Chen-Ricci inequality for biwarped product submanifolds in complex space forms

Abstract: The main objective of this paper is to achieve the Chen-Ricci inequality for biwarped product submanifolds isometrically immersed in a complex space form in the expressions of the squared norm of mean curvature vector and warping functions.The equality cases are likewise discussed. In particular, we also derive Chen-Ricci inequality for CR-warped product submanifolds and point wise semi slant warped product submanifolds.


Introduction
The accoplishment of warped product manifolds came into existent after the study of Bishop and O'Neill [1] on the manifolds of negative curvature. Examining the fact that a Riemannian product of manifolds can not have negative curvature, they constructed the model of warped product manifolds for the class of manifolds of negative (or non positive) curvature which is defined as follows: Let (U 1 , g 1 ) and (U 2 , g 2 ) be two Riemannian manifolds with Riemannian metrics g 1 and g 2 respectively and ψ be a positive differentiable function on U 1 . If ξ : U 1 × U 2 → U 1 and η : U 1 × U 2 → U 2 are the projection maps given by ξ(p, q) = p and η(p, q) = q for every (p, q) ∈ U 1 × U 2 , then the warped product manifold is the product manifold U 1 × U 2 equipped with the Riemannian structure such that g(V 1 , V 2 ) = g 1 (ξ * V 1 , ξ * V 2 ) + (ξ • π) 2 g 2 (η * V 1 , η * V 2 ), for all V 1 , V 2 ∈ T U. The function ψ is called the warping function of the warped product manifold. If the warping function is constant, then the warped product is trivial i.e., simply Riemannian product. On the basis of the fact that warped product manifolds admit a number of applications in Physics and theory of relativity [2], this has been a topic of extensive research. Warped products provide many fundamental solutions to Einstein field equations [2]. The concept of modelling of space-time near black holes adopts the idea of warped product manifolds [3]. Schwartzschild space-time is an example of warped product U × r K 2 , where the base U = R × R + is a half plane r > 0 and the fibre K 2 is the unit sphere. Under certain conditions, the Schwartzchild space-time becomes the black hole. A cosmological model to represent the universe as a space-time known as Robertson-Walker model is a warped product [4].
In [1] authors have studied some fundamental features of warped product manifolds. An extrinsic study on warped product submanifolds of the kaehler manifolds was performed by B. Y. Chen ( [5, 6]). Since then, many geometers have explored warped product manifolds in different settings like almost complex and almost contact manifolds and various existence results have been investigated (see the survey article [7]).
In 1999, Chen [8] discovered a relationship between Ricci curvature and squared mean curvature vector for an arbitrary Riemannian manifold. On the line of Chen a series of articles have been appeared to formulate the relationship between Ricci curvature and squared mean curvature in the setting of some important structures on Riemannian manifolds (see [9][10][11][12][13][14]). Recently, Mustafa et al. [15] proved a relationship between Ricci curvature and squared mean curvature for warped product submanifolds of a semi-slant submanifold of Kenmotsu space forms.
In this paper, our aim is to obtain a relationship between Ricci curvature and squared mean curvature for biwarped product submanifolds in the setting of complex space forms.

Preliminaries
LetŪ be an almost Hermitian manifold with an almost complex structure J and a Hermitian metric g, i.e., J 2 = −I and g(JV 1 , JV 2 ) = g(V 1 , V 2 ), for all vector fields V 1 , V 2 onŪ. If J is parallel with respect to the Levi-Civita connectionD onŪ, that mean is called a Kaehler manifold. A Kaehler manifoldŪ is called a complex space form if it has constant holomorphic sectional curvature denoted byŪ(c). The curvature tensor of the complex space formŪ(c) is given bȳ Let U be an n−dimensional Riemannian manifold isometrically immersed in a m−dimensional Riemannian manifoldŪ. Then the Gauss and Weingarten formulas areD Where D is the induced Levi-civita connection on U, ξ is a vector field normal to U, h is the second fundamental form of U, D ⊥ is the normal connection in the normal bundle T ⊥ U and A ξ is the shape operator of the second fundamental form. The second fundamental form h and the shape operator are associated by the following formula 3) The equation of Gauss is given by Where,R and R are the curvature tensors ofŪ and U respectively. For any V ∈ T U and N ∈ T ⊥ U, JV 1 and JN can be decomposed as follows and where PV 1 (resp. tN) is the tangential and FV 1 (resp. f N) is the normal component of JV 1 ( resp. JN).
