#### Abstract

We show that there exist no proper warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds such that totally geodesic submanifold and totally umbilical submanifold of the warped product are invariant and anti-invariant, respectively. Therefore, we consider warped product semi-invariant submanifolds in the form by reversing two factor manifolds and . We prove several fundamental properties of warped product semi-invariant submanifolds in an almost paracontact Riemannian manifold and establish a general inequality for an arbitrary warped product semi-invariant submanifold. After then, we investigate warped product semi-invariant submanifolds in a general almost paracontact Riemannian manifold which satisfy the equality case of the inequality.

#### 1. Introduction

It is well known that the notion of warped products plays some important role in differential geometry as well as physics. The geometry of warped product was introduced by Bishop and O'Neill [1]. Many geometers studied different objects/structures on manifolds endowed with an warped product metric (see [2β6]).

Recently, Chen has introduced the notion of CR-warped product in Kaehlerian manifolds and showed that there exist no proper warped product CR-submanifolds in the form in Kaehlerian manifolds. Therefore, he considered warped product CR-submanifolds in the form which is called CR-warped product, where is an invariant submanifold, and is an anti-invariant submanifold of Kaehlerian manifold (see [2, 7, 8]). Analogue results have been obtained for Sasakian ambient as the odd dimensional version of Kaehlerian manifold by Hasegawa and Mihai in [3] and Munteanu in [9].

Almost paracontact manifolds and almost paracontact Riemannian manifolds were defined and studied by at [10]. After then, many authors studied invariant and anti-invariant submanifolds of the almost paracontact Riemannian manifold with the structure , when is tangent to the submanifold, and is not tangent to the submanifold [11].

We note that CR-warped products in Kaehlerian manifold are corresponding warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds. In this paper, we showed that there exist no warped product semi-invariant submanifolds in the form in contrast to Kaehlerian manifolds (see Theorem 3.1). So, from now on we consider warped product semi-invariant submanifolds in the form , where is an anti-invariant submanifold, and is an invariant submanifold of an almost paracontact Riemannian manifold by reversing the two factor manifolds and and it simply will be called warped product semi-invariant submanifold in the rest of this paper (see Example 3.3 and Theorem 3.4).

#### 2. Preliminaries

Although there are many papers concerning the geometry of semi-invariant submanifolds of almost paracontact Riemannian manifolds (see [11β13]), there is no paper concerning the geometry of warped product semi-invariant submanifolds of almost paracontact Riemannian manifolds in literature so far. So the purpose of the present paper is to study warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds. We first review basic formulas and definitions for almost paracontact Riemannian manifolds and their submanifolds, which we shall use for later.

Let be an -dimensional differentiable manifold. If there exist on a type tensor field , a vector field and 1-form satisfying

then is said to be an almost paracontact manifold, where the symbol, denotes the tensor product. In the almost paracontact manifold, the following relations hold good:

An almost paracontactmanifold is said to be an almost paracontact metric manifold if Riemannian metric on satisfies

for all [14], where denotes the differentiable vector field set on . From (2.2) and (2.3), we can easily derive the relation

An almost paracontact metric manifold is said to be an almost paracontact Riemannian manifold with -connection if and , where denotes the connection on Since the vector field is also parallel with respect to [11, 13].

In the rest of this paper, let us suppose that is an almost paracontact Riemannian manifold with structure -connection.

Let be an almost paracontact Riemannian manifold, and let be a Riemannian manifold isometrically immersed in . For each , we denote by the maximal invariant subspace of the tangent space of . If the dimension of is the same for all in , then gives an invariant distribution on .

A submanifold in an almost paracontact Riemannian manifold is called semi-invariant submanifold if there exists on a differentiable invariant distribution whose orthogonal complementary is an anti-invariant distribution, that is, , where denotes the orthogonal vector bundle of in A semi-invariant submanifold is called anti-invariant (resp., invariant) submanifold if dim (resp., dim). It is called proper semi-invariant submanifold if it is neither invariant nor anti-invariant submanifold.

