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Geometry
Volume 2013 (2013), Article ID 654780, 11 pages
http://dx.doi.org/10.1155/2013/654780
Research Article

CR-Submanifolds of Generalized -Space Forms

Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India

Received 3 January 2013; Revised 28 February 2013; Accepted 3 March 2013

Academic Editor: Michel Planat

Copyright © 2013 Mahmood Jaafari Matehkolaee. 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.

Abstract

We study sectional curvature, Ricci tensor, and scalar curvature of submanifolds of generalized -space forms. Then we give an upper bound for foliate -horizontal (and vertical) CR-submanifold of a generalized -space form and an upper bound for minimal -horizontal (and vertical) CR-submanifold of a generalized -space form. Finally, we give the same results for special cases of generalized -space forms such as -space forms, generalized Sasakian space forms, Sasakian space forms, Kenmotsu space forms, cosymplectic space forms, and almost -manifolds.

Dedicated to my sister and my parents for their endless support, kind and sacrifices.

1. Introduction

In 1978, Bejancu introduced and studied CR-submanifolds of a Kähler manifold [1, 2]. Since then, many papers appeared on this topic with ambient manifold such as Sasakian space form [3], cosymplectic space form [4], and Kenmotsu space form [5, 6]. Recently Falcitelli and Pastore [7] introduced generalized globally framed -space forms. Globally framed -manifolds are studied from the point of view of the curvature and are introduced and the interrelation with generalized Sasakian and generalized complex space forms is pointed out. In this paper, we study CR-submanifolds of generalized -space forms.

The theory of a submanifold of a Sasaki manifold was investigated from two different points of view: one is the case where submanifolds are tangent to the structure vector and the other is the case where those are normal to the structure vector [8].

In the class of -structures introduced in 1963 by Yano [9], the so-called -structures with complemented frames, also called globally framed -structures or -structures with parallelizable kernel (briefly .-structures) [1013] are particulary interesting. An .-manifold is a manifold on which an -structure is defined, that is a -tensor field satisfying , of rank , such that the subbundle is parallelizable. Then, there exists a global frame , , for the subbundle , with dual 1-form , satisfying , from which , follow. An .-structure on a manifold is said to be normal if the tensor field vanishes, denoting the Nijenhuis torsion of . It is known that one can consider a Riemannian metric on associated with an .-structure , such that , for any , and the structure is then called a metric .-structure. Therefore, splits as complementary orthogonal sum of its subbundles and . We denote their respective differentiable distributions by and .

Let denote the 2-form on defined by , for any .

Several subclasses have been studied from different points of view [10, 11, 1416], also dropping the normality condition, and, in this case, the term almost precedes the name of the considered structures or manifolds. As in [10], a metric .-structure is said a -structure if it is normal and the fundamental 2-form is closed; a manifold with a -structure is called a -manifold. In particular, if , for all , the -structure is said to be an -structure and an -manifold. Finally, if for all , then the -structure is called a -structure and is said to be a -manifold. Obviously, if , a -manifold is a quasi Sasakian manifold, a -manifold is a cosymplectic manifold, and an -manifold is a Sasakian manifold.

The purpose of the present paper is to study Ricci tensor, sectional curvature, and scalar curvature of submanifolds of a generalized .-space form. In Section 2, we state definitions of .-space form, its curvature tensor, -horizontal CR-submanifold, and -vertical CR-submanifold. Section 3 is devoted to the study sectional curvature of submanifold of an .-space form. Finally, in Section 4, we investigate Ricci tensor and scalar curvature of submanifold of an .-space form and obtain upper bound for scalar curvature.

2. Preliminaries

We recall that the Levi-Civita connection of a metric .-manifold satisfies the following formula [10, 11]: where is given by .

Furthermore, for -manifolds we have , , [10]. Putting , is its dual form with respect to and We remark that (2) together with and , , characterizes the -manifolds among the metric .-manifolds.

