LIGHTLIKE SUBMERSIONS FROM AN INDEFINITE NEARLY KÄHLER MANIFOLD INTO A LIGHTLIKE MANIFOLD

As a generalization of almost Hermitian submersions, we introduce slant lightlike submersion from an indefinite nearly Kähler manifold into a lightlike manifold. We establish the existence theorems for these submersions and investigate the geometry of foliations which are arisen from the definition of lightlike submersion. We also find necessary and sufficient condition for the leaves of the distributions to be totally geodesic foliations in indefinite nearly Kähler manifold.


Introduction
Let (M, g M ) and (N, g N ) be two Riemannian manifolds.The idea of Riemannian submersion between two manifolds were introduced by O'Neill [5] and Gray [4].Later, such submersions were considered between manifolds with differentiable structures.As an analogue of holomorphic submanifolds, Watson defined almost Hermitian submersions between almost Hermitian manifolds [13].O' Neill introduced the semi-Riemannian submersions [6].
On the other hand, it is known that if M and N are Riemannian manifolds, then the fibres are always Riemannian manifolds.However, if M and N are semi-Riemannian manifolds, then the fibres may not be semi-Riemannian manifolds.Therefore, in [9], Sahin introduced a screen lightlike submersion from a lightlike manifold into a semi-Riemannian manifold.Later, Sahin and Gunduzulp in [10], introduced a lightlike submersion from a semi-Riemannian manifold into a lightlike manifold.
The geometry of lightlike submanifolds has extensive uses in mathematical physics and in particular in the theory of general relativity [2].It is also well known that semi-Riemannian submersions are of interest in physics, owing to their application in the Yang-Mills theory, Kaluza-Klein theory and supergravity and superstring theories [1,3,7,8].Moreover, we obtained the nonexistence of totally contact umbilical proper slant lightlike submanifolds of indefinite Sasakian manifold [11].Thus all these facts and results of above papers motivated us to work on the theory of lightlike submersions with slant lightlike submersions.From these facts we get the concept of slant lightlike submersion from an indefinite nearly Kähler manifold to lightlike manifold.
In the present paper we introduce slant lightlike submersion from an indefinite nearly Kähler manifold into lightlike manifold.
The paper is organized as follows: In section 2, we collect some basic information and notions needed for this paper.In section 3, we give definition of slant Riemannian submersions and investigate the geometry of leaves of distributions.We obtain necessary and sufficient conditions for such slant lightlike submersions to be totally geodesic.

Preliminaries
Let (M, g) be a real n-dimensional smooth manifold where g is a symmetric tensor field of type (0, 2).The radical space Rad T p M of T p M is defined by The dimension of Rad T p M is called the nullity degree of g.If the mapping defines a smooth distribution on M of rank r > 0, then Rad T M is known as the radical distribution of M and the manifold M is known as r-lightlike manifold if 0 < r ≤ n, see [12].