A semi-invariant submanifold of an almost paracontact Riemannian manifold is called a Riemannian product if the invariant distribution and anti-invariant distribution are totally geodesic distributions in . The geometry notion of the semi-invariant submanifolds has been studied by many geometers in the various type manifolds. Authors researched the fundamental properties of such submanifolds (see references).

Let and be two Riemannian manifolds with Riemannian metrics and , respectively, and a differentiable function on Consider the product manifold with its projection and . The warped product manifold is the manifold equipped with the Riemannian metric tensor such that

for any , where is the symbol for the tangent map. Thus we have , where is called the warping function of the warped product. The warped product manifold is characterized by the fact that and are totally geodesic and totally umbilical submanifolds of , respectively. Hence Riemannian products are special classes of the warped products [4].

In this paper, we define and study a new class of warped product semi-invariant submanifolds in an almost paracontact Riemannian manifolds, namely, we investigate the class of warped product semi-invariant submanifolds, and we establish the fundamental theory of such submanifolds.

Now, let be an isometrically immersed submanifold in an almost paracontact Riemannian manifold We denote by and the Levi-Civita connections on and respectively. Then the Gauss and Weingarten formulas are, respectively, defined by

for any , , where is the connection in the normal bundle is the second fundamental form of and is the shape operator. The second fundamental form and the shape operator are related by

Now, let be a differentiable manifold, and we suppose that is an isometrically immersed submanifold in almost paracontact Riemannian manifold . We denote by the metric tensor of as well as that induced on For any vector field tangent to , we put

where and denote the tangential and normal components of , respectively. For any vector field normal to , we also put

where and denote the tangential and normal components of , respectively. The submanifold is said to be invariant if is identically zero, that is, . On the other hand, is said to be anti-invariant submanifold if is identically zero, that is, .

We note that for any invariant submanifold of an almost paracontact Riemannian manifold , if is normal to , then the induced structure from the almost paracontact structure on is an almost product Riemannian structure whenever is nontrivial. If is tangent to then the induced structure on is an almost paracontact Riemannian structure.

Furthermore, we say that is a semi-invariant submanifold if there exist two orthogonal distributions and such that

(1) splits into the orthogonal direct sum ;(2)the distribution is invariant, that is, ;(3)the distribution is anti-invariant, that is,Given any submanifold of an almost paracontact Riemannian manifold , from (2.4) and (2.8) we have

for any ,

From now on we suppose that the vector field is tangent to

We recall the following general lemma from [1] for later use.

Lemma 2.1. *Let be a warped product manifold with warping function , then one has *(1)(2)(3)* where and denote the Levi-Civita connections on and , respectively.*

Let be a semi-invariant submaniold of an almost paracontact Riemannian manifold . We denote by the invariant distribution and anti-invariant distribution . We also denote the orthogonal complementary of in by , then we have direct sum

We can easily see that is an invariant subbundle with respect to .

#### 3. Warped Product Semi-Invariant Submanifolds in an Almost Paracontact Riemannian Manifold

Useful characterizations of warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds are given the following theorems.

Theorem 3.1. *If is a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold such that is an invariant submanifold and is an anti-invariant submanifold of , then is a usual Riemannian product.*

*Proof. *Let be normal to Taking into account that is symmetric and using (2.3), (2.6), (2.7), and considering Lemma 2.1, for and , we have
which implies that .

If is tangent to , then can be written as follows:
where and . Since , from the Gauss formulae, we have
for any . Considering Lemma 2.1, we get
for any and . If is identically zero, then from Lemma 2.1 we have
It follows that the warping function is a constant and is usual Riemannian product. Hence the proof is complete.

If the warping function is constant, then the metric on the βsecondβ factor could be modified by an homothety, and hence, the warped product becomes a direct product.

