Chen-Ricci Inequalities with a Quarter Symmetric Connection in Generalized Space Forms

Al-Khaldi, Ali H.; Aquib, Mohd.; Aslam, Mohd; Khan, Meraj Ali

doi:https://doi.org/10.1155/2021/3221643

Advances in Mathematical Physics

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Geometry of Warped Product Manifolds: Theory and Applications

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Volume 2021 | Article ID 3221643 | https://doi.org/10.1155/2021/3221643

Chen-Ricci Inequalities with a Quarter Symmetric Connection in Generalized Space Forms

Ali H. Al-Khaldi,¹Mohd. Aquib,²Mohd Aslam,³and Meraj Ali Khan⁴

Academic Editor: Antonio Scarfone

Received16 May 2021

Revised21 Sept 2021

Accepted22 Sept 2021

Published11 Oct 2021

Abstract

In this article, we obtain improved Chen-Ricci inequalities for submanifolds of generalized space forms with quarter-symmetric metric connection, with the help of which we completely characterized the Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian space form. We also discuss some geometric applications of the obtained results.

1. Introduction

One of the most basic problems in submanifold theory is to develop a simple relationship between the extrinsic invariants and the intrinsic invariants. The sectional curvature, the scalar curvature, and the Ricci curvature are the main intrinsic invariants while the squared mean curvature is the main extrinsic invariant.

Chen obtained the following important bound of the Ricci curvature in terms of the mean curvature for Lagrangian submanifolds in complex space forms [1]: where is the constant holomorphic sectional curvature of the complex space form.

Further, he discussed the geometry of a Lagrangian submanifold satisfying the equality case of the inequality under the condition that the dimension of the kernel of the second fundamental form is constant. The inequality (1) is known as the Chen-Ricci inequality. This inequality attracted many researchers due to its geometric importance [2–12].

Deng [13] improved the above inequality as

In [14], Deng further extended his result for Lagrangian submanifolds in quaternion space forms. In [15], Tripathi improved the inequality in the case of curvature-like tensors. In [6], Mihai and Radulescu obtained the same relation in Sasakian space forms using semisymmetric connection as

As the curvature invariants are of great interest in theoretical physics (see [16]), the above studies motivate us to obtain a complete characterization of Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian space form.

2. Preliminaries

Let be a Riemannian manifold and be a linear connection on . Then, is said to be a semisymmetric connection if its torsion tensor satisfies for a 1-form , then the connection is called a semisymmetric connection [17]. Let be a Riemannian metric on . If , then is called a semisymmetric metric connection on . The semisymmetric metric connection on is given by for any on , where denotes the Levi-Civita connection with respect to Riemannian metric and is a vector field. Further, is said to be a semisymmetric nonmetric connection if it satisfies

Moreover, the linear connection on a Riemannian manifold with Riemannian metric is said to be a quarter-symmetric connection if its torsion tensor is given by which satisfies such that is a 1-form given by where is a vector field and is a (1,1) tensor field.

Then, we can define a special quarter-symmetric connection by where and are real constants.

Remark 1. We notice from (5) that [18] (1)if , then a quarter symmetric connection becomes a semisymmetric metric connection(2)if and , then a quarter-symmetric connection becomes a semisymmetric nonmetric connection

Remark 2. It is also worthy to mention here that the quarter symmetric connections generalized several well-known connections.
The curvature tensor with respect to is In the same way, we can also define the curvature tensor .

Let are tensors. Then, the curvature tensor of is given by [19]

Let be an -dimensional submanifold in a Riemannian manifold . Let and be the induced quarter symmetric-metric connection and Levi-Civita connection, respectively, on . Then, the Gauss formulas are where is the second fundamental form that satisfies the relation where is the normal component of the vector field on .

Moreover, the equation of Gauss is defined by [19]

3. Characterization of Lagrangian Submanifold in Generalized Complex Space Form

A smooth manifold endowed with an almost complex structure and a Riemannian metric that is compatible with is called an almost Hermitian manifold. Further, for the Levi-Civita connection if , then an almost Hermitian manifold is said to be a Kaehler manifold. A Kaehler manifold of constant holomorphic curvature is called a complex space form. The curvature tensor of a complex space form is given by

However, an almost Hermitian manifold is called a generalized complex space form [20–22], denoted by , if for all vector fields , and on , the Riemannian curvature tensor satisfies where and are smooth functions on .

