Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications

: In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c , the Laplacian of the well-deﬁned warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given.


Introduction
In the submanifolds theory, creating a relationship between extrinsic and intrinsic invariants is considered to be one of the most basic problems.Most of these relations play a notable role in submanifolds geometry.The role of immersibility and non-immersibility in studying the submanifolds geometry of a Riemannian manifold was affected by the pioneering work of the Nash embedding theorem [1], where every Riemannian manifold realizes an isometric immersion into a Euclidean space of sufficiently high codimension.This becomes a very useful object for the submanifolds theory, and was taken up by several authors (for instance, see [2][3][4][5][6][7][8][9][10][11][12][13][14][15]).Its main purpose was considered to be how Riemannian manifolds could always be treated as Riemannian submanifolds of Euclidean spaces.Inspired by this fact, Nolker [16] classified the isometric immersions of a warped product decomposition of standard spaces.Motivated by these approaches, Chen started one of his programs of research in order to study the impressibility and non-immersibility of Riemannian warped products into Riemannian manifolds, especially in Riemannian space forms (see [11,[17][18][19]).Recently, a lot of solutions have been provided to his problems by many geometers (see [18] and references therein).
The field of study which includes the inequalities for warped products in contact metric manifolds and the Hermitian manifold is gaining importance.In particular, in [17], Chen observed the strong isometrically immersed relationship between the warping function f of a warped product M 1 × f M 2 and the norm of the mean curvature, which isometrically immersed into a real space form.Theorem 1.Let M(c) be a m−dimensional real space form and let ϕ : M = M 1 × f M 2 be an isometric immersion of an n−dimensional warped product into M(c).Then: where n i = dimM i , i = 1, 2, and ∆ is the Laplacian operator of M 1 and H is the mean curvature vector of M n .Moreover, the equality holds in (1) if, and only if, ϕ is mixed and totally geodesic and n 1 H 1 = n 2 H 2 such that H 1 and H 2 are partially mean curvatures of M 1 and M 2 , respectively.
In [2,5,[20][21][22][23][24][25][26][27][28][29][30][31][32], the authors discuss the study of Einstein, contact metrics, and warped product manifolds for the above-mentioned problems.Furthermore, in regard to the collections of such inequalities, we referred to [12] and references therein.The motivation came from the study of Chen and Uddin [33], which proved the non-triviality of warped-product pointwise bi-slant submanifolds of a Kaehler manifold with supporting examples.If the sectional curvature is constant with a Kaehler metric, then it is called complex space forms.In this paper, we consider the warped-product pointwise bi-slant submanifolds which isometrically immerse into a complex space form, where we then obtain a relationship between the squared norm of the mean curvature, constant sectional curvature, the warping function, and pointwise bi-slant functions.We will announce the main result of this paper in the following.Theorem 2. Let M 2m (c) be the complex space form and let ϕ : c) be an isometric immersion from warped product pointwise bi-slant submanifolds into M 2m (c).Then, the following inequality is satisfied: where θ 1 and θ 2 are pointwise slant functions along M 1 and M 2 , respectively.Furthermore, ∇ and ∆ are the gradient and the Laplacian operator on M n 1 1 , respectively, and H is the mean curvature vector of M n .The equality case holds in (2) if and only if ϕ is a mixed totally geodesic isometric immersion and the following satisfies where H 1 and H 2 are the mean curvature vectors along M n 1 1 and M n 2 2 , respectively.
As an application of Theorem 2 in a compact orientated Riemannian manifold with a free boundary condition, we prove that: 2 be a compact, orientate warped product pointwise bi-slant submanifold in a complex space form M 2m (c) such that M n 1 1 is a n 1 -dimensional and M n 2 2 is a n 2 −-dimensional pointwise slant submanifold M 2m (c).Then, M n is simply a Riemannian product if, and only if: where H is the mean curvature vector of M n .Moreover, θ 1 and θ 2 are pointwise slant functions.
By using classifications of pointwise bi-slant submanifolds which were defined in [33], we derived similar inequalities for warped product pointwise pseudo-slant submanifolds [34], warped product pointwise semi-slant submanifolds [35], and CR-warped product submanifolds [17] in a complex space form as well.