For any orthonormal basis {e 1 , e 2 , . . . , e k } of the tangent space T x U, the mean curvature vector H(x) and its squared norm are defined as follows where k is the dimension of U. If h = 0 then the submanifold is said to be totally geodesic and minimal if H = 0. If h(V 1 , V 2 ) = g(V 1 , V 2 )H for all V 1 , V 2 ∈ T U, then U is called totally umbilical. The scalar curvature ofŪ is denoted byτ(Ū) and is defined as whereκ pq =κ(e p ∧ e q ) and m is the dimension of the Riemannian manifoldM. Throughout this study, we shall use the equivalent version of the above equation, which is given by In a similar way, the scalar curvatureτ(L x ) of a L−plane is given bȳ Let κ pq andκ pq be the sectional curvatures of the plane sections spanned by e p and e q at x in the submanifold U k and in the Riemannian space formŪ m (c), respectively. Thus by Gauss equation, we have The global tensor field for orthonormal frame of vector field {e 1 , . . . , e k } on U k is defined as for all V 1 , V 2 ∈ T x U k . The above tensor is called the Ricci tensor. If we fix a distinct vector e u from {e 1 , . . . , e k } on U k , which is governed by χ. Then the Ricci curvature is defined by For a smooth function ψ on a Riemannian manifold U with Riemannian metric g, the gradient of ψ is denoted by ∇ψ and is defined as Let the dimension of U is k and {e 1 , e 2 , . . . , e k } be a basis of T U. Then as a result of (2.16), we get The Laplacian of ψ is defined by {(∇ e i e i )ψ − e i e i ψ}. (2.18)

Biwarped product submanifolds of a Kaehler manifold
B. Y. Chen and F. Dillen [16] generalize the definition of warped product submanifold to multiply warped product manifolds as follows.
Let {U i }, i = 1, 2, . . . , k be Riemannian manifolds with respective Riemannian metrics {g i } i=1,2,...,k and {ψ} i=2,3,...,k are positive valued functions on U 1 . Then the product manifold U = U 1 × U 2 × · · · × U k endowed with the Riemannian metric g given by is called multiply warped product manifold and denoted by U = U 1 × f 2 U 2 × · · · × f k U k where h i (i = 1, 2, . . . , k) are the projection maps of U onto U i respectively. The functions f i are known as the warping functions [16]. If the warping functions are constants, the warped product is simply Riemannian product of manifolds. As a paricular case of multiply warped product manifolds, we can define biwarped product manifolds for i = 3. For i = 2, multiply warped product manifold reduces to single warped product manifold. Consider the biwarped product manifold the Levi-civita connection of U i for i = 0, 1, 2. Now, we have the following result for biwarped product submanifold.
Recently, H. M. Tastan [18] studied biwarped submanifolds in the Kaehler manifolds and this was followed by M. A. Khan and K. Khan [19]. Basically, M. A. Khan and K. Khan explored biwarped product submnaifolds of the type U = U T × f 1 U ⊥ × f 2 U θ in the setting of complex space forms. Where U T , U ⊥ and U θ are the invarianat, totally real and pointwise slant submanifolds respectively. Throughout this study we consider k−dimensional biwarped product submanifold θ of a complex space form, where k 1 , k 2 , k 3 are the dimensions of the invariant, totally real and pointwise slant submanifolds. If U k 3 θ = {0} then the biwarped product submanifold becomes the CR-warped product submanifold. Similarly, if U k 2 ⊥ = {0} then the biwarped product submanifold reduced to pointwise semi-slant warped product submanifold.
For a biwarped product submanifold θ of a Riemannian manifold from Eq (3.5) of [16] one can conclude the following result Now, we have the following initial result.
θ be a biwarped product submanifold isometrically immersed in a Kaehler manifoldŪ. Then and N belongs to invariant subbundle of T ⊥ U. Proof. By using Gauss and Weingarten formulae in Eq (2.1), we have taking inner product with V 2 and using 3.1, we get the required result. In a similar way, we can prove the part (ii).
