CR-warped product submanifolds of a generalized complex space form

N ⟂ ×f NT such that N⟂ is totally real submanifold and NT is holomorphic submanifold of a Kaehler manifold and found that these warped product are simply CR-product as defined in Chen (1981). Therefore, Chen considered the warped product of the type NT ×f N⟂, and obtained an inequality for squared norm of second fundamental form, these types of warped product are called CR-warped product. Later on, the geometrical behavior of these type of submanifolds were studied by many researchers (c.f. Arslan, Ezentas, Mihai, & Murathan, 2005; Khan, Khan, & Uddin, 2009; Sahin, 2006).


Introduction
The notion of warped product of manifolds was introduced by Bishop and O'Neill (1965) in order to construct a large velocity of manifolds of negative curvature. The idea of warped product on submanifolds was introduced by Chen (2001). Basically, Chen considered warped product of the type N ⟂ × f N T such that N ⟂ is totally real submanifold and N T is holomorphic submanifold of a Kaehler manifold and found that these warped product are simply CR-product as defined in Chen (1981).
Therefore, Chen considered the warped product of the type N T × f N ⟂ , and obtained an inequality for squared norm of second fundamental form, these types of warped product are called CR-warped product. Later on, the geometrical behavior of these type of submanifolds were studied by many researchers (c.f. Arslan, Ezentas, Mihai, & Murathan, 2005;Khan, Khan, & Uddin, 2009;Sahin, 2006).
In Al-Luhaibi, Al-Solamy, and  investigated CR-warped product submanifolds in the setting of nearly Kaehler manifolds and obtained some basic results and finally worked out an estimation for squared norm of second fundamental form if ambient manifold is generalized complex space form. These types of warped products are also studied in different settings of almost Hermitian manifolds (c.f. Al-Luhaibi et al., 2009;Faghfouri & Majidi, 2015;Khan & Jamal, 2010;Sahin, 2009).

PUBLIC INTEREST STATEMENT
The warped product manifolds are the generalization of product manifolds and occur naturally. These types of manifolds have wide applications in Differential geometry, Relativity, Physics as well as in different branches of engineering. The present study predicts the geometric behavior of underlying warped product submanifolds. Further it is known that the warping function of a warped product manifolds is a solution of some partial differential equations and most of the physical phenomenons are described by partial differential equations. We hope that our study may find applications in Physics as well as in engineering.
The contact version of warped product manifolds were also studied in different settings (see Ateken, 2011Ateken, , 2013Sular & Ozgur, 2012). Recently, we also studied semi-invariant warped product submanifolds and obtained inequality for squared norm of second fundamental form Al-Solamy and Ali Khan (2012). Moreover, Aitceken (c.f. Ateken, 2011Ateken, , 2013 investigated an inequality for squared norm of second fundamental form which characterize the existence of contact CR-warped product submanifolds in the setting of Cosymplectic and Kenmotsu space forms. Motivated by Aitceken (2011Aitceken ( , 2013 and Sular and Ozgur (2012) studied the contact CR-warped product in more general setting namely trans-Sasakian generalized Sasakian space forms and obtained an inequality for existence of CR-warped product submanifolds. After reviewing the literature, we realized that characterizing inequality for existence of CR-warped product submanifolds is not yet investigated in the setting of generalized complex space forms. In this paper we obtained an estimation of second fundamental form in terms of Hessian of lnf, where f is a warping function and finally obtained an characterizing inequality for existence of CR-warped product submanifolds of generalized complex space forms and in particular for complex space forms.