Now, we give two examples for almost paracontact Riemannian manifold and their submanifolds in the form to illustrate our results. Firstly, we construct an almost paracontact metric structure on (see Example 3.2) and after give an example which is concerning its submanifold (see Example 3.3).

*Example 3.2. *Let
The almost paracontact Riemannian structure is defined on in the following way:
If , then we have
From this definition, it follows that
for an arbitrary vector field . Thus becomes an almost paracontact Riemannian manifold, where and denote usual inner product and standard basis of , respectively.

*Example 3.3. *Let be a submanifold in with coordinates given by
It is easy to check that the tangent bundle of is spanned by the vectors
We define the almost paracontact Riemannian structure of by
Then with respect to the almost paracontact Riemannian structure of , the space becomes
Since and are orthogonal to and , are tangent to , and can be taken subspace and subspace , respectively, where can be taken as for and . Furthermore, the metric tensor of is given by
Thus is a warped product semi-invariant submanifold with dimensional of almost paracontact manifold with warping function

Now, let be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold , where is an anti-invariant submanifold, and is an invariant submanifold of . If we denote the Levi-Civita connections on and by and , respectively, by using (2.6) and (2.8), we have

for any . Taking into account the tangential and normal components of (3.15), respectively, we have

where the derivatives of and are, respectively, defined by

Next, we are going to investigate the geometric properties of the leaves of the warped product semi-invariant submanifolds in an almost paracontact Riemannian manifold.

Theorem 3.4. *Let be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold Then the invariant distribution and the anti invariant distribution are always integrable.*

*Proof. *From (3.16) and considering lemma 2.1, we have
for any and . From the tangential and normal components of (3.19), respectively, we arrive at
By using (3.16) and (3.20) we get
Furthermore, by using the Gauss-Weingarten formulas and taking into account that is totally geodesic in and it is anti-invariant in , by direct calculations, it is easily to see that
which is also equivalent to
for any . Moreover, using (2.4) and (2.7) and making use of being self-adjoint, we obtain
which gives us
for any and . Thus from (3.24) and (3.26), we arrive at
for any . Furthermore, by using (2.6), (2.8), and (2.9) and considering Lemma 2.1, we have
for any , where denote the Levi-Civita connection on Taking into account the normal and tangential components of (3.28), respectively, we have
From (3.29), we can easily see that
for any . Finally, by using (3.17) and (3.31), we have
for any , that is, .

In the same way, making use of (3.16) and (3.27) for any , we conclude that
that is, . So we obtain the desired result.

Since the distributions and are integrable, we denote the integral manifolds of and by and , respectively.

Now, the following theorem characterizes (warped product or Riemannian product) semi-invariant submanifolds in almost paracontact manifolds.

Theorem 3.5. *Let be a submanifold of an almost paracontact Riemannian manifold . Then is a semi-invariant submanifold if and only .*

*Proof. *Let us assume that is a semi-invariant submanifold of an almost paracontact Riemannian manifold and by and we denote the projection operators on subspaces and , respectively, then we have
Moreover, by using (2.1), (2.8), and (2.9), if is tangent to , then we get
for any and . On the other hand, if is normal to , then (3.35) and (3.36) become, respectively,
From (2.8), we have
for any . From the tangential and normal components, we have
Since is invariant and is anti-invariant, we get
We have
by virtue of . Now by using the right-hand side to the second equation of (3.35) and using (3.40) and (3.41), we conclude that
which is also equivalent to

Conversely, for a submanifold of an almost paracontact Riemannian manifold , we suppose that . For any vector fields tangent to and normal to , by using (2.4) and (3.43), we have
for all . So we have . Since , it implies which is also equivalent to from (3.36). Since , we get . So, from (3.35) and (3.36), we conclude
Now, if we put
then we can derive that which show that and are orthogonal complementary projection operators and define complementary distributions and , respectively, where and denote the distributions which are belong to subspaces and , respectively. From (3.42), (3.45), and (3.46) we can derive
These equations show that the distribution is an invariant and the distribution is an anti-invariant. The proof is complete.

