**
Journal of Applied Mathematics and Physics** Vol.2 No.8(2014), Article
ID:47898,10
pages
DOI:10.4236/jamp.2014.28090

Some Properties of -Submanifolds of a Nearly Trans-Sasakian Manifold with a Semi Symmetric Non-Metric Connection

Lovejoy S. Das^{1}, Mobin Ahmad^{2}, Abdul Haseeb^{2}

^{1}Department of Mathematics, Kent State University, Kent, USA

^{2}Department of Mathematics, Integral University, Lucknow, India

Email: ldas@tusc.kent.edu, mobinahmad@rediffmail.com, malik_haseeb@rediffmail.com

Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 30 May 2014; revised 30 June 2014; accepted 11 July 2014

ABSTRACT

This paper deals with the study of -submanifolds of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Nijenhuis tensor, integrability conditions for some distributions on -submanifolds of a nearly trans-Sasakian manifold with a semi symmetric nonmetric connection are discussed.

**Keywords:**-Submanifolds, Nearly Trans-Sasakian Manifold, Semi Symmetric Non-Metric Connection, Distribution

1. Introduction

A. Bejancu defined and studied -submanifolds of a Kaehler manifold [1] . Later on, -submanifolds of a Sasakian manifold were studied by M. Kobayashi [2] , K. Yano and M. Kon [3] . J. A. Oubina introduced a new class of almost contact metric manifold known as trans-Sasakian manifold [4] . This class contains -Sasakian and -Kenmotsu manifold [5] . -submanifolds of a Kenmotsu manifold were studied by A. Bejancu and N. Papaghuic [6] . Geometry of -submanifolds of a trans-Sasakian manifold have been studied by M. H. Shahid in [7] [8] . -submanifolds of a nearly trans-Sasakian manifold were studied by Falleh R. Al-Solamy [9] . - submanifolds of an -Sasakian manifold with a semi-symmetric metric connection were studied by M. Ahmad et al. [10] . Motivated by the studies in [11] -[13] , in this paper we study -submanifolds of a nearly transSasakian manifold endowed with a semi symmetric non-metric connection.

Let be a linear connection in an -dimensional differentiable manifold. The torsion tensor of is given by

The connection is symmetric if torsion tensor vanishes, otherwise it is non-symmetric. The connection is metric connection if there is a Riemannian metric in such that, otherwise it is non-metric. It is well known that a linear connection is symmetric and metric if and only if it is the Levi-Civita connection.

In [14] , S. Golab introduced the idea of a semi-symmetric and quarter symmetric linear connections. A linear connection is said to be semi-symmetric if its torsion tensor is of the form

where is a 1-form and is a tensor field of the type (1,1).

We consider integrabilities of horizontal and vertical distributions of -submanifolds with a semi symmetric non-metric connection. We also consider parallel horizontal distributions of -submanifolds.

The paper is organized as follows: In Section 2, we recall some necessary details of nearly trans-Sasakian manifold. In Section 3, we study -submanifolds of a nearly trans-Sasakian manifold. In Section 4, some useful lemmas are proved. In Section 5, some basic results on parallel distribution are investigated. In Section 6, we calculated Nijenhuis tensor and studied integrability conditions of the distributions on -submanifolds of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection.

2. Nearly Trans-Sasakian Manifold

Let be an -dimensional almost contact metric manifold [15] with an almost contact metric structure, that is, is a (1,1) tensor field, is a vector field, is a 1-form and is a compatible Riemannian metric such that

(1)

(2)

(3)

for all vector fields,. There are two well known classes of almost contact metric manifolds, namely Sasakian and Kenmotsu manifolds. Sasakian manifolds are characterized by the tensorial relation

while Kenmotsu manifolds are given by the tensor equation

An almost contact metric structure on is called a trans-Sasakian structure [4] if belongs to the class of Gray-Hervella classification of almost Hermitian manifolds [16] , where is the almost complex structure on defined by

for all vector fields on and smooth function on. This may be expressed by the condition [17]

(4)

for some smooth functions and on and we say that the trans-Sasakian structure is of type.

