Research Article | Open Access
Some Results on Warped Product Submanifolds of a Sasakian Manifold
We study warped product Pseudo-slant submanifolds of Sasakian manifolds. We prove a theorem for the existence of warped product submanifolds of a Sasakian manifold in terms of the canonical structure .
The notion of slant submanifold of almost contact metric manifold was introduced by Lotta . Latter, Cabrerizo et al. investigated slant and semislant submanifolds of a Sasakian manifold and obtained many interesting results [2, 3].
The notion of warped product manifolds was introduced by Bishop and O'Neill in . Latter on, many research articles appeared exploring the existence or nonexistence of warped product submanifolds in different spaces (cf. [5–7]). The study of warped product semislant submanifolds of Kaehler manifolds was introduced by Sahin . Recently, Hasegawa and Mihai proved that warped product of the type in Sasakian manifolds is trivial where and are invariant and anti-invariant submanifolds of a Sasakian manifold, respectively .
In this paper we study warped product submanifolds of a Sasakian manifold. We will see in this paper that for a warped product of the type , if is any Riemannian submanifold tangent to the structure vector field of a Sasakian manifold then is an anti-invariant submanifold and if is tangent to then there is no warped product. Also, we will show that the warped product of the type of a Sasakian manifold is trivial and that the warped product of the type exists and obtains a result in terms of canonical structure.
Let be a dimensional manifold with almost contact structure defined by a tensor field , a vector field , and the dual form of , satisfying the following properties :
There always exists a Riemannian metric on an almost contact manifold satisfying the following compatibility condition:
An almost contact metric manifold is called Sasakian if
for all in , where is the Levi-Civita connection of on . From (2.3), it follows that
Let be submanifold of an almost contact metric manifold with induced metric and if and are the induced connections on the tangent bundle and the normal bundle of , respectively, then Gauss and Weingarten formulae are given by
for each and , where and are the second fundamental form and the shape operator (corresponding to the normal vector field ), respectively, for the immersion of into . They are related as
where denotes the Riemannian metric on as well as the one induced on .
For any , we write
where is the tangential component and is the normal component of .
Similarly, for any , we write
where is the tangential component and is the normal component of . We shall always consider to be tangent to . The submanifold is said to be invariant if is identically zero, that is, for any . On the other hand, is said to be anti-invariant if is identically zero, that is, , for any .
For each nonzero vector tangent to at , such that is not proportional to , we denote by the angle between and .
is said to be slant  if the angle is constant for all and . The angle is called slant angle or Wirtinger angle. Obviously, if is invariant and if is an anti-invariant submanifold. If the slant angle of is different from 0 and then it is called proper slant.
A characterization of slant submanifolds is given by the following.
Theorem 2.1 (see ). Let be a submanifold of an almost contact metric manifold , such that . Then is slant if and only if there exists a constant such that Furthermore, in such case, if is slant angle, then .
Following relations are straightforward consequences of (2.10)
for any tangent to
3. Warped and Doubly Warped Product Manifolds
Let and be two Riemannian manifolds and a positive differentiable function on . The warped product of and is the Riemannian manifold , where
A warped product manifold is said to be trivial if the warping function is constant. We recall the following general formula on a warped product :
where is tangent to and is tangent to .
Let be a warped product manifold then is totally geodesic and is totally umbilical submanifold of , respectively.
Doubly warped product manifolds were introduced as a generalization of warped product manifolds by Ünal . A doubly warped product manifold of and , denoted as is the manifold endowed with a metric defined as
where and are positive differentiable functions on and , respectively.
In this case formula (3.2) is generalized as
for each in and in .
If neither nor is constant we have a nontrivial doubly warped product . Obviously in this case both and are totally umbilical submanifolds of .
Now, we consider a doubly warped product of two Riemannian manifolds and embedded into a Sasakian manifold such that the structure vector field is tangent to the submanifold . Consider is tangent to , then for any we have
On comparing tangential and normal parts and using the fact that , and are mutually orthogonal vector fields, (3.6) implies that
This shows that is constant and is an anti-invariant submanifold of , if the structure vector field is tangent to .