Theorem 3.6. *Let be a semi-invariant submanifold of an almost paracontact Riemannian manifold . Then is a warped product semi-invariant submanifold if and only if the shape operator of satisfies
**
for some function on satisfying , .*

*Proof. *We suppose that is a warped product semi-invariant submanifold in an almost paracontact Riemannian manifold . Then from (3.22), we have
for any and . Since is the only function on , we can easily see that for all .

Conversely, let us assume that is a semi-invariant submanifold in an almost paracontact Riemannian manifold satisfying
for some function on satisying for all . Since the ambient space is an almost paracontact Riemannian manifold and making use of (2.4) and (3.27), we arrive at
for any and . Thus the anti-invariant distribution is totally geodesic in . In the same way, making use of being Levi-Civita connection and (3.22), we have
for any and , where . Since the invariant distribution of semi-invariant submanifold is always integrable (Theorem 3.4) and , for each , which implies that the integral manifold of is an extrinsic sphere in , that is, it is a totally umbilical submanifold and its mean curvature vector field is non-zero and parallel, thus we know that is a Riemannian warped product , where and denote the integral manifolds of the distributions of and , respectively, and is the warping function. So we obtain the desired result.

In the rest of this section, we are going to obtain an inequality for the squared norm of the second fundamental form by means of the warping function for warped product semi-invariant submanifolds of an almost paracontact Riemannian manifold. Now, we recall that semi-invariant is said to be mixed geodesic (resp., -geodesic and -geodesic) submanifold if the second fundamental form of satisfies , and (resp., , and , ).

Now, we are going to give the following lemma for later use.

Lemma 3.7. *Let be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold . Then one has*(1)(2)*, , *(3)*, for any and *(4)* if and only if is a usual Riemannian product, where and denote the leaves of and , respectively.*

*Proof. * For any , by using (2.4) and (3.27) and considering that the ambient space is an almost paracontact Riemannian manifold, we have

Making use of being Levi-Civita connection and Lemma 2.1(2.2), we get
for any , .β

In the same way, we have
for any and .

Considering Lemma 2.1 we derive
for any and .

Theorem 3.8. *Let be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold Then one has the following.*(1)* The squared norm of the second fundamental form of in satisfies
where denote the trace of mapping .*(2)* If the equality sign of (3.57) holds identically, then is a totally geodesic, is a totally umbilical submanifolds of and is a mixed geodesic submanifold in Furthermore, is a minimal submanifold if and only if or is a usual Riemannian product.*

*Proof. *Let be an orthonormal basis of an almost paracontact Riemannian manifold such that is tangent to is tangent to and is tangent to . Taking into account Lemma 3.7 and the basic linear algebra rules, by direct calculations, we have
for all and . Since
here by direct calculations, we get
So we conclude that
which proves our assertion.

Now we assume that the equality case of (3.57) holds identically, then from (3.58), respectively, we obtain
Since is totally geodesic submanifold in , the first condition in (3.62) implies that is totally geodesic submanifold in . Moreover, Lemma 2.1 shows that is totally umbilical submanifold in . Therefore, the second condition in (3.62) implies that is also totally umbilical submanifold in . On the other hand, (3.20) and (3.63) imply that is mixed geodesic submanifold in

Conclusion 3.9. * The geometry of the warped products in Riemannian manifolds is totally different from the geometry of the warped products in complex manifolds. Namely, in the complex manifolds, there exists no proper warped product CR-submanifold in the form (see [2, 8]) while there exists no proper warped product semi-invariant submanifold in the form in Riemannian manifolds (see Theorem 3.1). The first condition in (3.62) implies that warped product CR-submanifold is minimal in complex manifolds while it does not imply that warped product semi-invariant submanifold is minimal in Riemannian product manifolds.*

#### Acknowledgment

The author would like to thank the referees for valuable suggestions and comments, which have improved the present paper.