Abstract
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.
1. Introduction
The fundamental idea of warped product manifolds was first initiated in [1] with manifolds of negative curvature. Let and be two Riemannian manifolds endowed with Riemannian matrices and , respectively, such that is a positive smooth function on . Then, the warped product is characterized as the product manifold with the equipped metric . In particular, if , then turned to be a Riemannian product manifold; otherwise, is called a nontrivial warped product manifold. Let be a nontrivial warped product manifold. Then, for any vector fields and . If we consider a local orthonormal frame such that and , we have
In [2], Chen established the inequality for the squared norm of the mean curvature and the warping function of a CR-warped product , where is a totally real submanifold and is a holomorphic submanifold, isometrically immersed in a complex space form as follows.
Theorem 1 (see [2]). be a CR-warped product into a complex space form with constant sectional curvature . Then, where is the Laplacian operator of . Moreover, the equality holds if and only if is totally geodesic and is totally umbilical in .
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 be a contact CR-warped product submanifold of a generalized Sasakian space form such that the structure vector field is tangent to base manifold. Then, the following inequality is satisfied: where denotes the Laplace operator on . The equality holds if and only if is a totally geodesic submanifold of ; in this case, is a generalized Sasakian space form of .
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 [5–7], 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, 8–18] 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 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 CR-warped products of the form 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 . 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 [20–23] to explore new results on manifolds.
2. Preliminaries
An almost contact metric manifold is an odd-dimensional manifold , endowed with a field of an endomorphism on the tangent space, the Reeb vector field , a 1-form and admits Riemannian metric satisfying for any . An almost contact metric manifold is said to be trans-Sasakian manifold (cf. [12, 13]) if
for any , where is the Riemannian connection on . If we replace and in (8), we find that , which implies that . For a trans-Sasakian manifold, (8) implies
Remark 3. We classify a trans-Sasakian manifold in the following way: (a)If and in (8), a trans-Sasakian manifold becomes a cosymplectic manifold [7](b)If and in (8), it is a Sasakian manifold [5](c)If and in (8), it is a Kenmotsu manifold [6](d)-Sasakian manifold and -Kenmotsu manifold can be derived from the tans-Sasakian manifold when and in (8), respectively
Given an almost contact metric manifold , it is said to be a generalized Sasakian space form if there exist three functions , and on such that the curvature tensor is for any [24].
Remark 4. The characteristics are as follows: (a)If and , then is a Sasakian space form [25](b)If and , then is a Kenmotsu space form [6](c)If , then is a cosymplectic space form [26]
Let and be the induced Riemannian connections on the tangent bundle and the normal bundle of a submanifold of an almost contact metric manifold with the induced metric . Then, the Gauss and Weingarten formulas are given by for and , where and are the second fundamental form and the shape operator on . We have the relation: for and . For any tangent vector and normal vector , we have where and are tangential and normal components of , respectively. If is identically zero, then a submanifold is called a totally real submanifold. The Gauss equation with curvature tensors and on and , respectively, is defined by for any . The mean curvature vector for a local frame of the tangent space on is defined by
The scalar curvature for a Riemannian submanifold is given by where is the sectional curvature of section plane and spanned by and . Let be an -plane section on and let be a orthonormal basis of . Then, the scalar curvature of is given by
Similarly, we classify a Riemannian submanifold said to be totally umbilical and totally geodesic if and , respectively, for any .
Furthermore, if , then is minimal in . If preserves any tangent space of tangent to the structure vector field , i.e., , for each ; then, is called an invariant submanifold. Similarly, is called an anti-invariant submanifold tangent to the Reeb vector field if , for each . To generalize these definitions, we give the following definition.
Definition 5. A submanifold including the structure vector field of an almost contact metric manifold is characterized to be a contact CR-submanifold if the pair of orthogonal distributions and exists such that (i), where is -dimensional distribution spanned by (ii)the distribution is invariant, i.e., (iii)the distribution is anti-invariant, i.e.,
If the dimensions of invariant distribution and anti-invariant distribution of a contact CR-submanifold of are and , respectively, such that , then is invariant and anti-invariant if . It is called a proper contact CR-submanifold if neither nor . The normal bundle of a contact CR-submanifold with an invariant subspace under can be decomposed as
is a compact orientable Riemannian submanifold without boundary. Thus, we have where is the volume element of [27].
3. Main Inequalities of CR-Warped Products
We are mentioning that in the following study, we shall consider the structure field 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.
Lemma 6. Let be a CR-warped product submanifold in a trans-Sasakian manifold. Then, for , , and
Proof. From (11)(i), (8), and (5), we obtain Since is totally geodesic in with , (9) implies the results.☐
Lemma 7. Let be an isometric immersion from an -dimensional contact CR-warped product submanifold into a trans-Sasakian manifold such that is invariant submanifold of dimension tangent to . Then, is always -minimal submanifold of .
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.
Proposition 8. Assume that is an isometric immersion of an -dimensional contact CR-warped product submanifold into a trans-Sasakian manifold . Thus, (i)the squared norm of the second fundamental form of is satisfied:where is the dimension of anti-invariant submanifold and is the Laplacian operator of (ii)the equality holds in (22) if and only if is totally geodesic and is totally umbilical in . Moreover, is minimal submanifold of
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 be an isometric immersion from an -dimensional contact CR-warped product submanifold of a generalized Sasakian space form . Then, the second fundamental form is given by where , , and is the Laplacian operator on . The equality holds in (23) if and only if and are totally geodesic and totally umbilical submanifolds in , respectively, and hence, is a minimal submanifold of .
Proof. Substituting and in (10), we get Summing up along the orthonormal vector fields of , it can be derived from the above as As for an -dimensional CR-warped product submanifold tangent , one can derive from (15)(ii); we obtain On the other hand, by helping the frame field of , we have Similarly, we considered that is tangent to invariant submanifold . Then, using the frame vector fields of , we get from (24) 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.☐
4. Geometric Applications
Remark 10. Consider and in Theorem 9. It is the generalization of Theorem 4.6 in [4] for the result of contact CR-warped products in Kenmotsu space forms.
Remark 11. If we put and in Theorem 9, then it generalizes Corollary 4.6 in [5].
Remark 12. If in Theorem 9, then Theorem 9 coincides with Theorem 1.2 in [26].
Corollary 13. Let be a harmonic function on . Then there does not exist any CR-warped product submanifold into a generalized Sasakian space form with.
Corollary 14. Assume that is a nonnegative eigenfunction on with the corresponding nonzero positive eigenvalue. Then, there does not exist any CR-warped product submanifold into a generalized Sasakian space form with .
Theorem 15. Let be a compact orientated CR-warped product into a generalized Sasakian space form . Then, is a simply Riemannian product if where and .
Proof. From Theorem 9, we get We obtain Now, if Then, from (31), we find which is impossible for a positive integral function, and hence, , i.e., is a constant function on . Thus, by the definition of a warped product manifold, is trivial. The converse part is straightforward.☐
Corollary 16. Assume that is a CR-warped product submanifold in a generalized Sasakian space form . Let be a compact invariant submanifold and be nonzero eigenvalue of the Laplacian on . Then,
Proof. From the minimum principle property, we obtain From (23) and (35), we get the required result (34).☐
Data Availability
There is no data used for this manuscript.
Conflicts of Interest
The authors declare no competing interest.
Authors’ Contributions
All authors have equal contribution and finalized.
Acknowledgments
The authors extend their appreciation to the deanship of scientific research at King Khalid University for funding this work through a research group program under grant number R.G.P.2/71/41.