Abstract
In this study, we attain some existence characterizations for warped product pointwise semi slant submanifolds in the setting of Sasakian space forms. Moreover, we investigate the estimation for the squared norm of the second fundamental form and further discuss the case of equality. By the application of attained estimation, we obtain some classifications of these warped product submanifolds in terms of Ricci soliton and Ricci curvature. Further, the formula for Dirichlet energy of involved warping function is derived. A nontrivial example of such warped product submanifolds is also constructed. Throughout the paper, we will use the following acronyms: “WP” for warped product, “WF” for warping function, “AC” for almost contact, and “WP-PSS” for the warped product pointwise semi slant.
1. Introduction
In the field of geometry, an increased interest is observed for the exploration of the warped product manifolds due to its applicability in the theory of relativity and physics [1]. Earlier researchers have shown that warped products facilitate the computation of the Einstein field equations [1]. In addition, the concept of warped product manifolds is used in the notion of modelling of space-time near black holes [2], such as Schwarzschild space-time , where , the base is a half plane , and the unit sphere is the fibre. At some instances, Schwarzschild space-time also acts similar to the black hole. The Robertson-Walker model is an example of a warped product which is considered as a cosmological model adopted for the space-time modelling of the universe [3].
Identifying Dirichlet energy of smooth functions is considered as an integral part in the field of physics and engineering. Moreover, Dirichlet energy is found to be an analog of kinetic energy. Let be any real valued smooth function on a compact manifold , then the Dirichlet energy of is defined by where is the gradient of and is the volume element. It is clear that . It is well recognized that a product manifold cannot be used for the representation of nonzero (constant) curvature manifolds. Therefore, the concept of the warped product manifold emerged due to the nonexistence of negative curvature for the Riemannian product of manifolds. Therefore, Bishop and O’Neill [4] initiated the concept of warped product manifolds for devising the class of manifolds for the negative or nonpositive curvature. A warped product manifold (see definition in Section 2) is a generality of a product manifold. As the warping function of warped product manifolds is a positive valued smooth function, the focus of the present study is on examining the Dirichlet energy of such functions.
The study of Bishop and O’Neill [4] has enlightened a few intrinsic properties of the warped product manifolds. Initial extrinsic studies of the warped product manifold in the almost complex setting were performed by Chen [5, 6] while obtaining some results of existence for CR-submanifold as CR-warped product submanifold in Kaehler manifolds. Furthermore, contact CR-warped product submanifolds were investigated in the study of Hasegawa and Mihai [7] in the almost contact settings. Warped product manifolds are also investigated in the contact setting by many other geometers which have attained various existence results [8–11].
Warped product pointwise semi slant submanifold is another generalized class of warped product semi slant submanifolds and contact CR-warped product submanifolds. In [12], Park studied such warped product submanifolds. After that, Mihai and Uddin extended this study in Sasakian manifolds and acquired some optimal inequalities related to warping function and second fundamental form. Warped product pointwise semi slant submanifolds for almost contact and almost complex manifolds were explored in [13–15].
On the other hand, the gradient Ricci soliton is extensively investigated in physics, theory of relativity, and differential geometry. The classification results related to Ricci soliton and gradient Ricci solitons with the warped product structure have been established in [16–19]. Moreover, the Ricci curvature has a significant nature in Riemannian geometry; for example, Ricci flat is used to solve the Einstein field equation on a Riemannian manifold in which the cosmological constant vanishes. More clearly, in the general relativity theory, the Ricci tensor is correlated with Einstein’s field equation to study the material contents of the universe. So, in comparison with Riemannian curvature, the Ricci curvature is more significant in the theory of relativity and physics.
In this study, warped product pointwise semi slant submanifolds are studied in the setting of Sasakian space forms and some associated inequalities are attained. The study also provides estimates for the squared norm of the second fundamental form concerning the slant function and warping function and then addresses the case of equality. Some applications of the case of equality and the derived inequality are explored. More precisely, Dirichlet energy of the warping functions is computed via the obtained inequality. Finally, a classification of the WP-PSS submanifolds admitting the gradient Ricci soliton, in respect of Ricci curvature and second fundamental form, is obtained, and some existence results are also established.
