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
.