In 2000, C. Gherghe [18] introduced a nearly trans-Sasakian structure of the type An almost contact metric structure on is called a nearly trans-Sasakian structure [18] if

(5)

A trans-Sasakian structure is always a nearly trans-Sasakian structure. Moreover, a nearly trans-Sasakian structure of type is nearly Sasakian [19] .

Let be an -dimensional isometrically immersed submanifold of a nearly trans-Sasakian manifold and denote by the same the Riemannian metric tensor field induced on from that of.

3. -Submanifolds of Nearly Trans-Sasakian Manifolds

Definition 3.1 An -dimensional Riemannian submanifold of a nearly trans-Sasakian manifold is called a -submanifold if is tangent to and there exists on a differentiable distribution such that

(i) the distribution is invariant under, i.e., for each;

(ii) The orthogonal complementary distribution of the distribution on is antiinvarient under, i.e., for all, where and are tangent space and normal space of at respectively.

If dim (resp.,), then -submanifold is called an invariant (resp., anti-invariant). The distribution (resp.,) is called the horizontal (resp., vertical) distribution. The pair is called -horizontal (resp., -invariant) if (resp.,) for.

For any vector field tangent to, we put

(6)

where and belong to the distribution and respectively.

For any vector field normal to, we put

(7)

where (resp.,) denotes the tangential (resp., normal) component of.

Now, we remark that owing to the existence of the 1-form, we can define a semi symmetric non-metric connection in any almost contact metric manifold by

(8)

such that for any, where is the induced connection with respect to on.

By using (4) and (8), we get

(9)

Similarly, we have

On adding above equations, we obtain

(10)

This is the condition for with a semi symmetric non-metric connection to be nearly transSasakian manifold.

We denote by the metric tensor of as well as that induced on. Let be the semi symmetric non-metric connection on and be the induced connection on with respect to the unit normal. Then we have:

Theorem 3.2 (i) If is -horizontal, and is parallel with respect to, then the connection induced on a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection is also a semi symmetric non-metric connection.

(ii) If is -vertical, and is parallel with respect to, then the connection induced on a -submanifold of a nearly trans-Sasakian with a semi symmetric non-metric connection is also a semi symmetric non-metric connection.

(iii) The Gauss formula with respect to the semi symmetric non-metric connection is of the form

.

Proof. Let be the induced connection with respect to the unit normal on a -submanifold of a nearly trans-Sasakian manifold from a semi symmetric non-metric connection connection, then

(11)

where is a tensor field of the type (0,2) on -submanifold. If be the induced connection on -submanifold from Riemannian connection, then

(12)

where is a second fundamental form. By the definition of the semi symmetric non-metric connection, we have

Now, using (11) and (12) in above equation, we have

Using (6), the above equation can be written as

(13)

From (13), comparing the tangential and normal components from both the sides, we get

(14)

(15)

(16)

Using (14), the Gauss formula for a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection is

(17)

This proves (iii). In view of (15), if is -horizontal, and is parallel with respect to, then the connection induced on a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection is also a semi symmetric non-metric connection.

Similarly, using (16), if is -vertical, and is parallel withrespect to, then the connection induced on a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric nonmetric connection is also a semi symmetric non-metric connection.

Weingarten formula is given by

(18)

for, (resp.,) is the second fundamental form (resp., tensor) of in and denotes the operator of the normal connection. Moreover, we have

(19)

4. Some Basic Lemmas

Lemma 4.1 Let be a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(20)

(21)

(22)

for.

Proof. By direct covariant differentiation, we have

By virtue of (6), (9), (17) and (18), we get

Similarly, we have

On adding above equations, we have

Now using (6), (7) and equating horizontal, vertical and normal components in above equation, the lemma follows.

Lemma 4.2 Let be a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(23)

for any.

Proof. By the use of (17), we have

(24)

Also, we have

(25)

From above equations, we get

(26)

For a nearly trans-Sasakian manifold with a semi symmetric non-metric connection, we have

(27)

Combining (26) and (27), the lemma follows.