Preliminaries
Let M be an almost Hermitian manifold with almost complex structure J and a Hermitian metric g i.e. for all vector fields U, V ∈ TM. If almost complex structure J is parallel with respect to the Levicivita connection ∇ on M i.e. ∇ J = 0, then (M, J, g) is called Kaehler manifold. There is a more general structure on M , namely nearly Kaehler structure and characterized by the following equation equivalently, (2.2) can also be written as for all U, V ∈ TM.
A nearly Kaehler manifold M is Kaehler manifold if and only if Neijenhuis tensor of J vanish identically. Any four dimensional nearly Kaehler manifold is a Kaehler manifold. Six dimensional sphere S 6 is a typical example of a nearly Kaehler non Kaehler manifold. The complex structure J on S 6 is defined by vector cross product in the space of purely imaginary Cayley numbers. There is a more general class of almost Hermitian manifolds than nearly Kaehler manifold, this class is known as RK− manifolds. A generalized complex space form is an RK−manifold of constant holomorphic sectional curvature c and of constant is denoted by M (c, ). The sphere S 6 endowed with the standard nearly Kaehler structure is an example of a generalized complex space form which is not a complex (2.5)  The covariant differentiation of the tensors J, P, F, t and f are defined as respectively Furthermore, for any U, V ∈ TM, the tangential and normal parts of (∇ Moreover, it is easy to verify the following property On using equations (2.5)-(2.13) and (2.16), we may obtain that (2.17) (2.18) Similarly, for N ∈ T ⟂ M, denoting by  U N and  U N respectively the tangential and normal parts of (∇ U J)N, we find that On a submanifold M of a nearly Kaehler manifold by (2.2) and (2.16) for any U, V ∈ TM.
Now we have the following properties of  and , which can be verified very easily for all U, V, W ∈ TM and N ∈ T ⟂ M.
Let M be an almost Hermitian manifold with an almost complex structure J and Hermitian metric g and let M be a submanifold of M , M is said to be CR-submanifold if there exist two orthogonal complementary distributions D and D ⟂ such that D is holomorphic distribution i.e., JD ⊆ D and D ⟂ is totally real distribution i.e., JD ⟂ ⊆ T ⟂ M.
If is the invariant subspace of the normal bundle T ⟂ M, then in the case of CR-submanifold, the normal bundle T ⟂ M can be decomposed as follows A CR-submanifold M is called CR-product if the distribution D and D ⟂ are parallel on M. In this case M is foliated by the leaves of these distributions. In general, if N 1 and N 2 are Riemannian manifolds with Riemannian metrics g 1 and g 2 respectively, then the product manifold (N 1 × N 2 , g) is a Riemannian manifold with Riemannian metric g defined as where 1 and 2 are the projection maps of M onto N 1 and N 2 , respectively, and d 1 and d 2 are their differentials.
As a generalization of the product manifold and in particular of CR-product submanifold, one can consider warped product of manifolds which are defined as follows.
Definition 2.1 Let (B, g B ) and (C, g C ) be two Riemannian manifolds with Riemannian metric g B and g C respectively and ψ be a positive differentiable function on B. The warped product of B and C is the For a warped product manifold N 1 × N 2 , we denote by D 1 and D 2 the distributions defined by the vectors tangent to the leaves and fibers respectively. In other words, D 1 is obtained by the tangent vectors of N 1 via the horizontal lift and D 2 is obtained by the tangent vectors of N 2 via vertical lift. In case of CR-warped product submanifolds D 1 and D 2 are replaced by D and D ⟂ respectively.
The warped product manifold (B × C, g) is denoted by B × C. If U is the tangent vector field to M = B × C at (p, q) then Bishop and O'Neill (1965) proved the following Theorem 2.1 Let M = B × C be warped product manifolds. If X, Y ∈ TB and V, W ∈ TC then From above Theorem, for the warped product M = B × f C it is easy to conclude that for any X ∈ TB and V ∈ TC. grad is the gradient of and is defined as for all U ∈ TM.
Corollary 2.1 On a warped product manifold M = N 1 × N 2 , the following statements hold In what follows, N ⟂ and N T will denote a totally real and holomorphic submanifold respectively of an almost Hermitian manifold M .
A warped product manifold is said to be trivial if its warping function f is constant. More generally, a trivial warped product manifold M = N 1 × N 2 is a Riemannian product N 1 × N 2 where N 2 is the manifold with the Riemannian metric 2 g 2 which is homothetic to the original metric g 2 of N 2 . For example, a trivial CR-warped product is CR-product.
Let M be a m−dimensional Riemannian manifold with Riemannian metric g and let {e 1 , … , e m } be an orthogonal basis of TM. For a smooth function on M the Hessian of f are defined as for any U, V ∈ TM. The Laplacian of f is defined by It is eviedent from the above two equations that Laplacian is the negative of the Hessian. Moreover from the integration theory on manifolds, for a compact orientable Riemannian manifold M without boundary, we have (2.25) ‖U‖ 2 = ‖d 1 U‖ 2 + 2 (p)‖d 2 U‖ 2 . (2.26) where dV is the volume element of M (O' Neill, 1983).

CR-warped product submanifolds
In this section we consider warped product of the type M = N T × N ⟂ in a nearly Kaehler manifold, where N T is holomorphic submanifold and N ⟂ is totally real submanifold, these warped product submanifolds are called CR-warped product submanifolds. N T and N ⟂ are the integral submanifolds of the distributions D and D ⟂ .
Estimation of squared norm of second fundamental form for CR-warped product submnaifolds in the setting of almost Hermitian manifolds has been worked out by many authors (see Al-Luhaibi et al., 2009;Chen, 2003;Khan et al., 2009). Our aim in this paper is to obtain a characterizing inequality for squared norm of second fundamental form for CR-warped product submanifolds in the setting of generalized complex space form. Now, we obtain some basic results in the following Lemma.

Lemma 1 Let M = N T × N ⟂ be a CR-warped product submanifold of a nearly kaehler manifold M , then
for any U ∈ TN T and W ∈ TN ⟂ .
Proof Assume that M is a CR-warped product submanifold of a nearly Kaehler manifold. From (2.5) and (2.16), we can write using part (ii) of P 3 and (2.26), above equation gives By use of (2.16) and (2.17) in above equation we get replacing U by JU, we can find part (ii).
From Gauss formula and (2.16), we have Using (2.5), (2.26), (2.11) and (2.16), the above equation yields By use of part (i) of P 3 and some easy calculations, the second term of above equation becomes zero and we get the required result. Proof Let M be a CR-warped product submanifold of a nearly Kaehler manifold, then by Gauss formula we have By use of (2.11), (2.16), parts (ii) and (iii) of Lemma 3.1, above equation yiels Now, further calculating the third term on right hand side as follows In view of (2.26), the above equation becomes After using part (ii) of P 3 , (2.16) and (2.17), the second term on right hand side of above equation becomes zero. Further, on applying (2.5), (2.16) and (2.17), we get By use of Lemma 3.2 and (3.3) in (3.2), we get (3.1), this completes the proof. Now we will prove the following theorem for CR-warped product submnaifolds of a generalized complex space form. Since ∇ U U ∈ TN T , then we can replace U by ∇ U U as Applying Gauss formula, we find On using (2.11), (2.3) and (2.26) in the following equation one can conclude that h(U, U) ∈ , using this fact in (3.7), we get By use of (2.5) and (2.3), the above expression reduced to or Similarly, we can get Moreover, using (2.22)(b) as follows or using (2.17), it is easy to see that By making use of (3.8), (3.9), (3.10) in (3.6), we get By parts (iii) and (iv) of Lemma 3.1, and (2.28), we have Applying (3.1) and (3.5) in (3.11), we get which is the required result.
Finally, we will prove the main theorem.