Preliminaries and Notations
An almost complex structure J and a Riemannian metric g, such that J 2 = −I and g(JX, JY) = g(X, Y), for X, Y ∈ X( M), where I denotes the identity map and X( M) is the space containing vector fields tangent to M, then (M, J, g) is an almost Hermitian manifold.If the almost complex structure satisfied ( ∇ U J)V = 0, for any U, V ∈ X( M) and ∇ is a Levi-Cevita connection M. In this case, M is called the Kaehler manifold.A complex space form of constant holomorphic sectional curvature c is denoted by M 2m (c), and its curvature tensor R can be expressed as: for every U, V, Z, W ∈ X( M 2m (c)).A Riemannian manifold M m and its submanifold M, the Gauss and Weingarten formulas are defined by , respectively for each U, V ∈ X(M) and for the normal vector field ξ of M, where h and A ξ are denoted as the second fundamental form and shape operator.They are related as g(h(U, V), N) = g(A N U, V).Now, for any U ∈ X(M) and for the normal vector field ξ of M, we have: where PU(tξ) and FU( f ξ) are tangential to M and normal to M, respectively.Similarly, the equations of Gauss are given by: for all U, V, Z, W are tangent M, where R and R are defined as the curvature tensor of M m and M n , respectively.
The mean curvature H of Riemannian submanifold M n is given by A submanifold M n of Riemannian manifold M m is said to be totally umbilical and totally geodesic if h(U, V) = g(U, V)H and h(U, V) = 0, for any U, V ∈ X(M), respectively, where H is the mean curvature vector of M n .Furthermore, if H = 0, them M n is minimal in M m .
A new class called a "pointwise slant submanifold" has been studied in almost Hermitian manifolds by Chen-Gray [36].They provided the following definitions of these submanifolds: Definition 1. [36] A submanifold M n of an almost Hermitian manifold M 2m is a pointwise slant if, for any non-zero vector X ∈ X(T x M) and each given point x ∈ M n , the angle θ(X) between JX and tangent space T x M is free from the choice of the nonzero vector X.In this case, the Wirtinger angle become a real-valued function and it is non-constant along M n , which is defined on T * M such that θ : T * M → R.
Chen-Gray in [36] derived a characterization for the pointwise slant submanifold, where M n is a pointwise slant submanifold if, and only if, there exists a constant λ ∈ [0, 1] such that P 2 = − cos 2 θ I, where P is a (1,1) tensor field and I is an identity map.For more classifications, we referred to [36].
Following the above concept, a pointwise bi-slant immersion was defined by Chen-Uddin in [18], where they defined it as follows: Definition 2. A submanifold M n of an almost Hermitian manifold M 2m is said to be a pointwise bi-slant submanifold if there exists a pair of orthogonal distributions D θ 1 and D θ 2 , such that: Remark 1.A pointwise bi-slant submanifold is a bi-slant submanifold if each slant functions θ i : T * M → R f or i = 1, 2. are constant along M n (see [13]).
In this context, we shall define another important Riemannian intrinsic invariant called the scalar curvature of M m , and denoted at τ(T x M m ), which, at some x in M m , is given: where It is clear that the first equality ( 7) is congruent to the following equation, which will be frequently used in subsequent proof: Similarly, scalar curvature τ(L x ) of L−plan is given by: An orthonormal basis of the tangent space T x M is {e 1 , • • • e n } such that e r = (e n+1 , • • • e m ) belong to the normal space T ⊥ M.Then, we have: h r αβ = g(h(e α , e β ), e r ), g h(e α , e β ), h(e α , e β . Let K αβ and K αβ be the sectional curvatures of the plane section spanned by e α and e β at x in a submanifold M n and a Riemannian manifold M m , respectively.Thus, K αβ and K αβ are the intrinsic and extrinsic sectional curvatures of the span {e α , e β } at x. Thus, from the Gauss Equation (6) (i), we have: The following consequences come from ( 6) and ( 11), as: Similarly, we have: Assume that M n 1 1 and M n 2 2 are two Riemannian manifolds with their Riemannian metrics g 1 and g 2 , respectively.Let f be a smooth function defined on M n 1 1 .Then, the warped product manifold furnished by the Riemannian metric g = g 1 + f 2 g 2 [37].When considering that the is the warped product manifold, then for any X ∈ X(M 1 ) and Z ∈ X(M 2 ), we find that: Let {e 1 , • • • e n } be an orthonormal frame for M n ; then, summing up the vector fields such that: From (Equation (3.3) in [11]), the above equation implies that: 2 is said to be trivial or a simple Riemannian product manifold if the warping function f is constant.

