A General Inequality for CR-Warped Products in Generalized Sasakian Space Form and Its Applications

In the present paper, by considering the Gauss equation in place of the Codazzi equation, we derive new optimal inequality for the second fundamental form of CR-warped product submanifolds into a generalized Sasakian space form. Moreover, the inequality generalizes some inequalities for various ambient space forms.


Introduction
The fundamental idea of warped product manifolds was first initiated in [1] with manifolds of negative curvature. Let N 1 and N 2 be two Riemannian manifolds endowed with Riemannian matrices g 1 and g 2 , respectively, such that f : N 1 ⟶ ð0,∞Þ is a positive smooth function on N 1 . Then, the warped product M = N 1 × f N 2 is characterized as the product manifold N 1 × N 2 with the equipped metric g = g 1 + f 2 g 2 . In particular, if f = constant, then M turned to be a Riemannian product manifold; otherwise, M is called a nontrivial warped product manifold. Let M = N 1 × f N 2 be a nontrivial warped product manifold. Then, for any vector fields X, Y ∈ ΓðTN 1 Þ and Z ∈ ΓðTN 2 Þ. If we consider a local orthonormal frame fe 1 , e 2 ,⋯,e n g such that fe i g 1≤i≤n 1 ∈ N 1 and fe j g n 1 +1≤j≤n ∈ N 2 , we have In [2], Chen established the inequality for the squared norm of the mean curvature and the warping function f of a CR-warped product N T × f N ⊥ , where N ⊥ is a totally real submanifold and N T is a holomorphic submanifold, isometrically immersed in a complex space form as follows.
Theorem 1 (see [2]). N n 1 T × f N n 2 ⊥ be a CR-warped product into a complex space formMð4cÞ with constant sectional curvature c. Then, where Δ is the Laplacian operator of N T . Moreover, the equality holds if and only if N T is totally geodesic and N ⊥ is totally umbilical inMð4cÞ.
Moreover, Theorem 1 is extended to CR-warped product submanifolds in a generalized Sasakian space form by using the same technique.
Theorem 2 (see [3]). Let N n 1 T × f N n 2 ⊥ be a contact CR-warped product submanifold of a generalized Sasakian space form Mðλ 1 , λ 2 , λ 3 Þ such that the structure vector field ξ is tangent to base manifold. Then, the following inequality is satisfied: where Δ denotes the Laplace operator on N Furthermore, Mustafa et al. [4] recalled some fundamental problems of CR-warped products in Kenmotsu space forms as to simple relationships between the second fundamental form and the main intrinsic invariants by using the Gauss equation. In [5][6][7], some sharp inequalities are established for the sectional curvature of warped product pointwise semislant submanifolds in various space forms such as a Sasakian space form, a cosymplectic space form, a Kenmotsu space form, and a complex space form in terms of the Laplacian and the squared norm of a warping function with pointwise slant immersions. Afterward, several geometers [1,2,4,[8][9][10][11][12][13][14][15][16][17][18] obtained similar inequalities for different types of warped products in different kinds of structures.
Al-Ghefari et al. [3] proved the existence of CR-warped product submanifolds of type N T × f N ⊥ in trans-Sasakian manifolds. They obtained an inequality for the second fundamental form with constant sectional curvature in terms of a warping function. Moreover, the nonexistence of CRwarped products of the form N ⊥ × f N T in a generalized Sasakian space form was proved in [19].
In this paper, we shall establish a Chen-type inequality for CR-warped product submanifolds in a generalized Sasakian space form by considering the nontrivial case N T × f N ⊥ . We also find some applications of the inequality in the compact Riemannian manifold by using integration theory on manifolds. Our future work then is combining the work done in this paper with the techniques of singularity theory presented in [20][21][22][23] to explore new results on manifolds.
Remark 4. The characteristics are as follows: (a) If λ 1 = c + 3/4 and λ 2 = λ 3 = ðc − 1Þ/4, thenM is a Sasakian space form [25] (b) If λ 1 = ðc − 3Þ/4 and λ 2 = λ 3 = ðc + 1Þ/4, thenM is a Kenmotsu space form [6] (c) If λ 1 = λ 2 = λ 3 = c/4, thenM is a cosymplectic space form [26] Let ∇ and ∇ ⊥ be the induced Riemannian connections on the tangent bundle TM and the normal bundle T ⊥ M of a submanifold M of an almost contact metric manifold ðM, φ, η, ξÞ with the induced metric g. Then, the Gauss and Weingarten formulas are given by for U, V ∈ ΓðTMÞ and N ∈ ΓðT ⊥ MÞ, where h and A N are the second fundamental form and the shape operator on 2 Advances in Mathematical Physics M. We have the relation: for U, V ∈ ΓðTMÞ and N ∈ ΓðT ⊥ MÞ. For any tangent vector U ∈ ΓðTMÞ and normal vector N ∈ ΓðT ⊥ MÞ, we have where TUðtNÞ and FUð f NÞ are tangential and normal components of φUðφNÞ, respectively. If T is identically zero, then a submanifold M is called a totally real submanifold. The Gauss equation with curvature tensorsR and R onM and M, respectively, is defined bỹ for any U, V, Z, W ∈ ΓðTMÞ. The mean curvature vector H for a local frame fe 1 , e 2 , ⋯, e n g of the tangent space T M on M is defined by The scalar curvature τ for a Riemannian submanifold M is given by where Kðe i ∧ e j Þ is the sectional curvature of section plane and spanned by e i and e j . Let G r be an r-plane section on TM and let fe 1 , e 2 , ⋯, e r g be a orthonormal basis of G r . Then, the scalar curvature τðG r Þ of G r is given by Similarly, we classify a Riemannian submanifold M said to be totally umbilical and totally geodesic if hðU, VÞ = gðU, VÞH and hðU, VÞ = 0, respectively, for any U, V ∈ ΓðTMÞ.
M is a compact orientable Riemannian submanifold without boundary. Thus, we have ð where dV is the volume element of M [27].

