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

A. Bejancu defined and studied

Let

The connection

In [

where

We consider integrabilities of horizontal and vertical distributions of

The paper is organized as follows: In Section 2, we recall some necessary details of nearly trans-Sasakian manifold. In Section 3, we study

Let

for all vector fields

while Kenmotsu manifolds are given by the tensor equation

An almost contact metric structure

for all vector fields

for some smooth functions

In 2000, C. Gherghe [

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

Let

Definition 3.1 An

(i) the distribution

(ii) The orthogonal complementary distribution

and normal space of

If dim

For any vector field

where

For any vector field

where

Now, we remark that owing to the existence of the 1-form

such that

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

Similarly, we have

On adding above equations, we obtain

This is the condition for

We denote by

Theorem 3.2 (i) If

(ii) If

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

Proof. Let

where

where

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

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

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

Using (14), the Gauss formula for a

This proves (iii). In view of (15), if

Similarly, using (16), if

Weingarten formula is given by

for

Lemma 4.1 Let

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

for any

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

Also, we have

From above equations, we get

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

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

In particular, we have the following corollary.

Corollary 4.3 Let

for any

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

Lemma 4.4 Let

for any

Corollary 4.5 Let

for any

Lemma 4.6 Let

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.

Definition 5.1 The horizontal (resp., vertical) distribution

Now, we have the following proposition.

Proposition 5.2 Let

for all

Proof. By the parallelness of horizontal distribution

Therefore in view of (7), we have

From (22), we have

for any

Now, putting

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

Operating

for all

Now, for the distribution

Proposition 5.3 Let

Proof. By using Weingarten formula, we have

and

for

Using (10) and (17), we obtain

for any

If the distribution

or

which implies that

Definition 5.4 A

Definition 5.5 A normal vector field

Now, we have the following proposition.

Proposition 5.6 Let

In this section, we calculate the Nijenhuis tensor

Lemma 6.1 Let

for any

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

Also, we have

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

for any

On a nearly trans-Sasakian manifold with a semi symmetric non-metric connection

for any

From (46) and (50), we get

In view of (10), we have

Using above equation in (51), we obtain

for any

Proposition 6.2 Let

for any

Proof. The torsion tensor

Thus, we have

for any

Suppose that the distribution

for any

Replacing

Interchanging

Subtracting above equations, we get

for any

Now, we prove the following proposition.

Proposition 6.3 Let

for any

Proof. For

The above equation is true for all

Interchanging the vector fields

From (62) and (63), we get

for any

Proposition 6.4 Let

for

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

Corollary 6.5 Let

for