In particular, we have the following corollary.

Corollary 4.3 Let be a -vertical -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(28)

for any.

Similarly, by Weingarten formula, we can easily get the following lemma.

Lemma 4.4 Let be a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(29)

for any.

Corollary 4.5 Let be a -horizontal -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(30)

for any.

Lemma 4.6 Let be a CR-submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(31)

for any.

Proof. As we have

Now, by using Gauss and Weingarten formulae in above equation, we have

Also, we have

From above equations, we get

In view of (10) and above equation, the lemma follows.

5. Parallel Distributions

Definition 5.1 The horizontal (resp., vertical) distribution (resp.,) is said to be parallel [1] with respect to the semi symmetric non-metric connection on if (resp.,) for any (resp.,).

Now, we have the following proposition.

Proposition 5.2 Let be a -vertical -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(32)

for all.

Proof. By the parallelness of horizontal distribution, we have

(33)

, using the fact that, (21) gives

(34)

Therefore in view of (7), we have

(35)

From (22), we have

(36)

for any.

Now, putting and in (36), we get respectively

(37)

(38)

Hence from (37) and (38), we have

(39)

Operating on both sides of (39) and using, we get

(40)

for all.

Now, for the distribution, we have the following proposition.

Proposition 5.3 Let be a -vertical -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. If the distribution is parallel with a semi symmetric non-metric connection on. Then

(41)

Proof. By using Weingarten formula, we have

and

for. From above equations, we have

Using (10) and (17), we obtain

(42)

for any. Taking inner product with in (41), we get

(43)

If the distribution is parallel, then and for any. So from above equation, we get

(44)

or

(45)

which implies that.

Definition 5.4 A -submanifold with a semi symmetric non-metric connection is said to be mixed totally geodesic if for all and.

Definition 5.5 A normal vector field with a semi symmetric non-metric connection is called -parallel normal section if for all.

Now, we have the following proposition.

Proposition 5.6 Let be a mixed totally geodesic -vertical -submanifold of a nearly transSasakian manifold with a semi symmetric non-metric connection. Then the normal section is -parallel if and only if for all.

6. Integrability Conditions of Distributions

In this section, we calculate the Nijenhuis tensor on a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. For this, first we prove the following lemma.

Lemma 6.1 Let be a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(46)

for any.

Proof. From the definition of nearly trans-Sasakian manifold with a semi symmetric non-metric connection, we have

(47)

Also, we have

(48)

Now, using (48) in (47), we get

(49)

for any, which completes the proof of the lemma.

On a nearly trans-Sasakian manifold with a semi symmetric non-metric connection, Nijenhuis tensor is given by

(50)

for any.

From (46) and (50), we get

(51)

In view of (10), we have

Using above equation in (51), we obtain

(52)

for any.

Proposition 6.2 Let be a -vertical -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then the distribution is integrable if the following conditions are satisfied:

(53)

for any.

Proof. The torsion tensor of the almost contact metric structure is given by

(54)

Thus, we have

(55)

for any.

Suppose that the distribution is integrable. So for,. If, then from (52) and (54), we have

(56)

for any and.

Replacing by for, we get

(57)

Interchanging and for in (57), we have

(58)

Subtracting above equations, we get

(59)

for any and the assertion follows.

Now, we prove the following proposition.

Proposition 6.3 Let be a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then

(60)

for any.

Proof. For and, we have

(61)

The above equation is true for all, therefore transvecting the vector field both sides, we obtain

(62)

Interchanging the vector fields and, we get

(63)

From (62) and (63), we get

(64)

for any, which completes the proof.

Proposition 6.4 Let be a -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then the distribution is integrable if and only if

(65)

for.

Proof. Proof of the theorem is similar as proof of the theorem 5.4 of [2] .

Corollary 6.5 Let be a -horizontal -submanifold of a nearly trans-Sasakian manifold with a semi symmetric non-metric connection. Then the distribution is integrable if and only if

(66)

for.

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