The study consists of five sections in which the second section is dedicated to pave the way for the other sections by providing the fundamental definition, formulas, and preliminary outcomes needed for the study. Section 3 investigates the WP-PSS submanifold existence in Sasakian space forms and provides the proof of the main findings of the study. In addition, it gives a nontrivial example for the WP-PSS submanifolds in the setting of a Sasakian manifold. The formulas for Dirichlet energy of warping function is attained in Section 4 by using the derived inequality. The paper is concluded in Section 5, followed by its bibliography.
2. Preliminaries
A manifold of dimension is said to admit an AC structure if on a tensor field of type a vector field , and a 1-form fulfilling [20]
The manifold with the structure is called AC metric manifold. On an AC metric manifold , a Riemannian metric satisfying for all , where is the tangent bundle of .
An AC metric structure is referred to as Sasakian manifold if it fulfills the following relation [20]: for any where denotes the Riemannian connection of the metric . Then, we have
A Sasakian space form [20] is a Sasakian manifold with constant -holomorphic sectional curvature . Moreover, the following formula gives the curvature tensor of Sasakian space form, : vector fields on .
Let be a submanifold of an AC-metric manifold with induced metric , and the vector field is tangential to The Riemannian connection of induces canonically the connections and on the normal bundle and the tangent bundle of , resp. Then, the Weingarten and Gauss formulas are as follows: and where and are the shape operator and the second fundamental form, resp. For the immersion of into , and verify the relation
If is a submanifold of a Riemmanian manifold , then the equation of Codazzi is as follows: where denotes the normal component of the curvature tensor
If and denote the tangential and normal component of , respectively, then . Similarly, for any normal vector field ,
Thus, (resp., ) is a normal (resp., tangential) valued 1-form on (resp. ) and (resp., ) is 1-1 tensor field on (resp., ). The covariant derivatives of , , and are given by
From equations (4), (8), (7), (11), and (12), we have
Let be a local orthonormal basis of vector fields on -dimensional submanifold Hence, the mean curvature vector of is obtained from the formula and the squared norm of the second fundamental form is given by
A submanifold of is called totally umbilical if , and is defined as totally geodesic if , .
Definition 1 (see [21]). A submanifold of an AC metric manifold is slant if the angle between and is constant for any and . Then the constant angle s.t. is called slant angle of in
Definition 2 (see [21]). If , the submanifold is invariant submanifold and is anti-invariant submanifold if . If is neither nor , is proper slant submanifold.
Etayo [22] introduced the concept of pointwise slant submanifolds as a natural generality of the slant submanifolds of almost Hermitian manifolds. Later, Chen and Garay [23] investigated pointwise slant submanifolds in the setting of almost Hermitian manifolds and attained some primary results. A step forward, Park [12] extended the notion of pointwise slant submanifolds in the setting of AC-metric manifolds. Recently, Uddin and Al-Khalidi [24] modified the definition of pointwise slant submanifolds for AC-metric manifolds. More precisely, a submanifold of an AC-metric manifold is referred to as poinwise slant submanifold if such that the angle between and is independent of the choice of nonzero vector field . Hence, is dealt with as the function on which is known as a slant function of the pointwise slant function. Now, we attain the following characterizing theorem.
Theorem 3 (see [24]). Suppose that is a submanifold of an AC-metric manifold such that . Then, is pointwise slant if and only if where
Consequently, we have Recently, Mihai and Uddin [15] defined and studied pointwise semi slant submanifold of Sasakian manifolds.
Definition 4. A submanifold of an AC-metric manifold is referred to as a pointwise semi slant submanifold if two orthogonal complementary distributions and on such that (i)(ii) is invariant(iii) is pointwise slant with a slant function
The WP manifolds are the generalized form of the product manifolds and are defined as follows.
Let and be two Riemannian manifolds with the Riemannian metrics and , resp., and be a positive differentiable map on . Let be the projection maps given by and , The WP [4] is the manifold endowed with the Riemannian structure s.t. for all where denotes the tangential map (the differential on the tangent bundle). Here, is known as the WF of the WP manifold. is a trivial WP if the WF is constant.
Let be a vector field on and be a vector field on , then we induce from Lemma 7.3 of [3]: where is the Levi-Civita connection on . For , it can be seen that for and The gradient of is denoted by and is defined as
Let be a Riemannian manifold of dimension with as an orthogonal basis of and a Riemannian metric of . Then, as a result of (27), we get
The Laplacian of a smooth function on a Riemannian manifold is defined by the following equation:
For a smooth function on , the Hessian of is defined as for any
The Hessian tensor for a differential function is a symmetric covariant tensor of rank 2 and is defined as
Throughout the study, we used the following relation between the Hessian tensor and the Laplacian of a smooth function on a Riemannian manifold:
Hopf’s Lemma [25]. If is an -dimensional connected compact Riemannian manifold and is a differentiable function on s.t. everywhere on (or everywhere on ), then is a constant function.
For a compact orientable Riemannian manifold with or without boundary and as a consequence of the integration theory on manifolds, we have [26] where is a function on and is the volume element of .
The Ricci soliton idea was given by Hamilton [26]. It is regarded as the natural generality of Einstein metrics, and it is the self-similar solution of the Ricci flow If a smooth vector field such that the Ricci tensor meets the following condition: for any constant , where is the Lie derivative, then the metric on a complete Riemannian manifold is named as Ricci soliton. If , and , then the Ricci soliton is called expanding, steady, and shrinking, respectively. If we specify for a smooth function on , then admits gradient Ricci soliton with the potential function . For this case, (34) takes the form
Since the Laplacian and the gradient are related as , in terms of Hessian, (35) can be expressed as
Note 1. is an Einstein manifold if the potential function is constant on a gradient Ricci soliton.
3. Warped Product Pointwise Semi Slant Submanifolds
In [12], Park proved that the WP-PSS submanifolds of the type of the contact metric manifolds exist and achieved the result of the following lemma.
Lemma 5. Suppose that is a WP-PSS of a Sasakian manifold such that , where and are invariant and pointwise slant submanifolds of , respectively. Then, we have and .
In particular, if we replace by in the above equation, then one can conclude the following:
Now, let be a WP-PSS of a Sasakian manifold and we consider . If is invariant distribution and is proper pointwise slant distribution with as a slant function, then and are decomposed (resp.) as follows: where is the orthogonal complementary distribution of in . One can easily check that is an invariant subbundle of regarding .
In view of the above direct decomposition, can be written as for , where and are the components of in the normal subbundles and , respectively. Moreover, if is a local orthonormal basis of vector fields of , we have where
To ensure the existence, we construct an example of a WP-PSS of the type in the Sasakian manifold with tangent to
Example 1. We know that denotes a Sasakian manifold with the structure as follows: where , are the Cartesian coordinates. It is well known that is an orthonormal basis of
Now, consider the 9-dimensional Sasakian manifold and a submanifold isometrically immersed in as follows: and consider a basis of orthogonal vector fields on as
Hence, it can be seen that and define the invariant and pointwise slant distributions with as a slant function on the Sasakian manifold If and stand for the integral manifold of and , respectively, the metric on is defined by
Then, is a WP-PSS with the
In this section, for convention, we denote by and as the vector fields of respective tangent bundles of and . At first, some initial results need to be proved.
Lemma 6. Suppose that is a WP-PSS of a Sasakian manifold . Then, we have(i)(ii)(iii) and , where is the component of .
Proof. From (5), (8), and (26), it is easy to see that Moreover, replacing by in (38), using (2) and part (i), we get part (ii). To prove part (iii), on making use of (8) and (4), we get
By utilizing (26), the form of the above equation can be changed to as follows:
By comparing the normal parts, we get taking the inner product with , we obtain
Calculating the last term of the above equation by using (8), (4), and (22) as follows:
Utilizing part (ii), we get using (50), we obtain the required result.
Lemma 7. Suppose that is a WP-PSS submanifold of a Sasakian manifold . Then, for all ,
Proof. From part (ii) of Lemma 6, the following is attained: and . Replacing by and using the fact that and are perpendicular, the following is obtained: By using (15), (16), and (26), we get Now, taking the inner product with in the above equation, we have Interchanging and and subtracting the resultant from equation (57) lead to In particular, replacing by , we get Using (55) yields
Lemma 8. On a WP-PSS submanifold of a Sasakian manifold , we obtainwhere and are the frames of the orthonormal vector fields on and , respectively.
Proof. First, we expand the left hand term in the following way:
Using part (ii) of Lemma 6 and Lemma 7 and utilizing (28), we get
Replacing by in the above equation, we get
By subtracting the above two findings, the required result gets attained.
The next theorem provides the following characterization.
Theorem 9. Suppose that is a WP-PSS submanifold of a Sasakian space form such that is a compact submanifold. Then, is a Riemannian product submanifold if the following inequalities holdwhere stands for the component of in and and are the dimensions of and , respectively.
Proof. Let and be unit vector fields. Then, by making use of (2), (6), and (22), we have However, the Codazzi equation gives
Now, we compute the values of the terms involved in (67). First, we get
Using part (ii) of Lemma 6 leads to
Now, we compute the last term of (69) using (11) as follows:
By the use of (8) and (13), the above equation becomes
Using (4), (8), (26), and parts (ii) and (iii) of Lemma 6, we get
Utilizing (72) in (69), we obtain
Similarly, we can write
From the second part of Lemma 6, we get
Replacing by (using the totally geodesicness of , ) in the above equation, we have
By using (8), we obtain
Since is totally geodesic in , it can be ready to verify that for all . Using this and (13) in equation (77), we have
Similarly, we can write
By use of (26) and the second part of Lemma 6, we attain
Substituting values from (66), (73), (74), (78), (79), (80), and (81) in (67) yields
Let be the orthonormal frame on and be an orthonormal frame on . By using the decomposition (40) and formulas (42), (8), and (4) in the last term of the above equation using the orthonormal frame of , we get
Now, taking the sum of the above equation over and , using (28) and (29) and part (iii) of Lemma 6, one can get
From (84), if then ; hence, from Hopf’s Lemma, is constant which means that is constant, which proves the theorem.
The squared norm of with reference to the warping function and slant function is obtained in the next theorem as follows.
Theorem 10. Let be a -dimensional Sasakian space form and be an dimensional WP-PSS submanifold such that is a -dimensional invariant submanifold and is a dimensional proper pointwise slant submanifold of . Ifthen (ii)the necessary and sufficient conditions for the equality sign of (87) to be held identically are(a) is the totally geodesic invariant in . Furthermore, it is a Sasakian space form(b) is totally umbilical in (c)
Proof. From (84), we have
For the orthonormal frames and , in view of formulas (40) and (41) and part (ii) of Lemma 6, we get
Further using Lemma 7 and (28), the above equation is reduced to
From (88) and (90), we attain the desired inequality.
To prove part (ii), suppose that is the second fundamental form for the immersion of in . Then, and , the Gauss formula gives
Using (27), we attain
For the equality case of (87), we get
From the first equation of (94), we go to the conclusion that is totally geodesic in . Furthermore, one can readily check that , , . It leads to the conclusion that is totally geodesic in and it is a Sasakian space form as well. Moreover, the second equation of (94) together with (93) yields that is totally umbilical. This proves the theorem.
4. Some Applications
Theorem 10 motivates to compute the formula for Dirichlet energy for the warping function . Here, the Dirichlet energy of is denoted by and is calculated for a compact orientable WP-PSS submanifold in a Sasakian space form as follows.
Theorem 11. Suppose that is a compact orientable WP-PSS submanifold of a Sasakian space form , such that is a dimensional invariant submanifold tangent to the structure vector field and is a dimensional pointwise slant submanifold of . Then, for every , the necessary and sufficient conditions for the Dirichlet energies of the warping functions to fulfil the followingare the following: (i) is the totally geodesic invariant in , and it is a Sasakian space form(ii) is totally umbilical in (iii)
Proof. By integrating the case of equality of (87) and using the definition of Dirichlet energy and (33), we get the desired result.
If , the compact orientable WP-PSS submanifolds become contact CR-warped product submanifolds. The above theorem can be used for deducing the following results.
Corollary 12. Suppose that is a compact orientable contact CR-warped product submanifold of a Sasakian space form , such that is a dimensional invariant submanifold tangent to and is a -dimensional anti-invariant submanifold of Then, for every , the necessary and sufficient conditions for the relationto be held are the following: (i) is a Sasakian space form and totally geodesic invariant in (ii) is a totally umbilical anti-invariant in
If equality sign of (87) holds, then
The Laplacian of a smooth function is the trace of the Hessian of the function. In terms of Hessian, (98) can be written as follows:
Now, the study comprises the following classification theorem for the WP-PSS submanifolds admitting the gradient Ricci soliton fulfilling the equality case of (87).
Theorem 13. Suppose that is a -dimensional Sasakian space form and is a WP-PSS submanifold admitting a shrinking gradient Ricci soliton. Then, ifone among the following is true: (i), i.e., is a contact CR-warped product submanifold(ii) is constant, i.e., is trivial Riemannian product pointwise semi slant submanifold
Proof. Suppose that the WP-PSS submanifold satisfies the basic equation of the Ricci soliton, such that the potential function then for all Considering that is an orthonormal basis of the vector fields on . Now, taking summation over for in (101), we have
Replacing by in the above equation, we get
By the assumption that the equality in (87) holds, then by (99),
By assumption (100), we get
From the above equation, it is evident that or is constant, which proves the theorem.
If the submanifold admits the steady gradient Ricci soliton, then the following can be easily concluded from the above theorem.
Theorem 14. Suppose that is a -dimensional Sasakian space form and is a WP-PSS submanifold admitting a steady gradient Ricci soliton. If the relationholds, then one can be true from the following: (i), i.e., is a contact CR-warped product submanifold(ii) is constant, i.e., is trivial Riemannian product pointwise semi slant submanifoldIn terms of Ricci curvature, the study constitutes the following classification.
Theorem 15. Suppose that is a -dimensional Sasakian space form and is a WP-PSS submanifold with the equality in (87) holds. If the following holdsthen one among the listed statements must be correct: (i), i.e., is a contact CR-warped product submanifold(ii) is constant, i.e., is trivial Riemannian product pointwise semi slant submanifold
Proof. For a connection on a smooth manifold , there exists a tensor of type (1, 3) called the curvature tensor of the connection given by .
For a warping function from (109), we obtain
By the smoothness property of on and , then behaves like a derivative. Since is closed, then it is ready to check that , Now, for a local orthonormal basis on and for a fixed point such that for . If we specify for any , and taking trace with respect to and in the following equation: and utilizing (110), we have
Further solving the left hand side of (112), it becomes the above equation can be written in the form of an operator as follows:
Since is a compact orientable WP submanifold, then on integrating where is the volume element.
Since [26] and for any . So, it is easy to conclude that . Then,
Utilizing the above equation in (98), we have
By assumption (108), we get
From the above equation, it is evident that or is constant, which proves the theorem.
5. Conclusion
We obtain the inequalities characterizing the existence of WP-PSS submanifolds of Sasakian space forms via Hopf’s Lemma. Furthermore, we gave an estimation of the squared norm of the second fundamental form with regard to the slant function and warping function. To strengthen our results, we provided an example of a WP-PSS submanifold in a Sasakian manifold. Moreover, some applications in the form of the Dirichlet energy of the warping function are derived. The results obtained may be helpful in further studies on the Dirichlet energy of smooth functions.
Abbreviations
| WP: | Warped product |
| WF: | Warping function |
| AC: | Almost contact |
| WP-PSS: | Warped product pointwise semi slant submanifold. |
Data Availability
No data were used.
Conflicts of Interest
Both the authors declare that they have no competing interests.
Authors’ Contributions
Both of the authors worked equally on this paper.
Acknowledgments
The project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. D-139-662-1440. The authors, therefore, gratefully acknowledge the DSR technical and financial support.