N ) = ltr(T N) and the following decomposition where D ′ = D ⊕ ⊥ Rad(T N) and D ′ is invariant with respect to J. We denote the orthogonal complement to JD * in S(T N ⊥ ) by µ. Then, we have tr(T N) = ltr(T N) ⊕ ⊥ JD * ⊕ ⊥ µ.
Since Rad(T N) = Sp {F 1 , F 2 } and JF 1 = F 2 , one can easily see that N is a 2-lightlike submanifold. Now, we have D = Sp {F 5 , F 6 } such that J F 5 = F 6 which implies that D is invariant with respect to J. Also, we get D * = Sp W 1 = − cos t∂x 3 + sin t∂x 4 , W 2 = − cos l∂x 9 + sin l∂x 10 , W 3 = sin t∂x 3 + cos t∂x 4 + cos m∂x 5 + sin m∂x 6 .
Here we get JF 7 = W 2 , JN 1 = N 2 and µ = Sp {W 1 , W 3 } . Then, since we obtain that N is a SASI-lightlike submanifold of R 12 2 . Now, let P 0 , P 1 , P 2 , P 3 and S 1 be the projections on D, D * , D 0 , D 1 and Rad(T N) in T N, respectively. Similarly, let R 1 , R 2 and R 3 be the the projection on JD * , D 2 and D 3 in tr(T N), respectively. Then for any X ∈ Γ(T N ), we get where P X ∈ Γ(D ′ ), RX ∈ Γ(D * ) and QX ∈ Γ(D ′′ ).

Theorem 2. Let N be a SASI-lightlike submanifold of an indefinite Kaehler manifold N. The induced connection ∇ is a metric connection if and only if
Proof. For almost complex structure J, one can write from (2.18) where X, Y ∈ Γ(T N). Then, using (2.6) and (2.12), for X ∈ Γ(T N) and Y ∈ Γ(Rad(T N)), we have Taking the tangential parts of (3.8), we get According to (3.9), if T (A * JY X − ▽ * t X JY ) ∈ Γ(Rad(T N)) and Bh(X, J Y ) = 0, then ▽ X Y ∈ Γ(Rad(T N)). Consequently, the proof is completed.
Here for ∀X, Y ∈ Γ(D ′ ), obviously wX = wY = 0. From the equations (3.6) and (3.7), we obtain Proof. Since D ′′ is neither an invariant nor an anti-invariant distribution on N, we have T Z = 0 = T U and wZ = 0 = wU. Using (3.5)-(3.6)-(3.7), we obtaine Then, using symmetry property of the second fundamental form h, we complete the proof. On the other hand, as a result of the necessary calculations, we get Then, form (3.11) and (3.12), we prove that for Y ∈ Γ(D) and Y ∈ Γ(Rad(T N )) the conditions (i) and (ii) are true. The converse of the proof of is obvious.
Proof. If D ′ is a totally geodesic foliation in N , then for ∀X, Y ∈ Γ(D ′ ), we get ∇ X Y ∈ Γ(D ′ ). For ∀ξ ∈ Γ(Rad(T N)), V ∈ Γ(D * ), Z ∈ Γ(D ′′ ) and E ∈ Γ(S(T N ⊥ )), we have Then we obtain the following results: Thus, it is clear that N is D ′ -geodesic, which implies that D ′ is parallel respect to ∇ on N . Then, the (i) and (ii) conditions are proved.
Proof. Getting Z instead of V in the previous theorem and its proof, one can prove the theorem.
CONCLUDING REMARK. One can recall that a lightlike hypersurface M of a semi-Riemannian manifolds is totally geodesic if only if Rad(T M) is a Killing distribution on M [9]. In our study, Teorem 9, Teorem 10 and Teorem 11 of Section 3 are important in this sense. Also, Teorem 2, Teorem 6, Teorem 7 and Teorem 8 are about the screen shape operator or the second fundamental form which are related to the notion of Killing horizon of general relativity [11]. Indeed, "Solutions of the highly non-linear Einstein's field equations require the assumption that they admit Killing or homothetic vector fields. This is due to the fact that the Killing symmetries leave invariant the metric connection, all the curvature quantities and the matter tensor of the Einstein field equations of a spacetime. Most explicit solutions (see [14] pp. 267) have been found by using one or more Killing vector fields. Related to the theme of this section, here we show that the Killing symmetry has an important role in the most active area of research on Killing horizons in general relativity [14]." And it is known that the theory of lightlike geometry has an important relation with general relativity.