Ricci Curvature of Contact CR-Warped Product Submanifolds in Generalized Sasakian Space Forms Admitting a Trans-Sasakian Structure

. The objective of this paper is to achieve the inequality for Ricci curvature of a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form admitting a trans-Sasakian structure in the expressions of the squared norm of mean curvature vector and warping function. We provide numerous physical applications of the derived inequalities. Finally, we prove that under a certain condition the base manifold is isometric to a sphere with a constant sectional curvature.


Introduction
Let (N 1 , 1 ) and (N 2 , 2 ) be two Riemannian manifolds with Riemannian metrics 1 and 2 respectively and ψ be a positive differentiable function on N 1 . If π : N 1 ×N 2 → N 1 and η : N 1 ×N 2 → N 2 are the projection maps given by π(p, q) = p and η(p, q) = q for every (p, q) ∈ N 1 × N 2 , then the warped product manifold is the product manifold N 1 × N 2 equipped with the Riemannian structure such that (X, Y) = 1 (π * X, π * Y) + (ψ • π) 2 2 (η * X, η * Y), for all X, Y ∈ TM. The function ψ is called the warping function of the warped product manifold [24]. If the warping function is constant, then the warped product is trivial i.e., simply Riemannian product. On the grounds that warped product manifolds admit a number of applications in Physics and theory of relativity [33], this has been a topic of extensive research. Warped products provide many basic solutions to Einstein field equations [33]. The concept of modelling of space-time near black holes adopts the idea of warped product manifolds [34]. Schwartzschild space-time is an example of warped product P × r S 2 , where the base P = R × R + is a half plane r > 0 and the fibre S 2 is the unit sphere. Under certain conditions, the Schwartzchild space-time becomes the black hole. A cosmological model to model the universe as a spacetime known as Robertson-Walker model is a warped product [35].
Some natural properties of warped product manifolds were studied in [24]. B. Y. Chen ([1], [2]) performed an extrinsic study of warped product submanifolds in a Kaehler manifold. 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 [10]).
In 1999, Chen [6] 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 [12], [13], [16], [17], [18], [38]). Recently Ali et al. [20] established a relationship between Ricci curvature and squared mean curvature for warped product submanifolds of a sphere and provide many physical applications.
In this paper our aim is to obtain a relationship between Ricci curvature and squared mean curvature for contact CR-warped product submanifolds in the setting of generalized Sasakian space form admitting a trans-Sasakian structure. Further, we provide some applications in terms of Hamiltonians and Euler-Lagrange equation. In the last we also worked out some applications of Obata's differential equation.
Let M be an n−dimensional Riemannian manifold isometrically immersed in a m−dimensional Riemannian manifoldM. Then the Gauss and Weingarten formulas are∇ X Y = ∇ X Y + h(X, Y) and∇ X N = −A N X + ∇ ⊥ X N respectively, for all X, Y ∈ TM and N ∈ T ⊥ M. Where ∇ is the induced Levi-civita connection on M, N is a vector field normal to M, h is the second fundamental form of M, ∇ ⊥ is the normal connection in the normal bundle T ⊥ M and A N is the shape operator of the second fundamental form. The second fundamental form h and the shape operator are associated by the following formula The equation of Gauss is given by for all X, Y, Z, W ∈ TM. Where,R and R are the curvature tensors ofM and M respectively.
For any X ∈ TM and N ∈ T ⊥ M, φX and φN can be decomposed as follows and where PX (resp. tN) is the tangential and FX (resp. f N) is the normal component of φX ( resp. φN).
For any orthonormal basis {e 1 , e 2 , . . . , e n } of the tangent space T x M, the mean curvature vector H(x) and its squared norm are defined as follows (h(e i , e i ), h(e j , e j )), where n is the dimension of M. If h = 0 then the submanifold is said to be totally geodesic and minimal if H = 0. If h(X, Y) = (X, Y)H for all X, Y ∈ TM, then M is called totally umbilical.
The scalar curvature ofM is denoted byπ(M) 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 curvatureπ(L x ) of a L−plane is given bȳ Let {e 1 , . . . , e n } be an orthonormal basis of the tangent space T x M and if e r belongs to the orthonormal basis {e n+1 , . . . e m } of the normal space T ⊥ M, then we have and (h(e p , e q ), h(e p , e q )).
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 M n and in the Riemannian space formM m (c), respectively. Thus by Gauss equation, we have The global tensor field for orthonormal frame of vector field {e 1 , . . . , e n } on M n is defined as for all X, Y ∈ TM n . The above tensor is called the Ricci tensor. If we fix a distinct vector e u from {e 1 , . . . , e n } on M n , which is governed by χ. Then the Ricci curvature is defined by Consider the warped product submanifold N 1 × ψ N 2 . Let X be a vector field on M 1 and Z be a vector field on M 2 , then from Lemma 7.3 of [24], we have where ∇ is the Levi-Civita connection on M. For a warped product M = M 1 × ψ M 2 it is easy to observe that for X ∈ TM 1 and Z ∈ TM 2 .
∇ψ is the gradient of ψ and is defined as for all X ∈ TM. Let M be an n−dimensional Riemannian manifold with the Riemannian metric and let {e 1 , e 2 , . . . , e n } be an orthogonal basis of TM. Then as a result of (21), we get The Laplacian of ψ is defined by The Hessian tensor for a differentiable function ψ is a symmetric covariant tensor of rank 2 and is defined as For the warped product submanifolds N n 1 1 × ψ N n 2 2 , we have following well known result [9] n 1 p=1 n 2 q=1 κ(e p ∧ e q ) = n 2 ∆ψ ψ = n 2 (∆lnψ − ∇lnψ 2 ), where n 1 and n 2 are the dimensions of the submanifolds N n 1 1 and N n 2 2 respectively. Now, we state the Hopf's Lemma.
Hopf's Lemma [3]. If M is a m−dimensional connected compact Riemannian manifold. If ψ is a differentiable function on M s. t. ∆ψ ≥ 0 everywhere on M (or ∆ψ ≤ 0 everywhere on M), then ψ is a constant function.
For a compact orientable Riemannian manifold M with or without boundary and as a consequences of the integration theory of manifolds, we have where ψ is a function on M and dV is the volume element of M.

Contact CR-warped product submanifolds of a trans-Sasakian manifold
Suppose M be a n−dimensional submanifold isometrically immersed in an almost contact metric man-ifoldM(φ, ξ, η, ) such that the structure vector field ξ is tangent to M. The submanifold M is called contact CR-submanifold if it admits an invariant distribution D whose orthgonal complementary distribution D ⊥ is anti-invariant such that TM = D ⊕ D ⊥ ⊕ ξ , where φD ⊆ D, φD ⊥ ⊆ T ⊥ M and ξ is the 1-dimensional distribution spanned by ξ. If µ is the invariant subspace of the normal bundle T ⊥ M, then in the case of contact CR-submanifold, the normal bundle T ⊥ M can be decomposed as T ⊥ M = µ ⊕ φD ⊥ . A contact CR-submanifold is called contact CR-product submanifold if the distributions D and D ⊥ are parallel on M. As a generalization of the product manifold submanifolds one can consider warped product submanifolds. I. Hesigawa and I. Mihai [12] extended the study of Chen for the contact CR-warped product submanifolds of the Sasakian manifolds. Moreover, I. Mihai [13] obtained the estimation for the squared norm of second fundamental form in terms of the warping function for contact CR-warped product submanifolds in the setting of Sasakian space form. Further, K. Arslan et al. [14] extended the study of I. Mihai and Chen and they established an inequality for second fundamental form in terms of warping function for the contact CR-warped product submanifolds of a Kenmotsu space form. Using different techniques and methodology M. Atceken ([38], [39]) proved the inequalities for existence of contact CR-warped product submanifolds for Kenmotsu and cosymplectic space forms. Later Sibel Sular and CihanÖzgür [40] generalized these inequalities for contact CR-warped product submanifolds of generalized Sasakian space form admitting trans-Sasakian structure.
It is well known that two classes of almost contact metric manifolds namely Sasakian and Kenmotsu manifolds are quit different from each other and it has always been interesting to explore that how far a submanifold of a Sasakian manifold differ or resemble with that of Kenmotsu manifold. The setting of trans-Sasakian manifolds in a way unifies the two classes of manifolds. By studying the contact CRwarped product submanifolds of a generalized Sasakian space form admitting trans-Sasakian structure one clearly find out the deviations in the geometric behavior of a contact CR-warped product submanifold in the Sasakian and Kenmotsu space forms. Throughout, this study we consider n−dimensional contact CR-warped product submanifold M n = N n 1 T × ψ N n 2 ⊥ , such that the structure vector field ξ is tangential to N T , where n 1 and n 1 are the dimensions of the invariant and anti-invariant submanifold respectively. Now, we have the following initial result ⊥ be a contact CR-warped product submanifold isometrically immersed in a trans-Sasakian manifoldM, then for any X, Y ∈ TN T , Z ∈ TN ⊥ and N ∈ µ.
Proof. By using Gauss and Weingarten formulae in equation (3), we have taking inner product with Y and using (6), we get the required result.
To prove (ii), for any X ∈ TN T we have∇ using Gauss formula and (3), we get taking inner product with φN, above equation yields interchanging X by φX and using (4), the above equation gives From (26) and (27), we get the required result.
By the Lemma 3.1 it is evident that the isometric immersion N n 1 T × ψ N n 2 ⊥ into a trans-Sasakian manifold M is D− minimal. The Dminimality property provides us a useful relationship between the CR-warped product submanifold N T × ψ N ⊥ and the equation of Gauss.

Definition 3.1
The warped product N 1 × ψ N 2 isometrically immersed in a Riemannian manifoldM is called N i totally geodesic if the partial second fundamental form h i vanishes identically. It is called N i -minimal if the partial mean curvature vector H i becomes zero for i = 1, 2.
From Lemma 3.1, it is easy to conclude that m r=n+1 Thus it follows that the trace of h due to N T becomes zero. Hence in view of the Definition 3.1, we obtain the following important result.
Theorem 3.2. Let M n = N n 1 T × ψ N n 2 ⊥ be a contact CR-warped product submanifold isometrically immersed in a trans-Sassakian manifold. Then M n is D− minimal.
So, it is easy to conclude the following where H 2 is the squared mean curvature.

Ricci curvature for contact CR-warped product submanifold
In this section, we investigate Ricci curvature in terms of the squared norm of mean curvature and the warping function as follows ⊥ be a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space formM( f 1 , f 2 , f 3 ) admitting a trans-Sasakian structure . Then for each orthogonal unit vector field χ ∈ T x M orthogonal to ξ, either tangent to N T or N ⊥ we have (1) The Ricci curvature satisfies the following inequality.
(i) If χ is tangent to N n 1 T , then (2) If H(x) = 0 for each x ∈ M n , then there is a unit vector field X which satisfies the equality case of (1) if and only if M n 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 (30) holds identically for all unit vector fields tangent to N T at each x ∈ M n if and only if M n is mixed totally geodesic and D−totally geodesic contact CR-warped product submanifold in The equality case of (31) holds identically for all unit vector fields tangent to N ⊥ at each x ∈ M n if and only if M is mixed totally geodesic and either M n is D ⊥ -totally geodesic contact CR-warped product or The equality case of (1) holds identically for all unit tangent vectors to M n at each x ∈ M n if and only if either M n is totally geodesic submanifold or M n is a mixed totally geodesic totally umbilical and D− totally geodesic submanifold with dim N ⊥ = 2.
Where n 1 and n 2 are the dimensions of N n 1 T and N n 2 ⊥ respectively.
Proof. Suppose that M = N n 1 T × ψ N n 2 ⊥ be a contact CR-warped product submanifold of a generalized Sasakian space form. From Gauss equation, we have Let {e 1 , . . . , e n 1 , e n 1 +1 , . . . , e n } be a local orthonormal frame of vector fields on M n such that {e 1 , . . . , e n 1 } are tangent to N T and {e n 1 +1 , . . . , e n } are tangent to N ⊥ . So, the unit tangent vector χ = e A ∈ {e 1 , . . . , e n } can be expanded (32) as follows The above expression can be written as follows In view of the Lemma 3.1, the preceding expression takes the form By equation (16) Substituting the values of equation (35) in (34), we discover Since, M n = N n 1 T × ψ N n 2 ⊥ , then from (13), the scalar curvature of M n can be defined as follows The usage of (13) and (24), we derive Utilizing (38) together with (5) in (36), we have Considering unit tangent vector χ = e a , we have two choices: χ is either tangent to the base manifold N n 1 T or to the fibre N n 2 ⊥ .
Now, in a similar way as in case i using (51), we have Using similar steps of case i, the above inequality takes the form The last term of the above inequality can be written as Moreover, the fifth term of (57) can be expanded as Using last two values in (57), we have or equivalently On applying similar techniques as in the proof of case i, we arrive (h r nn − (h r n 1 +1n 1 +1 + · · · + h r nn )) 2 , which gives the inequality (31).
Next, we explore the equality cases of the inequality (30). First, we redefine the notion of the relative null space N x of the submanifold M n in the generalized Sasakian space formM m ( f 1 , f 2 , f 3 ) at any point x ∈ M n , the relative null space was defined by B. Y. Chen [6], as follows For A ∈ {1, . . . , n} a unit vector field e A tangent to M n at x satisfies the equality sign of (30) identically if and only if such that r ∈ {n + 1, . . . m} the condition (i) implies that M n is mixed totally geodesic contact CR-warped product submanifold. Combining statements (ii) and (iii) with the fact that M n is contact CR-warped 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 contact CR-warped product submanifold, the inequality sign of (30) holds identically for all unit tangent vector belong to N T at x if and only if where p ∈ {1, . . . , n 1 } and r ∈ {n + 1, . . . , m}. Since M n is contact CR-warped product submanifold, the third condition implies that h r pp = 0, p ∈ {1, . . . , n 1 }. Using this in the condition (ii), we conclude that M n is D−totally geodesic contact CR-warped product submanifold inM m ( f 1 , f 2 , f 3 ) and mixed totally geodesicness follows from the condition (i). Which proves (a) in the statement (3).
If the first case of (64) satisfies, then by virtue of condition (ii), it is easy to conclude that M n is a D ⊥ − totally geodesic contact CR-warped product submanifold inM m (c). This is the first case of part (b) of statement (3). For the other case, assume that M n is not D ⊥ −totally geodesic contact CR-warped product submanifold and dim N ⊥ = 2. Then condition (ii) of (64) implies that M n is D ⊥ − totally umbilical contact CR-warped product submanifold inM m ( f 1 , f 2 , f 3 ), which is second case of this part. This verifies part (b) of (3).
To prove (c) using parts (a) and (b) of (3), we combine (63) and (64). For the first case of this part, assume that dimN ⊥ 2. Since from parts (a) and (b) of statement (3) we conclude that M n is D−totally geodesic and D ⊥ − totally geodesic submanifold inM m ( f 1 , f 2 , f 3 ). Hence M n is a totally geodesic submanifold inM m (c).
For another case, suppose that first case does not satisfy. Then parts (a) and (b) provide that M n is mixed totally geodesic and D− totally geodesic submanifold ofM m ( f 1 , f 2 , f 3 ) with dimN ⊥ = 2. From the condition (b) it follows that M n is D ⊥ −totally umbilical contact CR-warped product submanifold and from (a) it is D−totally geodesic, which is part (c). This proves the theorem.
In view of (24), we have another version of the theorem 4.1 as follows Theorem 4.2. Let M = N n 1 T × ψ N n 2 ⊥ be a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space formM m ( f 1 , f 2 , f 3 ) admitting a admitting a trans-Sasakian structure. Then for each orthogonal unit vector field χ ∈ T x M orthogonal to ξ, either tangent to N T or N ⊥ . Then the Ricci curvature satisfies the following inequality.
(i) If χ is tangent to N T , then The equality cases are similar as in Theorem 4.1.

Some geometric applications in Mechanics
In this section, we investigate some applications of our attained inequalities, this section is divided in different subsections as follows

Application of Hopf's Lemma
In this subsection, we shall consider that the submanifold M n is a compact such that ∂M = φ. In the next theorem, we will see the application of Hopf's lemma for contact CR-warped product submanifold Theorem 5.1. Let M n = N n 1 T × ψ N n 2 ⊥ be a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space formM m ( f 1 , f 2 , f 3 ) admitting a admitting a trans-Sasakian structure. If the unit tangent vector χ orthogonal to ξ is tangent to either N T or N ⊥ , then M n is a simply Riemannian product submanifold if the Ricci curvature satisfy one of the following inequalities.
(i) the unit vector field χ is tangent to N T and (ii) the unit vector field χ is tangent to N ⊥ and Proof. Suppose that inequality (68) holds then from (30), we get ∆ψ ψ ≤ 0, which implies ∆ψ ≤ 0, on using Hopf's Lemma, we observe that the warping function is constant and the submanifold M n is Riemannian product. Similar result can be proved by using inequality (69).

First eigenvalue of the warping function
The lower bound of Ricci curvature contains numerous geometric properties. Suppose the submanifold M n is complete non-compact and x be a any arbitrary point on M n . For the Riemannian manifold M n , λ 1 (M n ) denotes the first eigenvalue of the following Dirichlet boundary value problem for a smooth function τ on M n ∆τ = λτ in M n and τ = 0 on ∂M n , where ∆ denotes the Laplacian on M n and defined as ∆τ = −div(∇φ). By the principle of monotonicity one has r < t which indicates that τ 1 (M n r ) > λ 1 (M n t ) and Lim r→∞ λ 1 (D r ) exists and first eigenvalue is defined as λ 1 (M) = Lim r→∞ λ 1 (D r ).
The relation between Ricci curvature and first eigenvalue is derived in the following theorem Theorem 5.2. Let M n = N n 1 T × ψ N n 2 ⊥ be a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space formM m ( f 1 , f 2 , f 3 ) admitting a trans-Sasakian structure. Suppose that the warping function lnψ is an eigen function of the Laplacian of M n associated to the first eigenvalue λ 1 (M n ) of the problem (70), then the following inequalities hold The Euler-Lagrange equation for L τ is given by Considering that the contact CR-warped product submanifold M n = N n 1 T × ψ N n 1 ⊥ is a compact orientable without boundary such that ∂M n = φ. Then in the following theorem we have a relation between Dirichlet energy, Ricci curvature and mean curvature vector Theorem 5.3. Let M n = N n 1 T × ψ N n 2 ⊥ be contact CR-warped product submanifold of a generalized Sasakian space form admitting a admitting a trans-Sasakian structure. Then we have the following inequalities for the Dirichlet energy of the warping function lnψ (i) If the unit vector field χ is tangent to N T then (ii) If the unit vector field χ is tangent to N ⊥ then The equality cases are similar as in Theorem 4.1.
Proof. For a positive valued differentiable function τ defined on a compact orientable Riemannian manifold without boundary, by theory of integration on Riemannian manifold we have M n ∆φdV = 0. On applying this fact for the warping function lnψ, we have Integrating inequality (30) with respect to volume element dV on contact CR-warped product submanifold M n , which is compact and orientable without boundary, we get Using the formula (78) and after some computation, the required inequality is derived. In a similar method, we can prove the inequality (81) Further, in the following theorem we will compute the Lagrangian for the warping function lnψ Theorem 5.4. Let M n = N n 1 T × ψ N n 2 ⊥ be a compact orientable contact CR-warped submanifold isometrically immersed in a generalized Sasakian space formM( f 1 , f 2 , f 3 ) admitting a trans-Sasakian structure such that the warping function lnψ satisfies the Euler-Lagrangian equation, then (i) If the unit vector field χ is tangent to N T , then L lnψ ≥ 1 2n 2 Ric(χ) − n 2 8n 2 H 2 − 1 2n 2 [(n+n 1 n 2 − 1) f 1 + 3 f 2 2 − (n 2 + 1) f 3 ].
(85) (ii) If the unit vector field χ is tangent to N ⊥ , then Where L lnψ is the Lagrangian of the warping function defined in (79). The equality cases are same as theorem. 4.1 Proof. The proof follows immediately on using (79) and (80) in theorem 30.
Further, the Hamiltonian for a local orthonormal frame at any point x ∈ M n is expressed as follows [23] H(p, x) = 1 2 n i=1 p(e i ) 2 .
On replacing p by a differential operator dφ, then from (22), we get In the next result we obtain a relation between Hamiltonian of warping function, Ricci curvature and squared norm of mean curvature vector Theorem 5.5. Let M n = N n 1 T × ψ N n 2 ⊥ be a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form admitting a trans-Sasakian structure, then the Hamiltonian of the warping function satisfy the following inequalities The equality cases are same as theorem 4.1 Proof. By the application of (88) in theorem 4.1, we get the required results.

Application of Obata's differential equation
This subsection is based on the study of Obata [22]. Basically, Obata characterized a Riemannian manifolds by a specific ordinary differential equation and derived that an n−dimensional complete and connected Riemannian manifold (M n , ) to be isometric to the n−sphere S n if and only if there exists a non-constant smooth function τ on M n that is the solution of the differential equation H τ = −cτ , where H τ is the Hessian of τ. Inspired by the work of Obata [22], we obtain the following characterization