To prove (iii), for any V 1 ∈ T U T we haveD using Gauss formula and (2.1), we get taking inner product with JN, above equation yields interchanging V 1 by JV 1 the above equation gives From (3.3) and (3.4), we get the required result.
From Lemma 3.2, it is easy to conclude that Thus it follows that the trace of h due to U k 1 T becomes zero. Hence in view of the Definition 3.1, we obtain the following important result.
So, it is easy to conclude the following where H 2 is the squared mean curvature.

Ricci curvature for biwarped product submanifold
In this section, we investigate Ricci curvature in terms of the squared norm of mean curvature and the warping functions as follows.
θ be a biwarped product submanifold isometrically immersed in a complex space formŪ(c). Then for each orthogonal unit vector field χ ∈ T x U, either tangent to U k 1 T , U k 2 ⊥ or U k 3 θ , we have (1) The Ricci curvature satisfy the following inequalities (2) If H(x) = 0, then each point x ∈ U k there is a unit vector field χ which satisfies the equality case of (1) if and only if U k is mixed totally geodesic and χ lies in the relative null space N x at x. (3) For the equality case we have (a) The equality case of (4.1) holds identically for all unit vector fields tangent to U k 1 T at each x ∈ U k if and only if U k is mixed totally geodesic and S −totally geodesic biwarped product submanifold inŪ m (c). (b) The equality case of (4.2) holds identically for all unit vector fields tangent to U k 2 ⊥ at each x ∈ U k if and only if U is mixed totally geodesic and either U k is S ⊥ -totally geodesic biwarped product or U k is a S ⊥ totally umbilical inS m (c) with dim S ⊥ = 2. (c) The equality case of (4.3) holds identically for all unit vector fields tangent to U k 3 θ at each x ∈ U k if and only if U is mixed totally geodesic and either U k is S θ -totally geodesic biwarped product submanifold or U k is a S θ totally umbilical inŪ m (c) with dim S θ = 2. (d) The equality case of (1) holds identically for all unit tangent vectors to U k at each x ∈ U k if and only if either U k is totally geodesic submanifold or U k is a mixed totally geodesic totally umbilical and S − totally geodesic submanifold with dim U θ = 2 and dim U ⊥ = 2 where k 1 , k 2 , and k 3 are the dimensions of U k 1 T , U k 2 ⊥ , and U k 3 θ respectively.
θ be a biwarped product submanifold of a complex space form. From Gauss equation, we have (4.4) Let {e 1 , . . . , e k 1 , e k 1 +1 , . . . , e k 2 , . . . e k } be a local orthonormal frame of vector fields on U k such that {e 1 , . . . , e k 1 } are tangent to U k 1 T , {e k 1 +1 , . . . , e k 2 } are tangent to U k 2 ⊥ and {e k 2 +1 , . . . , e k } are tangent to U k 3 θ . So, the unit tangent vector χ = e A ∈ {e 1 , . . . , e k } can be expanded (4.4) as follows The above expression can be written as follows In view of the Lemma 3.2, the preceding expression takes the form (4.6) Considering unit tangent vector χ = e A , we have three choices χ is either tangent to the base manifold U k 1 T or to the fibers U k 2 ⊥ and U k 3 θ . Case 1: If χ is tangent to U k 1 T , then we need to choose a unit vector field from {e 1 , . . . , e k 1 }. Let χ = e 1 .
Then from (2.14) and (3.5) we have (4.7) Putting V 1 , V 4 = e i and V 2 , V 3 = e j in the formula (2.2), we have Using these values in (4.7), we get (4.9) In view of Lemma 3.1 Utilizing in (4.9), we have (4.10) The third term on the right hand side can be written as h r 11 h r nn . (4.11) Combining above two expressions, we have ). (4.12) Or equivalently which gives the inequality (i) of (1). Case 2. If χ is tangent to U k 2 ⊥ , we chose the unit vector from {e k 1 +1 , . . . , e k 2 }. Suppose χ = e k 2 , then from (4.6), we deduce (4.14) From (2.2) by putting V 1 , V 4 = e i and V 2 , V 3 = e j , one can compute Using these values together with (4.8) in (4.14) and applying similar techniques as in Case 1, we obtain (4.15) By the Lemma 3.1, one can conclude h r ii h r nn = 0.
