International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2012, Article IDΒ 947640, 18 pages

http://dx.doi.org/10.1155/2012/947640

## Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection

Department of Mathematics, Kuvempu University, Shankaraghatta, Karnataka, Shimoga 577451, India

Received 30 March 2012; Accepted 2 July 2012

Academic Editor: V. R.Β Khalilov

Copyright Β© 2012 B. S. Anitha and C. S. Bagewadi. 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

The object of this paper is to study invariant submanifolds of Sasakian manifolds admitting a semisymmetric nonmetric connection, and it is shown that *M* admits semisymmetric nonmetric connection. Further it is proved that the second fundamental forms
and with respect to Levi-Civita connection and semi-symmetric nonmetric connection coincide. It is shown that if the second fundamental form is recurrent, 2-recurrent, generalized 2-recurrent, semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel and *M* has parallel third fundamental form with respect to semisymmetric nonmetric connection, then *M* is totally geodesic with respect to Levi-Civita connection.

#### 1. Semisymmetric Nonmetric Connection

The geometry of invariant submanifolds of Sasakian manifolds is carried out from 1970βs by M. Kon [1], D. Chinea [2], K. Yano and M. Kon [3] and B.S. Anitha and C.S. Bagewadi [4]. The aurthor [1] has proved that invariant submanifold of Sasakian structure also carries Sasakian structure. In this paper we extend the results to invariant submanifolds of Sasakian manifolds admitting Semisymmetric Nonmetric connection.

We know that a connection on a manifold is called a metric connection if there is a Riemannian metric on if ; otherwise it is Nonmetric. Further it is said to be Semisymmetric if its torsion tensor , where is a 1-form. A study of Semisymmetric connection on a Riemannian manifold was initiated by Yano [5]. In 1992, Agashe and Chafle [6] introduced the notion of Semisymmetric Nonmetric connection. If denotes Semisymmetric Nonmetric connection on a contact metric manifold, then it is given by [6] where .

The covariant differential of the th order, of a -tensor field denoted by , defined on a Riemannian manifold with the Levi-Civita connection . The tensor is said to be *recurrent* [7], if the following condition holds on :
respectively.

Consider where . From (1.2) it follows that at a point , if the tensor is nonzero, then there exists a unique -form , respectively, a -tensor , defined on a neighborhood of such that respectively.

The following
holds on , where denotes the norm of and . The tensor is said to be *generalized *2*-recurrent* if
holds on , where is a -form on . From this it follows that at a point if the tensor is nonzero, then there exists a unique -tensor , defined on a neighborhood of , such that
holds on .

#### 2. Isometric Immersion

Let be an isometric immersion from an -dimensional Riemannian manifold into -dimensional Riemannian manifold , , . We denote and as Levi-Civita connection of and , respectively. Then the formulas of Gauss and Weingarten are given by
for any tangent vector fields and the normal vector field on , where , , and are the second fundamental form, the shape operator, and the normal connection, respectively. If the second fundamental form is identically zero, then the manifold is said to be *totally geodesic*. The second fundamental form and is related by
for tangent vector fields . The first and second covariant derivatives of the second fundamental form are given by
respectively, where is called the *van der Waerden-Bortolotti connection* of [8]. If , then is said to have *parallel second fundamental form* [8]. We next define endomorphisms and of by
respectively, where and is a symmetric -tensor.

Now, for a -tensor field and a -tensor field on , we define the tensor by Putting into consideration the previous formula β and ,β we obtain the tensors and .

#### 3. Sasakian Manifolds

An -dimensional differential manifold is said to have an almost contact structure if it carries a tensor field of type , a vector field , and 1-form on , respectively, such that

Thus a manifold equipped with this structure is called an almost contact manifold and is denoted by . If is a Riemannian metric on an almost contact manifold such that where are vector fields defined on , then is said to have an almost contact metric structure , and with this structure is called an almost contact metric manifold and is denoted by .

If on the exterior derivative of 1-form satisfies then is said to be a contact metric structure and together with manifold is called contact metric manifold and is a 2-form. The contact metric structure is said to be normal if

If the contact metric structure is normal, then it is called a Sasakian structure and is called a Sasakian manifold. Note that an almost contact metric manifold defines Sasakian structure if and only if

*Example of Sasakian Manifold*

Consider the -dimensional manifold , where are the standard coordinates in . Let be linearly independent global frame field on given by
Let be the Riemannian metric defined by
The is given by
The linearity property of and yields
for any vector fields on . By definition of Lie bracket, we have
Let be the Levi-Civita connection with respect to previously mentioned metric and be given by Koszula formula
Then, we have
The tangent vectors and to are expressed as linear combination of ; that is, and , where and are scalars. Clearly and satisfy (3.1), (3.2), (3.5), and (3.6). Thus is a Sasakian manifold. Further the following relations hold:
for all vector fields, and where denotes the operator of covariant differentiation with respect to is a tensor field, is the Ricci tensor of type , and is the Riemannian curvature tensor of the manifold.