Similarly, if is tangent to and for any we have
That is, is constant and is an anti-invariant submanifold of .
Note. From the above conclusion we see that for warped product submanifolds of a Sasakian manifold , if the structure vector field is tangent to the first factor then second factor is an anti-invariant submanifold. On the other hand the warped product is trivial if the structure vector field is tangent to .
To study the warped product submanifolds with structure vector field tangent to , we have obtained the following lemma.
Lemma 3.1 (see ). Let be a proper warped product submanifold of a Sasakian manifold , with , where and are any Riemannian submanifolds of . Then (i), (ii), (iii), (iv) for any and .
4. Warped Product Pseudoslant Submanifolds
The study of semislant submanifolds of almost contact metric manifolds was introduced by Cabrerizo et.al. . A semislant submanifold of an almost contact metric manifold is a submanifold which admits two orthogonal complementary distributions and such that is invariant under and is slant with slant angle , that is, and makes a constant angle with for each . In particular, if , then a semislant submanifold reduces to a contact CR-submanifold. For a semislant submanifold of an almost contact metric manifold, we have
Similarly we say that is an pseudo-slant submanifold of if is an anti-invariant distribution of , that is, and is slant with slant angle . The normal bundle of an pseudo-slant submanifold is decomposed as
where is an invariant subbundle of .
From the above note, we see that for warped product submanifolds of a Sasakian manifold , one of the factors is an anti-invariant submanifold of . Thus, if the manifolds and are slant and anti-invariant submanifolds of Sasakian manifold , then their possible warped product pseudo-slant submanifolds may be given by one of the following forms:(a), (b).
The above two types of warped product pseudo-slant submanifolds are trivial if the structure vector field is tangent to and , respectively. Here, we are concerned with the other two cases for the above two types of warped product pseudo-slant submanifolds and when is in and in , respectively.
For the warped product of the type , we have
Theorem 4.1. There do not exist the warped product Pseudo-slant submanifolds where is an anti-invariant and is a proper slant submanifold of a Sasakian manifold such that is tangent to .
Proof. For any and , we have Using (2.3), (2.5), (2.6), and the fact that is tangent to , we obtain Comparing tangential and normal parts, we get Equation (4.5) takes the form on using (3.2) as Taking product with , the left hand side of the above equation is zero using the fact that and are mutually orthogonal vector fields. Then Using (2.7), (2.11) and the fact that is tangent to , we get As , then interchanging by and taking account of (2.10), we obtain or Adding equations (4.8) and (4.10), we get The right hand side of the above equation is zero by Lemma 3.1(iv); then Since is proper slant and is nonnull, then In particular, for , Lemma 3.1 (i) implies that . This means that is constant on . Hence the theorem is proved.
Now, the other case is dealt with in the following theorem.
Theorem 4.2. Let be a warped product submanifold of a Sasakian manifold such that is an invariant submanifold tangent to and is an anti-invariant submanifold of . Then lies in the invariant normal subbundle for each and .
Proof. As is a warped product submanifold with tangent to , then by (2.3),
for any and . Using this fact in the formula
for each , thus, we obtain
Then from (2.5) and (2.6), we get
Which on using (2.8) and (2.9) yields
From the normal components of the above equation, formula (3.2) gives
Taking the product in (4.19) with for any , we get
Then from (2.2), we have
On the other hand, we have
Taking the product in (4.23) with for any and using (4.22), (2.2), (3.2), and the fact that is tangential to , we obtain that
for any and .
Now, if then using the formula (4.23), we get
As is an invariant submanifold, then for any , thus using the fact that the product of tangential component with normal is zero, we obtain that for any and . Thus from (4.24) and (4.26), it follows that . Thus the proof is complete.
The authors are thankful to the referee for his valuable suggestion and comments which have improved this paper.
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