The second and seventh terms on right hand side of (4.15) can be solved as follows h r kk h r j j . Utilizing these two values in (4.15), we arrive (4.17) By using similar steps as in Case 1, the above inequality can be written as The last inequality leads to inequality (ii) of (1). Case 3. If χ is tangent to U k 3 θ , then we choose the unit vector field from {e k 2 +1 , . . . , e k }. Suppose the vector χ is e k . Then from (4.6) By usage of these values together with (4.8) in (4.19) and analogous to case 1 and case 2, we obtain (4.20) On applying the Lemma 3.1, it is easy to verify Using in (4.20), we obtain (4.22) The third and seventh terms on the right hand side of (4.22) in a similar way as in case 1 and case 2 can be simplified as h r kk h r j j . (2h r kk − (h r k 1 +1k 1 +1 + · · · + h r kk )) 2 (4.24) The last inequality leads to inequality (iii) in (1). Next, we explore the equality cases of (1). First, we redefine the notion of the relative null space N x of the submanifold U k in the complex space formŪ m (c) at any point x ∈ U k , the relative null space was defined by B. Y. Chen [8], as follows For A ∈ {1, . . . , k} a unit vector field e A tangent to U k at x satisfies the equality sign of (4.1) identically if and only if holds for r ∈ {k + 1, . . . m}, which implies that U k is mixed totally geodesic biwarped product submanifold. Combining statements (ii) and (iii) with the fact that U k is biwarped product submanifold, we get that the unit vector field χ = e A belongs to the relative null space N x . The converse is trivial, this proves statement (2). For a biwarped product submanifold, the equality sign of (4.1) holds identically for all unit tangent vector belong to U T at x if and only if where p ∈ {1, . . . , k 1 } and r ∈ {k + 1, . . . , m}. Since U k is biwarped product submanifold, the third condition implies that h r pp = 0, p ∈ {1, . . . , k 1 }. Using this in the condition (ii), we conclude that U k is S −totally geodesic biwarped product submanifold inŪ m (c) and mixed totally geodesicness follows from the condition (i). Which proves (a) in the statement (3).
For a biwarped product submanifold, the equality sign of (4.2) holds identically for all unit tangent vector fields tangent to U ⊥ at x if and only if If the first case of (4.27) satisfies, then by virtue of condition (ii), it is easy to conclude that U k is a S ⊥ − totally geodesic biwarped product submanifold inŪ m (c). This is the first case of part (b) of statement (3). For a biwarped product submanifold, the equality sign of (4.3) holds identically for all unit tangent vector fields tangent to U k 3 θ at x if and only if (i) If the first case of (4.29) satisfies, then by virtue of condition (ii), it is easy to conclude that U k is a S θ − totally geodesic biwarped product submanifold inŪ m (c). This is the first case of part (c) of statement (3). For the other case, assume that U k is not S θ −totally geodesic biwarped product submanifold and dim U θ = 2. Then condition (ii) of (4.29) implies that U k is S θ − totally umbilical biwarped product submanifold inŪ(c), which is second case of this part. This verifies part (c) of (3).
For another case, suppose that first case does not satisfy. Then parts (a), (b) and (c) provide that U k is mixed totally geodesic and S − totally geodesic submanifold ofŪ m (c) with dimU ⊥ = 2 and dimU θ = 2. From the conditions (b) and (c) it follows that U k is S ⊥ − and D θ −totally umbilical biwarped product submanifolds and from (a) it is S −totally geodesic, which is part (d). This proves the theorem.
If U k 2 ⊥ = {0}, then the biwarped product submanifold becomes the Point wise semi-slant warped product submanifold that is U k = U k 1 T × f 2 U k 3 θ . Now, we have the following corollary which can be deduced from the Theorem 4.2.
Corollary 4.2. Let U k = U k 1 T × f 3 U k 3 θ be a pointwise semi-slant warped product submanifold isometrically immersed in a complex space formŪ(c). Then for each orthogonal unit vector field χ ∈ T x U, either tangent to U k 1 T or U k 3 θ , we have (1) The Ricci curvature satisfy the following inequalities (i) If χ is tangent to U k 1 T , then ). (4.31) (ii) χ is tangent to U k 3 θ , then (4.32)