International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 947640, 18 pages

doi:10.1155/2012/947640

Research Article

Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection

B. S. Anitha and C. S. Bagewadi

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

Received 30 March 2012; Accepted 2 July 2012

Academic Editor: V. R. Khalilov

Copyright © 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 1970’s 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 “ and ,” 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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