Main Inequalities of CR-Warped Products
We are mentioning that in the following study, we shall consider the structure field ξ tangent to the base manifold of warped product manifold. In this main section, we classify the contact CR-warped product submanifolds in a trans-Sasakian manifold.

Advances in Mathematical Physics
warped product submanifold into a trans-Sasakian manifold ðM, φ, η, ξÞ such that N T is invariant submanifold of dimension n 1 = 2d 1 + 1 tangent to ξ. Then, N T is always ℓ-minimal submanifold ofM.
Proof. We skip the proof of the above lemma due to the similar proof of Theorem 4.2 in [4]. ☐ By helping the above lemma, the following result can be obtained as follows.
where n 2 is the dimension of anti-invariant submanifold N ⊥ and Δ is the Laplacian operator of N T (ii) the equality holds in (22) if and only if N T is totally geodesic and N ⊥ is totally umbilical inM. Moreover, M is minimal submanifold ofM Proof. It can be easily proven as the proof of Theorem 4.4 in [4] if we consider a Riemannian submanifold as a CR-warped product submanifold, and the base manifold is a trans-Sasakian manifold instead of a Kenmotsu manifold. Now, we prove our main theorem using Proposition 8 for a generalized Sasakian space form. ☐ Theorem 9. Let ℓ : M = N T × f N ⊥ ⟶Mðλ 1 , λ 2 , λ 3 Þ be an isometric immersion from an n -dimensional contact CRwarped product submanifold of a generalized Sasakian space formMðλ 1 , λ 2 , λ 3 Þ. Then, the second fundamental form is given by where n 1 = dim N T , n 2 = dim N ⊥ , and Δ is the Laplacian operator on N T . The equality holds in (23) if and only if N T and N ⊥ are totally geodesic and totally umbilical submanifolds inMðλ 1 , λ 2 , λ 3 Þ, respectively, and hence, M is a minimal submanifold ofMðλ 1 , λ 2 , λ 3 Þ.
Proof. Substituting X = W = e i and Y = Z = e j in (10) Summing up along the orthonormal vector fields of M, it can be derived from the above as As for an n-dimensional CR-warped product submanifold tangent ξ, one can derive kTk 2 = n − 1 from (15) On the other hand, by helping the frame field of TN ⊥ , we have Similarly, we considered that ξ is tangent to invariant submanifold N T . Then, using the frame vector fields of T N T , we get from (24) 2τ Therefore, using (26), (27), and (28) in Proposition 8, we get the required result. The equality case follows from Proposition 8. Thus, the proof is completed. ☐

Corollary 13.
Let ln f be a harmonic function on N T . Then there does not exist any CR-warped product submanifold N T × f N ⊥ into a generalized Sasakian space formMðλ 1 , λ 2 , λ 3 Þ with c ≤ −λ 1 .

Corollary 14.
Assume that ln f is a nonnegative eigenfunction on N T with the corresponding nonzero positive eigenvalue. Then, there does not exist any CR-warped product submanifold N T × f N ⊥ into a generalized Sasakian space form Mðλ 1 , λ 2 , λ 3 Þ with c ≤ −λ 1 .
Proof. From the minimum principle property, we obtain From (23) and (35), we get the required result (34). ☐

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Conflicts of Interest
The authors declare no competing interest.