A metric .-manifold has pointwise constant (p.c.) -sectional curvature if at any , does not depend on the -section spanned by , for any unit . Several results involving the pointwise constancy of the -sectional curvatures of an almost contact metric manifold (i.e., for ) are recently obtained in [1719]. We refer to [20] for a systematic exposition of the classical curvature results on contact metric manifolds.

We recall some known results.

Proposition 1 (see [6]). A Sasaki manifold has p.c. -sectional curvature if and only if its curvature tensor field verifies for any tangent to .

A Sasaki manifold with constant -sectional curvature is called a Sasakian space form and denoted by . It is well known that, if , a Sasaki manifold with p.c. -sectional curvature is a Sasakian space form. As examples of Sasakian space forms, we mention and , with standard Sasakian structures [14].

Definition 2 (see [10]). An almost contact metric manifold is a generalized Sasakian space form, denoted by , if it admits three smooth functions , , such that its curvature tensor field verifies that, for any

Remark 3. Any generalized Sasakian space form has p.c. -sectional curvature . Obviously, a Sasaki manifold of p.c. -sectional curvature satisfies (4) with and . A cosymplectic manifold with p.c. -sectional curvature satisfies (4) with .

Proposition 4 (see [10, 21]). An -manifold has p.c. -sectional curvature if and only if its curvature tensor field verifies for any tangent to .

An -manifold with constant -sectional curvature is called an -space-form and denoted by . Moreover, it is also well known that if , then an -manifold with p.c. -sectional curvature is an -space form. We remark that for (5) reduces to (3).

Definition 5. In [22], Oubiña introduced the notion of a trans-Sasakian manifold. An almost contact metric manifold is called trans-Sasakian manifold if there exist two functions and on such that [2224] for vector fields on . From (6) it is easy to see that In particular, if , then is said to be an -Sasakian manifold. Sasakian manifolds appear as examples of -Sasakian manifolds with .
On the other hand, if , then is said to be a -Kenmotsu manifold. Kenmotsu manifolds, defined in [25], are particular examples with .
Another important kind of trans-Sasakian manifolds is that of cosymplectic manifolds obtained for .

Proposition 6 (see [25]). An almost contact metric manifold is said to be an almost -manifold if its Riemannian curvature tensor verifies for vector fields , , , and on , where is a real number. Moreover, if such a manifold has constant -sectional curvature equal to , then its curvature tensor is given by and so, it is a generalized Sasakian space form with and .

Let denote any set of smooth function on such that for any .

Definition 7 (see [7]). A generalized .-space form, denoted by , is a metric .-manifold which admits smooth functions , , and such that its curvature tensor field verifies
For , we obtain a generalized Sasakian space form with , and . In particular, if the given structure is either Sasakian, Kenmotsu, or possibly cosymplectic, then (10) holds with , and in the first case, , , and in the second case, and , , , and in the last case.

Definition 8. Let be an -dimensional submanifold immersed in . is said to be an invariant submanifold if for any and for any . On the other hand, it is said to be an anti-invariant submanifold if for any .
An -dimensional Riemannian submanifold of a generalized .-space form is called a CR-submanifold if 's are tangent to (so, ) and there exist two differentiable distributions and on satisfying(i) (direct sum),(ii) the distribution is invariant under , that is, for any ,(iii) the distribution is anti-invariant under , that is, for any .
We denote by and the real dimensions of and , respectively, for any . Then, if , we have an anti-invariant submanifold tangent to , and if , we have an invariant submanifold.
As an example, it is easy to prove that each hypersurface of which is tangent to inherits the structure of CR-submanifold of . Also, pseudoumbilical, totally contact umbilical, totally contact geodesic, totally umbilical, and totally geodesic hypersurfaces of a generalized -space form are also generalized -space forms, and, moreover, the bundle space of a principal toroidal bundle over a Kählerian manifold and the warped product of R times a generalized -space form are generalized -space forms, too [26].