Main Inequality for Warped Product Pointwise Bi-Slant Submanifolds
To obtain similar inequalities like Theorem 1, for warped product pointwise bi-slant submanifolds of complex space forms, we need to recall the following lemma.
Then from ( 6) and the scalar curvature for the complex space form (11), we get: Now from ( 23) and ( 26), we have: Using (19) in the above equation and relation we derive: which implies inequality.The equality sign holds in (2) if, and only if, the leaving terms in (23) and (24) imply that: and where H 1 and H 2 are partially mean curvature vectors on M n 1 1 and M n 2 2 , respectively.Moreover, also from (23), we find that This shows that ϕ is a mixed, totally geodesic immersion.The converse part of ( 30) is true in a warped product pointwise bi-slant into the complex space form.Thus, we reached our promised result.

Consequences of Theorem 2
Inspired by the research in [6,35] and using the Remark 3 in Theorem 2 for pointwise semi-slant warped product submanifolds, we obtained: c) be an isometric immersion from the warped product pointwise semi-slant submanifold into a complex space form M 2m (c), where M n 1 1 is the holomorphic and M n 2 2 is the pointwise slant submanifolds of M 2m (c).Then, we have the following inequality: where n i = dimM i , i = 1, 2. Furthermore, ∇ and ∆ are the gradient and the Laplacian operator on M n 1 1 , respectively, and H is the mean curvature vector of M n .The equality sign holds in (31) if, and only if, n 1 H 1 = n 2 H 2 , where H 1 and H 2 are the mean curvature vectors along M n 1 1 and M n 2 2 , respectively, and ϕ is a mixed, totally geodesic immersion.
From the motivation studied in [14,35], we present the following consequence of Theorem 2 by using the Remark 2 for a nontrivial warped product pointwise pseudo-slant submanifold of a complex space, such that: c) be an isometric immersion from a warped product pointwise pseudo-slant submanifold into a complex space form M 2m (c), such that M n 1 1 is a totally real and M n 2 2 is a pointwise slant submanifold of M 2m (c).Then, we have the following inequality: where n i = dimM i , i = 1, 2. Furthermore, ∇ and ∆ are the gradient and the Laplacian operator on M n 1 1 , respectively, and H is the mean curvature vector of M n .The equality condition holds in (32) if, and only if, the following satisfies: where H 1 and H 2 are the mean curvature vectors along M n 1 1 and M n 2 2 , respectively, and ϕ is a mixed, totally geodesic isometric immersion.Corollary 3. Let ϕ : M n = M n 1 1 × f M n 2 2 → M 2m (c) be an isometric immersion from a warped product pointwise pseudo-slant submanifold into a complex space form M 2m (c), such that M n 1 1 is a pointwise slant and M n 2 2 is a totally real submanifold of M 2m (c).Then, we have the following: where n i = dimM i , i = 1, 2. Furthermore, ∇ and ∆ are the gradient and the Laplacian operator on M n 1 1 , respectively, and H is the mean curvature vector of M n .This equally holds in (33) if, and only if, ϕ is a mixed, totally geodesic isometric immersion and the following satisfies, where H 1 and H 2 are the mean curvature vectors along M n 1 1 and M n 2 2 , respectively.
Similarly, using Remark 4 and from [17], we got the following result from Theorem 2: