Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds

In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of the warped product semi-slant submanifolds of the type $N_\theta\times{_{f}N_T}$ is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type $N_T\times{_{f}N_\theta}$. We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds [16, 18, 19, 24], while by a new technique, the second inequality is constructed to express the relation between extrinsic invariant (second fundamental form) and intrinsic invariant (scalar curvatures). The equality cases are also discussed.


Introduction
In a natural way, warped products appeared in differential geometry generalizing the class of Riemannian product manifolds to a much larger one, called warped product manifolds, which are applied in general relativity to model the standard space time, especially in the neighborhood of massive stars and black holes [20,21]. These manifolds were introduced by Bishop and O'Neill [3]. They defined warped products as follows: Let N 1 and N 2 be two Riemannian manifolds with Riemannian metrics g 1 and g 2 , respectively, and f > 0 be a differential function on N 1 . Consider the product manifold N 1 × N 2 with its projections π 1 : N 1 × N 2 → N 1 and π 2 : N 1 × N 2 → N 2 . Then their warped product manifold M = N 1 × f N 2 is the Riemannian manifold N 1 × N 2 = (N 1 × N 2 , g) equipped with the Riemannian structure such that X 2 = π 1⋆ (X) 2 + (f • π 1 ) 2 π 2⋆ (X) 2 , for any vector field X tangent to M , where ⋆ is the symbol for the tangent maps. A warped product manifold M = N 1 × N 2 is said to be trivial or simply Riemannian product if the warping function f is constant. For the survey on warped products as Riemannian submanifolds we refer to [12].
A (2m + 1)−dimensional C ∞ manifold (M , g, φ, ξ, η) is said to have an almost contact structure if there exist onM a tensor field φ of type (1, 1), a vector field ξ, a 1−form η and a Riemannian metric g satisfying η(X) = g(X, ξ), g(φX, φY ) = g(X, Y ) − η(X)η(Y ). (1.2) where X and Y are vector fields onM [5]. We shall use the symbol Γ(TM) to denote the Lie algebra of vector fields on the manifoldM . In the classification of almost contact structures, D. Chinea and C. Gonzalez [14] divided this structure into twelve well known classes, one of the classes that appears in this classification is denoted C 1 ⊕ C 5 ⊕ C 6 , according to their classification an almost contact metric manifold is a nearly trans-Sasakian manifold if it belongs to this class. Another line of thought C. Gherghe introduced nearly trans-Sasakian structure of type (α, β), which generalizes trans-Sasakian structure in the same sense as nearly Sasakian generalizes Sasakian ones, in this sense an almost contact metric structure (φ, ξ, η, g) onM is called a nearly trans-Sasakian structure if for any X, Y ∈ Γ(TM ). Moreover, a nearly trans-Sasakian of type (α, β) is nearly-Sasakian, or nearly Kenmotsu, or nearly cosymplectic according as β = 0 or α = 0 or α = β = 0. J. S. Kim et.al [17] initiated the study of semi-invariant submanifolds of nearly trans-Sasakian manifolds and obtained many results on the extrinsic geometric aspects of these submanifolds, whereas the slant submanifolds were studied in the setting of nearly trans-Sasakian manifolds by Al-Solamy and V.A. Khan [1]. Recently, we have initiated the study of CR-warped product in nearly trans-Sasakian manifolds [19]. In the present paper we consider warped product of proper slant and invariant submanifolds of nearly trans-Sasakian manifolds, called warped product semi-slant submanifolds. The paper is organized as follows: Section 2 is devoted to provide the basic definitions and formulas which are useful to the next section. In section 3, general and special non-existence results are proved. In section 4, the necessary lemmas for the two inequalities and some geometric obstructions are obtained. In section 5, a general inequality which generalizes obtained inequalities in [16,18,19,24] is established. In section 6, we develop a new technique by means of Gauss equation and apply to construct a general inequality for the second fundamental form in terms of the scalar curvatures of submanifolds and the warping function.

Preliminaries
Let M be a n-dimensional Riemannian manifold isometrically immersed in any Riemannian manifoldM . Then, Gauss and Weingarten formulae are respectively given by∇ for all X, Y ∈ Γ(T M ), where ∇ is the induced Riemannian connection on M , N is a vector field normal toM , 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. They are related as where g denotes the Riemannian metric onM as well as the metric induced on M . For any X ∈ Γ(T M ), we decompose φX as follows where P X and F X are the tangential and normal components of φX, respectively. For a submanifold M of an almost contact manifoldM , if F is identically zero then M is invariant and if P is identically zero then M is anti−invariant.
For any orthonormal basis {e 1 , · · · , e n } of the tangent space T x M , the mean curvature vector H(x) is given by Let (M, g) be a submanifold of a Riemannian manifoldM equipped with a Riemannian metric g. The equation of Gauss is given by for all X, Y, Z, W ∈ Γ(T M ), whereR and R are the curvature tensors ofM and M respectively, and h is the second fundamental form.
Definition 2.1 [11]. An immersion ϕ : N 1 × f N 2 →M is called N i -totally geodesic if the partial second fundamental form h i vanishes identically. It is called N iminimal if the partial mean curvature vector H i vanishes, for i = 1, 2.
The scalar curvature τ (x) of M is defined by where K(e i ∧ e j ) is the sectional curvature of the plane section spanned by e i and e j at x ∈ M . Let Π k be a k-plane section of T x M and {e 1 , · · · , e k } any orthonormal basis of Π k . The scalar curvature τ (Π k ) of Π k is given by [11] τ The scalar curvature of τ (x) of M at x is identical with the scalar curvature of the tangent space T is the scalar curvature of the image exp x (Π k ) of Π k at x under the exponential map at x. If Π 2 is a 2-plane section, τ (Π 2 ) is simply the sectional curvature K(Π 2 ) of Π 2 , [11,12,13]. Now, let us put h r ij = g(h(e i , e j ), e r ), where i, j ∈ {1, · · · , n}, and r ∈ {n + 1, · · · , 2m + 1}. Then, in view of the equation of Gauss, we have where K(e i ∧ e j ) andK(e i ∧ e j ) denote to the sectional curvature of the plane section spanned by e i and e j at x in the submanifold M and in the ambient manifoldM respectively. Taking the summation over the orthonormal frame of the tangent space of M in the above equation, we obtain whereτ (T x M ) = 1≤i<j≤nK (e i ∧ e j ) denotes the scalar curvature of the nplane section T x M , for each x ∈ M in the ambient manifoldM .
There are different classes of submanifolds which we introduce briefly such as slant submanifolds, CR-submanifolds and semi-slant submanifolds. We shall always consider ξ to be tangent to the submanifold M . For a slant submanifold M , there is a non zero vector X tangent to M at x, such that X is not proportional to ξ x , we denote by 0 ≤ θ(X) ≤ π/2, the angle between φX and T x M is called the slant angle. If the slant angle θ(X) is constant for all X ∈ T x M − ξ x and x ∈ M , then M is said to be a slant submanifold [7]. Obviously, if θ = 0, M is invariant and if θ = π/2, M is an anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant submanifold.
We recall the following result for a slant submanifold of an almost contact metric manifold.
Theorem 2.1 [7]. Let M be a submanifold of an almost contact metric manifold M , such that ξ ∈ Γ(T M ). Then M is slant if and only if there exists a constant λ ∈ [0, 1] such that Furthermore, if θ is slant angle, then λ = cos 2 θ. Following relations are straightforward consequence of equation (2.10) The idea of semi-slant submanifolds of almost Hermitian manifolds was given by N. Papaghuic [22]. In fact, semi-slant submanifolds were defined on the line of CR-submanifolds. These submanifolds are defined and investigated by Cabrerizo et.al for almost contact manifolds [9]. They defined these submanifolds as follows: Definition 2.2 [9]. A submanifold M of an almost contact manifoldM is said to be a semi-slant submanifold if there exist two orthogonal distributions D and D θ such that In the above definition, if θ = π/2 then M is contact CR-submanifold of M . If ν is the invariant subspace of the normal bundle T ⊥ M , then in case of semi-slant submanifolds, the normal bundle T ⊥ M can be decomposed as follows (2.13) For differential function ψ on M , the gradient gradψ and the Laplacian ∆ψ of ψ are defined respectively by for any vector field X tangent to M , where ∇ denotes the Riemannian connection on M .

Warped product submanifolds
In this section, we study warped product submanifolds of nearly trans-Sasakian manifolds. We recall the following results on warped products for later use.
Lemma 3.1 [21]. Let M = N 1 × f N 2 be a warped product manifold with the warping function f. Then for any X, Y ∈ Γ(T N 1 ) and Z, W ∈ Γ(T N 2 ), where ∇ and ∇ N2 denote the Levi-Civita connections on M and N 2 , respectively and gradf is the gradient of f .
In the following, we prove the non-existence of warped products of the form M = N 1 × f N 2 in a nearly trans-Sasakian manifold such that ξ is tangent to N 2 .
Theorem 3.1. LetM be a nearly trans-Sasakian manifold that is not nearly Sasakian and M = N 1 × f N 2 be a warped product submanifold ofM such that ξ is tangent to N 2 , then M is simply a Riemannian product of N 1 and N 2 , where N 1 and N 2 are any Riemannian submanifolds ofM .
Proof. For any X ∈ Γ(T N 1 ), we have (∇ X φ)ξ + (∇ ξ φ)X = −αX − βφX. By a simple calculation, this relation gives Taking the inner product with φX in (3.1) and using the fact that ξ is tangent to N 2 , the above equation takes the form Now, since X and ξ are orthogonal vectors, then by (1.2) we can write ξg(φX, φX) = ξg(X, X), which is equivalent to In view of Lemma 3.1 (ii), the right hand side of the above equation vanishes, hence From (3.2) and (3.3), we getβ X 2 = 0, this means that the first factor of the warped product vanishes, which proves the theorem completely.
In view of the above theorem we get a non-existence result about the warped product semi-slant submanifolds in a nearly trans-Sasakian manifold, i.e. there do not exist warped product semi-slant submanifolds N θ × f N T and N T × f N θ of a nearly trans-Sasakian manifold when the characteristic vector field ξ is a tangent to the second factor. Now we are going to show that the warped product N θ × f N T is also a Riemannian product if ξ is tangent to the first factor. Theorem 3.2. There do not exist warped product semi-slant submanifolds of the type M = N θ × f N T of a nearly trans-Sasakian manifoldM such that ξ is tangent to N θ , unlessM is nearly β-Kenmotsu.
Proof. Consider X as an arbitrary tangent vector to N T , then making use of (1.3) it follows (∇ X φ)ξ +(∇ ξ φ)X = −αX −βφX. This relation can be simplified as −φ∇ X ξ +∇ ξ φX − φ∇ ξ X = −αX − βφX. (3.4) Taking the inner product with X in (3.4), we get By orthogonality of X and ξ and Lemma 3.1 (ii), the left hand side of (3.5) vanishes identically, hence we reach α X 2 = 0, this means that the first factor of the warped product N θ × f N T vanishes, which proves the theorem. From the above discussion, we conclude that there do not exist warped product semi-slant submanifolds of type N θ × f N T in a nearly trans-Sasakian manifoldM in both the cases either ξ is tangent to the first factor or to the second. Also, the warped product N T × f N θ is just a Riemannian product when the characteristic vector field ξ is tangent to N θ . Now, we discuss the warped product submanifolds N T × f N θ such that ξ is tangent to N T .
First, we prove a key lemma characterizing geometric properties of warped product submanifolds N T × f N θ of a nearly trans-Sasakian manifoldM . Lemma 3.2. Let M = N T × f N θ be a warped product semi-slant submanifold of a nearly trans-Sasakian manifoldM such that ξ is tangent to N T . Then the following hold for any X, Y ∈ Γ(T N T ), Z, W ∈ Γ(T N θ ) and ζ ∈ Γ(ν).
Proof. The first three parts can be proved by the same way as we have proved for contact CR-warped products in [19]. Now, as we consider ξ is tangent to N T , then for any X ∈ Γ(T N T ) and Z ∈ Γ(T N θ ), we have Taking the inner product with Z, we obtain (3.6) Also, we have . Taking the inner product with Z and using Lemma 3.1 (ii), we obtain g((∇ X φ)Z, Z) = 0. (3.7) Similarly, we can obtain Then from (3.6), (3.7) and (3.8), we obtain part (iv) of the lemma. Now, from the structure of nearly trans-Sasakian manifolds and Lemma 3.1 (ii), we have for any X ∈ Γ(T N T ) and Z ∈ Γ(T N θ ) such that ξ is tangent to N T . Again, by Lemma 3.1 (ii) and Gauss-Weingarten formulas, we obtain g((∇ X φ)P Z, Z) = g(h(X, P Z), F Z) − g(h(X, Z), F P Z) (3.10) and g((∇ P Z φ)X, Z) = g(h(X, P Z), F Z) + (X ln f ) cos 2 θ Z 2 . Interchanging Z by P Z in (3.12), we obtain −2g(h(X, P Z), F Z) + g(h(X, Z), F P Z) = {βη(X) − (X ln f )} cos 2 θ Z 2 .

An inequality for warped product submanifolds
In the setting of almost contact structures, many authors have proved general inequalities in terms of the squared norm of the second fundamental form and the gradient of the warping function in various structures [16,18,19,24]. In fact, all these inequalities are the extension of the original inequality constructed by Chen in the almost Hermitian setting [10]. However, no one proved this relation for warped product semi-slant submanifolds. For this reason, our inequality generalizes the inequalities obtained for CR-warped products in almost contact setting. Another reason is that a nearly trans-Sasakian structure includes all almost contact structures as a special case. From now on, we shall follow the following orthonormal basis frame of the ambient manifoldM for the warped product semi-slant submanifold M = N T × f N θ such that ξ is tangent to N T . We shall denote D and D θ for the tangent spaces of N T and N θ , respectively instead of T N T and T N θ .
Note. In the inequality (5.1), if α = 0 and β = 1, then it reduces to which is the inequality for nearly Kenmotsu manifolds. Also, If α = 1 and β = 0, then the inequality reduces for the nearly Sasakian manifolds. The equality cases can also be discussed.

Another inequality for warped products
Let ϕ : M = N 1 × f N 2 −→M be an isometric immersion of a warped product N 1 × f N 2 into a Riemannian manifoldM of constant sectional curvature c. Denote by n 1 , n 2 , n the dimensions of N 1 , N 2 , N 1 × f N 2 , respectively. Then for unit vector fields X, Z tangent to N 1 , N 2 , respectively, we have If we choose the local orthonormal frame e 1 , · · · , e n such that e 1 , · · · , e n1 are tangent to N 1 and e n1+1 , · · · , e n are tangent to N 2 , then we have for each j = n 1 + 1, · · · , n. In this section, our aim is to develop a new method which is giving a useful formula for the squared norm of the mean curvature vector H under ϕ, Geometrically, this formula declares the N T -minimality of ϕ.
We know that Taking in consideration that (n = n 1 + n 2 ), where n 1 and n 2 are the dimensions of N T and N θ , respectively, then it follows (h r 11 + · · · + h r n1n1 + h r n1+1n1+1 + · · · + h r nn ) 2 .
By the end of this discussion, we can state the following lemma.
i.e., ϕ is N T -minimal immersion, where H is the mean curvature vector and n 1 , n 2 , n and (2m + 1) are the dimensions of N T , N θ , M andM , respectively. From the Gauss equation and the above key Lemma 5.1, we are able to state and prove the following general inequality.
Theorem 5.1. Let ϕ : M = N T × f N θ −→M be an isometric immersion from a warped product semi-slant submanifold into a nearly trans-Sasakian manifold M such that ξ is tangent to N T . Then, we have (ii) If the equality sign in (i) holds identically, then N T and N θ are totally geodesic and totally umbilical submanifolds inM , respectively.
Proof. We start by recalling (2.9) as a consequence of (2.5) as Making use of (2.6) in the above equation, we deduce K(e i ∧ e j ) − 2τ (T N T ) − 2τ (T N θ ) + 2τ (T M ) + n 2 H 2 .
The above equation is equivalent to the following form The above equation takes the following form when we add and subtract the same term on the right hand side Similarly, we can add and subtract the same term for the sixth term in the above equation and finally, we derive (h r n1+1n1+1 + · · · + h r nn ) 2 + n 2 H 2 .
Taking account of Lemma 5.1, we get the inequality (i). For the equality case, from the last relation we get From (5.6) and (5.7), we obtain that the immersion ϕ : M →M is totally geodesic. Also, from Corollary 3.1 we know that the immersion N T → M is totally geodesic and the immersion N θ → M is totally umbilical, hence the result (ii).