Definition 9. The -sectional curvature of determined by a unit vector orthogonal to 's is the sectional curvature of the plane section spanned by and . Also, we denote by (and ) Ricci tensor (and sectional curvature) determined by (orthonormal) vector fields , respectively.

Definition 10. A CR-submanifold of a generalized .-space form is said to be -totally geodesic (resp., - totally geodesic) if for any (resp., ), and it is said to be -mixed totally geodesic if for any , .
Also, CR-submanifold is said to be minimal if , where is the mean curvature vector, defined by .

Definition 11. Let be a CR-submanifold with horizontal distribution and vertical distribution . The pair is called -horizontal if for any , and in a similar way the pair is called -vertical if for any .

Definition 12. Let and be a local field of orthonormal frames on such that in case when is -horizontal, is a local frame field on and is a local frame field on .
Let be an -horizontal CR-submanifold of . The mean curvature vector field of in is defined by If , then is said to be minimal. Now, we will define If , then the CR-submanifold is said to be -minimal, and if , then it is said to be -minimal. Similar definitions can be given for -vertical CR-submanifolds.

We denote by and the projection morphisms of on and , respectively. We call (resp., ) the horizontal (resp., vertical) distribution. Then for any vector field tangent to , we have: where and belong to the distribution (horizontal part) and (vertical part), respectively. Also, for a vector field normal to , we put: where and denote the horizontal and normal component of , respectively.

Definition 13. Let be a CR-submanifold of an ambient manifold , with horizontal distribution . Then is called involutive (or integrable) if for any where is Lie bracket of . Also, is a foliate if is involutive (or integrable).

Let be an -dimensional submanifold immersed in a generalized .-space form . The Gauss-Weingarten formulas are given by where is the connection in the normal bundle, is the second fundamental form of and the Weingarten endomorphism associated with . Then and are related by:

We denote by and the curvature tensor fields associated with and , respectively. The Gauss equation is given by where and belong to .

3. Sectional Curvature of Submanifolds

Let be a submanifold of a generalized .-space form . Then from the equation of Gauss, we have for any and tangent to .

Let be the sectional curvature determined by orthonormal vectors and . Then from (18), we have Thus we have the following theorem.

Theorem 14. Let be a submanifold of a generalized .-space form . Then the sectional curvature of determined by orthonormal tangent vectors is given by

From this we have the following corollaries for the sectional curvature of submanifold determined by orthonormal tangent vectors .

Corollary 15. The sectional curvature of a submanifold of an -space form is given by

Proof. We will get the result by using , for all ; ; in (20).

Corollary 16. The sectional curvature of a submanifold of a generalized Sasakian space form is given by

Proof. We will get the result by using , , , and in (20).

Corollary 17. The sectional curvature of a submanifold of a Sasakian space form is given by

Proof. We get the result by using , in (22).

Corollary 18. The sectional curvature of a submanifold of a Kenmotsu space form is given by

Proof. We get the result by using and in (22).

Corollary 19. The sectional curvature of a submanifold of a cosymplectic space form is given by

Proof. By taking in (22), we obtain the above.

Corollary 20. The sectional curvature of a submanifold of an almost -manifold is given by

Proof. By getting , in (22), we obtain (26).

Proposition 21. If is a -horizontal CR-submanifold of a generalized .-space form , then the sectional curvature of determined by is given by

Proof. From (20) and by replacing ; and we get the result immediately.