#### 4. Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection

If is a Sasakian manifold with structure tensors , then we know that its invariant submanifold has the induced Sasakian structure .

A submanifold of a Sasakian manifold with a Semisymmetric Nonmetric connection is called an invariant submanifold of with a Semisymmetric Nonmetric connection, if for each , . As a consequence, becomes tangent to . For an invariant submanifold of a Sasakian manifold with a Semisymmetric Nonmetric connection we have for any vector tangent to .

Let be a Sasakian manifold admitting a Semisymmetric Nonmetric connection .

Lemma 4.1. *Let be an invariant submanifold of contact metric manifold which admits Semisymmetric Nonmetric connection , and let and be the second fundamental forms with respect to Levi-Civita connection and Semisymmetric Nonmetric connection; then (1) admits Semisymmetric Nonmetric connection and (2) the second fundamental forms with respect to and are equal. *

* Proof. *We know that the contact metric structure on induces on invariant submanifold. By virtue of (1.1), we get
By using (2.1) in (4.2), we get
Now Gauss formula (2.1) with respect to Semisymmetric Nonmetric connection is given by
Equating (4.3) and (4.4), we get (1.1) and

Now we introduce the definitions of semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel with respect to Semisymmetric Nonmetric connection.

*Definition 4.2. *An immersion is said to be semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel with respect to Semisymmetric Nonmetric connection, respectively, if the following conditions hold for all vector fields tangent to :
where denotes the curvature tensor with respect to connection . Here and are functions depending on .

Lemma 4.3. *Let be an invariant submanifold of contact manifold which admits Semisymmetric Nonmetric connection. Then Gauss and Weingarten formulae with respect to Semisymmetric Nonmetric connection are given by
*

*Proof. * The Riemannian curvature tensor on with respect to Semisymmetric Nonmetric connection is given by
Using (1.1) and (2.1) in (4.9), we get
Comparing tangential and normal part of (4.10), we obtain Gauss and Weingarten formulae (4.7) and (4.8).

Lemma 4.4. *Let be an invariant submanifold of contact manifold which admits Semisymmetric Nonmetric connection. If is semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel with respect to Semisymmetric Nonmetric connection, then we have
**
for all vector fields , and tangent to , where
*

*Proof. * We know, from tensor algebra, that
Replacing by in (4.9), we get
In view of (1.1), (2.1), and (2.2), we have the following equalities:
Similarly
Substituting (4.15), (4.16) and (4.17) into (4.14), we get
By virtue of (4.10) in and , we get
Substituting (4.18), (4.19) and (4.20) into (4.13), we get (4.11).

#### 5. Recurrent Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection

We consider invariant submanifolds of a Sasakian manifold when is recurrent, 2-recurrent, and generalized 2-recurrent and has parallel third fundamental form with respect to Semisymmetric Nonmetric connection. We write (2.4) and (2.5) with respect to Semisymmetric Nonmetric connection, and they are given by We prove the following theorems.

Theorem 5.1. *Let be an invariant submanifold of a Sasakian manifold admitting a Semisymmetric Nonmetric connection. Then is recurrent with respect to Semisymmetric Nonmetric connection if and only if it is totally geodesic with respect to Levi-Civita connection. *

* Proof. * Let be recurrent with respect to Semisymmetric Nonmetric connection; from (1.4) we get
where is a -form on ; in view of (5.1) and putting in the above equation, we have
By virtue of (4.1) in (5.4), we get
Using (1.1), (3.1), (3.6), and (4.1) in (5.5), we get
Replacing by and by virtue of (3.1) and (4.1) in (5.6), we get
Adding (5.6) and (5.7), we obtain . Thus is totally geodesic. The converse statement is trivial. This proves the theorem.

Theorem 5.2. *Let be an invariant submanifold of a Sasakian manifold admitting a Semisymmetric Nonmetric connection. Then has parallel third fundamental form with respect to Semisymmetric Nonmetric connection if and only if it is totally geodesic with respect to Levi-Civita connection. *

*Proof. * Let have parallel third fundamental form with respect to Semisymmetric Nonmetric connection. Then we have
Taking and using (5.2) in the above equation, we have
In view of (4.1) and by virtue of (5.1) in (5.9), we get
Using (1.1), (3.1), (3.6), and (4.1) in (5.10), we get
Putting and using (3.1), (3.6), and (4.1) in (5.11), we get
Replacing by and by virtue of (3.1) and (4.1) in (5.12), we get
Multiplying (5.12) by 1 and (5.13) by 3 and adding these two equations, we obtain . Thus is totally geodesic. The converse statement is trivial. This proves the theorem.