Consider a smooth submersion
The kernel of f * at the point p is given by and (ker f * ) ⊥ is given by ) is a semi-Riemannian vector space, (ker f * ) ⊥ may be not complementary to (ker f * ).Hence, we assume that Thus, we get the following four cases of submersions: Since ker f * is a real lightlike vector space and S(ker f * ) is the complementary non degenerate subspace of ∆ in S(Ker f * ) and we obtain Similarly, we have where S(ker f * ) ⊥ is a complementary non degenerate subspace of ∆ in (ker f * ) ⊥ .
Since S(ker f * ) ⊥ is non-degenerate in T p (M 1 ), we get where (S(ker f * )) ⊥ is the complementary subspace of S(ker f * ) in T p (M 1 ).Since S(ker f * ) and (S(ker f * )) ⊥ are non-degenerate in T p (M 1 ), we get Thus, from [2] , a quasi-orthonormal basis of M 1 along (ker f * ) can be constructed.Therefore, we obtain Using equation (2.1) , it is clear that ltr(ker f * ) and Ker f * are not orthogonal to each other.
Denote by V = ker f * the vertical space of T p (M 1 ) and H = tr(ker f * ) the horizontal space of T p (M 1 ).Thus we have Definition 1.1.Let (M 1 , g 1 ) be a semi-Riemannian manifold and let (M 2 , g 2 ) be an r-lightlike manifold.Let f : M 1 → M 2 be a submersion such that: . Then f is called an r-lightlike submersion. Case and f is called a totally lightlike submersion.
Definition 1.2.Let (M, g, J) be an indefinite almost Hermitian manifold and be the Levi-Civitia connection on M with respect to g such that for X,Y on M. Then M is called an indefinite nearly Kähler manifold if (4) It is well known that every Kähler manifold is a nearly Kähler manifold but converse is not true.
Note: Whatever it is need we have suppose the horizontal vector field to be basic.For any arbitrary tangent vector fields V and W on M, we have ( 5) where P V W and Q V W denote the horizontal and vertical component of ( V J)W respectively.
For a Kähler manifold M we have If M is a nearly Kähler manifold, then it can be easily seen that both P and Q are antisymmetric in V and W , hence We need the statement of following theorem to define a slant lightlike submersion from an indefinite nearly Kähler manifold into a lightlike manifold.
Theorem 2.1.Let f : M 1 → M 2 be an r-lightlike submersion from an indefinite almost Hermitian manifold (M 1 , g 1 , J) where g 1 is a semi-Riemannian metric of index 2r, to an r-lightlike Let J∆ be a distribution on M such that ∆ ∩ J∆ = 0. Then any distribution complementary to J∆ ⊕ Jltr(ker f * ) in S(ker f * ) is Riemannian.

SLANT LIGHTLIKE SUBMERSION
Let M be an r-lightlike submanifold of an indefinite Hermitian manifold M of index 2r.Then M is a slant lightlike submanifold of M if the following conditions are satisfied: (b) For any non zero vector field tangent to D for p ∈ U ⊂ M, the angle θ (X) between JX and the vector space D p is constant i.e., it is independent of the choice of p ∈ U ⊂ M and X ∈ D p , where D is the distribution complementary to JRadT M ⊕ Jltr(T M) in the screen distribution