Corollary 22. If is a -horizontal CR-submanifold of an -space form , then the sectional curvature of determined by is given by

Corollary 23. If is a -horizontal CR-submanifold of a generalized Sasakian space form , then the sectional curvature of determined by is given by

Corollary 24. If is a -horizontal CR-submanifold of a Sasakian space form , then the sectional curvature of determined by is given by

Corollary 25. If is a -horizontal CR-submanifold of a Kenmotsu space form , then the sectional curvature of determined by is given by

Corollary 26. If is a -horizontal CR-submanifold of a cosymplectic space form , then the sectional curvature of determined by is given by

Corollary 27. If is a -horizontal CR-submanifold of a -manifold , then the sectional curvature of determined by is given by

Proposition 28. If is a -vertical CR-submanifold of a generalized .-space form , then the sectional curvature of determined by is given by

Corollary 29. If is a -vertical CR-submanifold of an -space form , then the sectional curvature of determined by is given by

Corollary 30. If is a -vertical CR-submanifold of space form , then the sectional curvature of determined by ,(i) where is a generalized Sasakian space form, is given by: (ii) where is a Sasakian space form, is given by (iii) where is a Kenmotsu space form, is given by (iv) where is a cosymplectic space form, is given by (v) where is a -manifold, is given by

Proposition 31. The -sectional curvature of a CR-submanifold of a generalized .-space form , determined by , is given by

Proof. By using , for all in (20), we will get the result.

Proposition 32. The -sectional curvature of a CR-submanifold of a generalized Sasakian space form , determined by is given by

Corollary 33. The -sectional curvature of a CR-submanifold of either an -space form, a Sasakian space form, a Kenmotsu space form, a cosymplectic space form, or an almost -manifold , determined by , is given by:

We recall the following Lemma [27].

Lemma 34. Let be a foliate -horizontal CR-submanifold of a -space form ; then

Proposition 35. If is a foliate -horizontal CR-submanifold of a -space form ; then and the equality holds if and only if is -totally geodesic.

Corollary 36. If is a foliate -horizontal CR-submanifold of a generalized Sasakian space form , then and the equality holds if and only if is -totally geodesic.

Corollary 37. If M is a foliate -horizontal CR-submanifold of either a Sasakian space form, a Kenmotsu space form, a cosymplectic space form, or an almost -manifold form , then and the equality holds if and only if is -totally geodesic.

Proposition 38. If be a -minimal -horizontal CR-submanifold of a generalized .-space form ; then is -totally geodesic iff for any .

Proof. Let is -minimal -horizontal CR-submanifold of generalized .-space form , then by definition of -minimal, we have: where is a local frame field on . Therefore, for any On the other hand, from (20), we have for Hence, is -totally geodesic if and only if for any

Proposition 39. If is a -minimal -vertical CR-submanifold of a generalized .-space form , then is -totally geodesic if and only if for any with .

Proposition 40. If is a -horizontal CR-submanifold and -mixed totally geodesic of a generalized .-space form , then for any and .

Proof. By using , for all and and , we arrive at the aforementioned equation, easily.

Proposition 41. If is a -vertical CR-submanifold and -mixed totally geodesic of a generalized .-space form , then for any and .

Proof. By using , for all ,    and (20), we arrive at the abovementioned equation, easily.

4. The Ricci Tensor and Scalar Curvature of a Submanifold

Let be a submanifold of a generalized .-space form . Then it is straightforward to calculate the Ricci tensor of as follows Also, the scalar curvature of a submanifold of is then given by Thus, we obtain the following.

Theorem 42. Let be a submanifold of a generalized .-space form . Then the Ricci tensor and scalar curvature of (resp.) are given by

Theorem 43. Let be a minimal CR-submanifold of a generalized .-space form . Then is negative semidefinite and

Theorem 44. Let be a -horizontal (resp., -vertical) CR-submanifold of a generalized .-space form . Then the Ricci tensor of for any (resp., ) is given by

Theorem 45. Let be minimal -horizontal (resp., -vertical) CR-submanifold of a generalized .-space form . Then for any (resp., ) is negative semidefinite.

Corollary 46. One has for Ricci tensor of minimal -horizontal CR-submanifold Table 1.

tab1
Table 1

Corollary 47. One has for scalar curvature of minimal -horizontal CR-submanifold Table 2.

tab2
Table 2

Remark 48. Similar results can be written for minimal -vertical CR-submanifolds, easily.

Acknowledgment

The author would like to express his sincere thanks to referees for their precious suggestions.

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