Corollary 5.3. *Let be an invariant submanifold of a Sasakian manifold admitting a Semisymmetric Nonmetric connection. Then is 2-recurrent with respect to Semisymmetric Nonmetric connection if and only if it is totally geodesic with respect to Levi-Civita connection. *

* Proof. * Let be 2-recurrent with respect to Semisymmetric Nonmetric connection; from (1.5), we have
Taking and using (5.2) in the above equation, we have
In view of (4.1) and by virtue of (5.1) in (5.15), we get
Using (1.1), (3.1), (3.6), and (4.1) in (5.16), we get
Putting and using (3.1), (3.6), (4.1) in (5.17), we get
Replacing by and by virtue of (3.1) and (4.1) in (5.18), we get
Multiplying (5.18) by 1 and (5.19) by 3 and adding these two equations, we obtain . Thus is totally geodesic. The converse statement is trivial. This proves the theorem.

Theorem 5.4. *Let be an invariant submanifold of a Sasakian manifold admitting a Semisymmetric Nonmetric connection. Then is generalized -recurrent with respect to Semisymmetric Nonmetric connection if and only if it is totally geodesic with respect to Levi-Civita connection. *

*Proof. * Letting be generalized -recurrent with respect to Semisymmetric Nonmetric connection, from (1.7), we have
where and are -recurrent and -form, respectively. Taking in (5.20) and using (4.1), we get
Using (4.1) and (5.2) in above equation, we get
In view of (4.1) and by virtue of (5.1) in (5.22), we get
Using (1.1), (3.1), (3.6), and (4.1) in (5.23), we get
Putting and using (3.1), (3.6), (4.1) in (5.24), we get
Replacing by and by virtue of (3.1) and (4.1) in (5.25), we get
Multiplying (5.25) by 1 and (5.26) by 3 and adding these two equations, we obtain . Thus is totally geodesic. The converse statement is trivial. This proves the theorem.

#### 6. Semiparallel, Pseudoparallel, and Ricci-Generalized Pseudoparallel Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection

We consider invariant submanifolds of Sasakian manifolds admitting Semisymmetric Nonmetric connection satisfying the conditions .

Theorem 6.1. *Let be an invariant submanifold of a Sasakian manifold admitting a Semisymmetric Nonmetric connection. Then we prove that is semiparallel with respect to Semisymmetric Nonmetric connection if and only if . *

*Proof. * Let be semiparallel . Putting and by virtue of (3.1), (3.6), and (4.1) in (4.11), we get
Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.1), we get
By definition is a vector-valued covariant tensor, and so is a vector. Therefore is a vector, and hence by (4.1), we have
Then from (6.2), we get
Interchanging and in (6.4), we get
Adding these tow equations, (6.4) and (6.5), we get

Theorem 6.2. *Let be an invariant submanifold of a Sasakian manifold admitting a Semisymmetric Nonmetric connection. Then we prove that is pseudoparallel with respect to Semisymmetric Nonmetric connection if and only if . *

* Proof. * Let be pseudoparallel . Putting and by virtue of (3.1), (3.6), and (4.1) in (2.7), (4.11), we get
Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.7), we get
Now by using (6.3) in (6.8), we get
Interchanging and in (6.9), we get
Adding (6.9) and (6.10), we get

Theorem 6.3. *Let be an invariant submanifold of a Sasakian manifold admitting a Semisymmetric Nonmetric connection. Then we prove that is Ricci-generalized pseudoparallel with respect to Semisymmetric Nonmetric connection if and only if . *

*Proof. *Let be Ricci-generalized pseudoparallel . Putting and by virtue of (3.1), (3.6), (3.16), and (4.1) in (2.7), (4.11), we get
Using (1.1), (2.1), (3.6), (3.15), (4.1), and (5.1) in (6.12), we get
Now by using (6.3) in (6.13), we get
Interchanging and in (6.14), we get
Adding (6.14) and (6.15), we get
Writting the above equation, we have

*Remark 6.4. * Let be an invariant submanifold of a Sasakian manifold which admits Semisymmetric Nonmetric connection. If is semiparallel, pseudoparallel, and Ricci-generalized pseudoparallel, then we have obtained conditions connecting , , , and . These conditions need further investigation and are to be interpreted geometrically.

Using Theorems 5.1 to 5.4 and corollary 5.3, we have the following result.

Corollary 6.5. *Let be an invariant submanifold of a Sasakian manifold admitting a Semisymmetric Nonmetric connection. Then the following statements are equivalent: *(1)*is recurrent, *(2)* is 2-recurrent, *(3)* is generalized 2-recurrent, *(4)* has parallel third fundamental form.*

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