In fact, we have following fundamental result from Tricerri and Vanhecke [20].

Theorem 3 (see [20]). Let be a connected almost Hermitian manifold with real dimension and Riemannian curvature is of the form (18) such that is not identically zero. Then, is a complex space form.

Remark 4. From (18), we notice that if , then we recover the complex space form.

From (13) and (18), we have

Lemma 5 (see [13]). Let be a function on defined by If then and the equality holds if and only if , where is a constant.

Lemma 6 (see [13]). Let be a function on defined by If then and the equality holds if and only if and , where is a constant.

Let be an -dimensional submanifold of an almost Hermitian manifold . Then, is said to be totally real if

Then, we have the following relations [23]: or equivalently, where is the shape operator with respect to and

Remark 7. A totally real submanifold which is of maximal dimension is known as the Lagrangian submanifold [24].

Definition 8 (see [25]). A nontotally geodesic Lagrangian submanifold of a complex space form is called -umbilical if its second fundamental form satisfies for some functions and with respect to an orthonormal frame , where is the complex structure of .

Theorem 9. Let be a totally real submanifold of maximal dimension in a connected complex space form of dimension with a quarter-symmetric metric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality holds in (29) identically if and only if either (1) is totally geodesic, provided that , or(2) and is a -umbilical Lagrangian surface with

Proof. As is tangent to , we have Let us assume an orthonormal basis and at point with unit vector . Then, by combining (16) and (19) and substituting and , for , we get Taking the summation over , we find which implies From the above equation and (26), it is easy to see that Putting and combining Lemma 5 with the relation , we obtain Then, by Lemma 6 for , we get We find from (34), (36), and (37) that which is the desired inequality (29).☐

Now, we discuss the equality cases.

Case 1. For if , then for all . Therefore, using Lemma 6, we derive Further, Lemma 5 yields In (33), we see that . In the same way, by deriving and making use of the equality, we conclude that In consequence, we find We see that the equality holds for every unit tangent vectors. The above conclusion is also valid for . Thus, Then, the only possible nonzero entries for (resp., for ) are Substituting and in (16), we derive On the other hand, if we substitute and in (16), we get Using (46) and (47), we find Moreover, the equality case of (29) implies that Using the fact by (48) and (49), it is easy to see that . This implies that is a totally geodesic in .

Case 2. In case , is nontotally geodesic, then together with . This proves that is -umbilical surface.

The above theorem gives the following results.

Corollary 10. Let be a totally real submanifold of maximal dimension in a connected complex space form of dimension with a semisymmetric metric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality holds in (50) identically if and only if either (1) is totally geodesic, provided , or(2) and is a -umbilical Lagrangian surface with

Proof. Using the fact together with Theorem 9, the result directly follows.☐

Remark 11. It is worthy to mention here that Corollary 10 together with Remark 4 is the main result for complex case of the paper [26].

Corollary 12. Let be a totally real submanifold of maximal dimension in a connected complex space form of dimension with a semisymmetric nonmetric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality holds in (51) identically if and only if either (1) is totally geodesic, provided , or(2) is a -umbilical Lagrangian surface with

Proof. Using the fact and together with Theorem 9, the result directly follows.☐

4. Characterization of Legendrian Submanifold in Generalized Sasakian Space Form

Let a -dimensional almost contact metric manifold furnished with the almost complex structure , where is a (1,1) tensor field, is the structure vector field, the 1-form, and is the Riemannian metric on . Then, following relations hold good:

It also follows from the above relations that for all vector fields on .

Let be an almost contact metric manifold whose curvature tensor satisfies [27]

for all vector fields on where are differentiable functions on . Then, is said to be a generalized Sasakian space form.

Remark 13. The generalized Sasakian space forms are [27] (1)Sasakian space forms if (2)Kenmotsu space forms if and (3)cosymplectic space forms if

From (13) and (54), we have

A submanifold of an almost contact manifold normal to is called a -totally real submanifold. On such a submanifold, maps any tangent vector to at into the normal space . In particular, if , i.e., has maximum dimension, then it is a Legendrian submanifold. For a Legendrian submanifold , if and be tangent orthonormal frame and normal orthonormal frame, respectively, on . One has or equivalently, where is the shape operator with respect to and

Definition 14 (see [28]). A nontotally geodesic Legendrian submanifold of a Sasakian space form is called -umbilical if its second fundamental form satisfies for some functions and with respect to an orthonormal frame , where is the contact structure of .

Theorem 15. Let be a totally real submanifold of maximal dimension in a generalized Sasakian space form of dimension with a quarter-symmetric metric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality holds in (60) identically if and only if either (1) is totally geodesic, provided , or(2) and is a -umbilical Legendrian surface with

Proof. As is tangent to , we have Let us assume an orthonormal basis and at point with unit vector . Then, by combining (16) and (55) and substituting and and summing over , we compute From (62) and (57), we deduce Putting and by using the fact together with the Lemma 5, we see that Application of Lemma 6 for gives Equations (65), (67), and (68) yield the following relation Thus, we derive which is the desired inequality (60).☐

Now, we discuss the equality cases.

Case 1. For if . Then, for all . Therefore, using Lemma 6, we derive Further, Lemma 5 yields In (63), we see that . In the same way, by deriving and making use of the equality, we conclude that In consequence, we find We see that the equality holds for every unit tangent vectors. The above conclusion is also valid for . Thus, Then, the only possible nonzero entries for (resp., for ) are Substituting and in (16), we obtain On the other hand, if we put and in (16), we get From (77) and (78), it follows that Moreover, using the equality case of (29), we see that Indeed with (81) and (84), we find . This implies that is a totally geodesic in .

Case 2. In the case , is nontotally geodesic, then together with . This proves that is -umbilical surface.

Remark 16. If we consider the structure vector field tangential to the submanifold , then we have the following result.

Theorem 17. Let be a totally real submanifold of maximal dimension in a generalized Sasakian space form of dimension with a quarter-symmetric metric connection such that the vector field is tangent to . Then, for any unit tangent vector to , and the equality holds in (81) identically if and only if either (1) is totally geodesic, provided , or(2) and is a -umbilical Legendrian surface with

Proof. We obtain the proof on the same lines of the proof for Theorem 15 additionally assuming an orthonormal basis such that .
As a consequence of Theorem 15, we obtain the following results.☐

Corollary 18. Let be a totally real submanifold of maximal dimension in a Sasakian space form of dimension with a quarter-symmetric metric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality holds in (83) identically if and only if either (1) is totally geodesic, provided , or(2) and is a -umbilical Legendrian surface with

Proof. The proof follows immediately from Theorem 15 by putting .☐

Corollary 19. Let be a totally real submanifold of maximal dimension in a Kenmotsu space form of dimension with a quarter-symmetric metric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality holds in (84) identically if and only if either (1) is totally geodesic, provided , or(2) and is a -umbilical Legendrian surface with

Proof. The proof follows immediately from Theorem 15 by replacing and .☐

Corollary 20. Let be a totally real submanifold of maximal dimension in a cosymplectic space form of dimension with a quarter-symmetric metric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality holds in (85) identically if and only if either (1) is totally geodesic, provided , or(2) and is a -umbilical Legendrian surface with

Proof. The proof follows immediately from Theorem 15 by substituting .☐

Corollary 21. Let be a totally real submanifold of maximal dimension in a generalized Sasakian space form of dimension with a semisymmetric metric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality in (60) holds identically if and only if either (1) is totally geodesic, provided , or(2) and is a -umbilical Legendrian surface with

Proof. Using the fact that together with Theorem 15, the result directly follows.☐

Remark 22. It is worthy to mention here that Corollary 21 together with Remark 13 (1) is the main result of the paper [6].

Corollary 23. Let be a totally real submanifold of maximal dimension in a generalized Sasakian space form of dimension with a semisymmetric nonmetric connection such that the vector field is tangent to . Then, for any unit tangent vector to and the equality in (87) holds identically if and only if either (1) is totally geodesic, provided , or(2) and is a -umbilical Legendrian surface with

Proof. Using the fact that and together with Theorem 15, the result directly follows.☐

Remark 24. All the above cases for Theorem 15 can be seen in the case of Theorem 17 as well.

Remark 25. Examples of totally geodesic submanifolds, -umbilical Lagrangian submanifolds, and -umbilical Legendrian submanifolds, i.e., examples of submanifolds attaining the equality case of the inequalities stated in this article, can be found in [25, 29, 30].

Data Availability

No data is used for the research

Conflicts of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Acknowledgments

The first author (A. H. Al-Khaldi) extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Research Groups Program under grant number R.G.P2/74/42.

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Copyright © 2021 Ali H. Al-Khaldi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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