S(T M).
This constant angle θ (X) is called the slant angle of the distribution D. If D = {0} and θ = 0, π 2 then a slant lightlike submanifold is said to be proper slant lightlike submanifold.Definition 3.1.Let (M 1 , g 1 , J) be a real 2m-dimensional indefinite nearly Kähler manifold, where g 1 is a semi-Riemannian metric of index 2r, 0 < r < m and any other manifold (M 2 , g 2 ) which is an r-lightlike manifold.Let f be an r-lightlike submersion, f : M 1 → M 2 .Then f is said to be slant lightlike submersion if following conditions are satisfied: (d) The angle θ (X) between JX and D is constant for each non zero vector field X tangent to D , where D is the distribution complementary to J∆ ⊕ Jltr(ker f * ) in S(ker f * ).
Hence, we get where µ is the orthogonal subbundle complementary to f (D) in S(ker f * ).Let f be a slant lightlike submersion from an indefinite nearly Kähler manifold (M 1 , g 1 , J) into an r−lightlike manifold (M 2 , g 2 ).Then any X ∈ v p can be written as (7) JX = φ X + ωX, where φ X and ωX are the tangential and transverse components of JX, respectively.Similarly, for any Z ∈ H p , we get where BZ and C Z are the tangential and transversal component of JZ, respectively.Denote P 1 , P 2 , Q 1 and Q 2 the projections onto the distributions ∆, J∆, J ltr(ker f * ) and D respectively.
Thus, we can express X as (9) for any X ∈ V p .Applying J to (9), we get for any X ∈ V p .Then, clearly, Therefore, we can write Since the geometry of Riemannian submersions is characterized by O'Neill's tensors T and A, Sahin [9] defined these tensors for lightlike submersions as follows: (12) for vector fields E and F on M 1 , where ∇ is the Levi-Civitia connection of g 1 .It should be noted that T and A are skew-symmetric tensors in Riemannian submersions but not in lightlike submersions because the horizontal and vertical subspaces are not orthogonal to each other.
The tensors T and A both reverse the horizontal and vertical subspaces and, moreover, T has the symmetric property ie.( 14) Lemma 3.1.Let (M 1 , g 1 , J) be an indefinite nearly Kähler manifold, where g 1 is a semi-Riemannian metric of index 2r and let (M 2 , g 2 ) be an r-lightlike manifold.If f be a slant lightlike submersion such as f : Lemma 3.2.Let (M 1 , g 1 , J) be an indefinite nearly Kähler manifold, where g 1 is a semi-Riemannian metric of index 2r and let (M 2 , g 2 ) be an r-lightlike manifold.If f be a slant lightlike submersion such as f : where for any X,Y ∈ Γ(ker f * ).
Proof.For any X,Y ∈ Γ(ker f * ), using equations (5), ( 7), ( 8), ( 15) and (16), we get From above equation, we have Comparing vertical and horizontal parts, we obtain Theorem 3.1.Let (M 1 , g 1 , J) be an indefinite nearly Kähler manifold, where g 1 is a semi-Riemannian metric of index 2r and let (M 2 , g 2 ) be an r-lightlike manifold.Let f be a slant lightlike submersion such as f : M 1 → M 2 ,then f is a proper slant lightlike submersion if and only if moreover, in this case λ = −cos 2 θ .
Proof.Let f be a slant lightlike submersion.Then J∆ is a distribution on S(TM).Hence by virtue of Theorem (1), J (ltr(ker f * )) is a distribution on M 1 .Further the slant angle between JQ 2 X and D p is constant and given by cos and also the cos θ (Q 2 X) is also given by Hence using ( 22) and (23), we obtain Since the angle θ (Q 2 X) is constant on D, we have where λ = −cos 2 θ .(a) implies that J∆ is a distribution on S(ker f * ).Hence, in view of theorem (2.1), any distribution complementary to J∆ ⊕ Jltr(ker f * ) in S(ker f * ) is Riemannian.
Corollary 3.1.Let (M 1 , g 1 , J) be an indefinite nearly Kähler manifold, where g 1 is a semi-Riemannian metric of index 2r and let (M 2 , g 2 ) be an r-lightlike manifold.Let f be a proper slant lightlike submersion such as f : M 1 → M 2 with slant angle θ , then for any X,Y ∈ Γ(ker f * ) Theorem 3.2.Let (M 1 , g 1 , J) be an indefinite nearly Kähler manifold, where g 1 is a semi-Riemannian metric of index 2r and let (M 2 , g 2 ) be an r−lightlike manifold.Let f be a slant lightlike submersion such as f : Proof.Let f be a slant lightlike submersion, then J (ltr(ker f * )) is a distribution on M 1 .Using equations (3), ( 7), ( 8) to (10), we obtain Comparing the components of the distribution D on both sides of last equation, we get Hence by using (21), we get Conversely, by virtue of equations (3.20) and (3.21), we obtain Further, we prove that the orthogonal complement subbundle µ of f (D) in S(ker f * ) ⊥ is holomorphic with respect to J and determine the dimension.
Theorem 3.3.Let (M 1 , g 1 , J)be an indefinite nearly Kähler manifold, where g 1 is a semi-Riemannian metric of index 2r and let (M 2 , g 2 ) be an r−lightlike manifold.Let f be a slant lightlike submersion such as f : M 1 → M 2 ,then µ is invariant under J.
Theorem 3.7.Let (M 1 , g 1 , J) be an indefinite nearly Kähler manifold, where g 1 is a semi-Riemannian metric of index 2r and let (M 2 , g 2 ) be an r-lightlike manifold.Let f be a proper slant lightlike submersion such as f : M 1 → M 2 .Then the distribution H defines a totally geodesic foliation on M 1 if and only if for any V,W ∈ Γ(H ).
then f is a proper slant lightlike submersion if and only if (a) J(ltr(ker f * )) is a distribution on M 1 ; (b) for any vector field tangent to M 1 , there exists a constant v ∈ [−1, 0] such that (26) BωQ 2 X = vQ 2 X, where v = −sin 2 θ .