Abstract

The present paper focuses on the study of noninvariant hypersurfaces of a nearly trans-Sasakian manifold equipped with -structure. Initially some properties of this structure have been discussed. Further, the second fundamental forms of noninvariant hypersurfaces of nearly trans-Sasakian manifolds and nearly cosymplectic manifolds with -structure have been calculated provided is parallel. In addition, the eigenvalues of have been found and proved that a noninvariant hypersurface with -structure of nearly cosymplectic manifold with contact structure becomes totally geodesic. Finally the paper has been concluded by investigating the necessary condition for totally geodesic or totally umbilical noninvariant hypersurface with -structure of a nearly trans-Sasakian manifold.

1. Introduction

A hypersurface such that transform of a tangent vector of the hypersurface by the structure tensor field defining the almost contact structure is never tangent to the hypersurface was studied by Goldberg and Yano in 1970 [1]. Yano and Okumura [2] introduced -structure and termed it as a noninvariant hypersurface of almost contact metric manifold. In this paper, they showed that there always exists a -structure and gave the results that there does not exist an invariant hypersurface of a contact manifold. As a consequence, an invariant hypersurface of (almost) cosymplectic manifold is (almost) Kähler manifold. Further, they proved that there is no noninvariant hypersurfaces of a Sasakian manifold. In 1990, Chen [3] introduced slant submanifold which is a generalization of invariant and anti-invariant submanifold of almost Hermite manifold. Recently, Prasad [4] studied the noninvariant hypersurface of trans-Sasakian manifold. In this paper, we study noninvariant hypersurface to the setting of nearly trans-Sasakian manifold.

2. Preliminaries

Let be an almost contact metric manifold with almost contact metric structure ; that is, is tensor field, is -form, and is a compatible Riemannian metric such that for all .

An almost contact metric manifold is called a Sasakian manifold if there exists a killing vector field of unit length on so that the tensor field of type , defined by , satisfies the condition for any pair of vector fields and on .

The notion of trans-Sasakian structure is given by Oubina [5]. In 1990, Blair and Oubina [6] find the condition for all vector fields , on and where and are smooth functions. In this case we say that the trans-Sasakian structure is of type .

Gherghe introduced a nearly trans-Sasakian structure of type . An almost contact metric manifold with almost contact metric structure is said to be a nearly trans-Sasakian manifold [7] if for all vector fields , on , where and are smooth functions on and is the operator of covariant differentiation with respect to .

Moreover, a nearly trans-Sasakian structure of type is nearly Sasakian or nearly Kenmotsu or nearly cosymplectic according to , , or , , or , respectively.

From (4), we have The Gauss and Weingarten formulae are given by for all , where and are the Riemannian and induced Riemannian connections in and , respectively, and is the unit normal vector in the normal bundle . In this formula is the second fundamental form on related to by Let be a hypersurface of an almost contact metric manifold; then we define the following: For , we get an induced -structure [2, 8] on the noninvariant hypersurface satisfying for all ; .

Therefore we see that every transversal hypersurface of an almost contact Riemannian manifold also admits a -structure.

3. Noninvariant Hypersurfaces with -Structure

A noninvariant hypersurface of an almost contact manifold [1] is a hypersurface such that the transform of a tangent vector of the hypersurface by a linear transformation field of type acting in each tangent space of , defining the almost contact structure, is never tangent to the hypersurface. Let be a tangent vector of noninvariant hypersurface and then will never be tangent to the hypersurface defined by (8).

A hypersurface of an almost contact manifold does not in general possess an almost complex structure. Goldberg and Yano [1] showed that there does not exist an invariant hypersurface of a contact manifold. This statement says that it is impossible to imbed a manifold as an invariant hypersurface of a contact space. It is well known that a hypersurface (real codimension 1) of an almost complex manifold admits an almost contact structure. However, this hypersurface clearly is not invariant, since the real codimension is 1; otherwise it admits an almost complex structure.

Consider noninvariant hypersurfaces of almost contact manifolds . These again admit almost complex structures but, in addition, there is a distinguished -form induced by the contact form of .

Lemma 1. Let be a noninvariant hypersurface with -structure of a nearly trans-Sasakian manifold . Then for all .

Proof. Consider by using (6), (8), and (9). Then from (16) and (17), we have (13).
Next, Therefore Similarly From (19) and (20), we get (14).
Further, consider which proves (15).

Proposition 2. Let be a noninvariant hypersurface with -structure of a nearly trans-Sasakian manifold , and then for all .

Proof. From (5) and (15), we have Equating tangential and normal parts, we have

Now we find some results on totally geodesic noninvariant hypersurfaces.

Theorem 3. Let be a totally umbilical noninvariant hypersurface with -structure of a nearly trans-Sasakian manifold. Then it is totally geodesic if Also, if nearly trans-Sasakian manifold admits contact structure, then

Proof. Consider Then, we have From (5) and (28), we calculate Equating normal part, we have If is totally umbilical, then , where is Kählerian metric [9] and we know the relation of on related to by Therefore , and then (30) gives If is totally geodesic, that is, , then (32) gives (25). If nearly trans-Sasakian manifold has contact structure, then from (25) we have (26).

Theorem 4. Let be a noninvariant hypersurface with -structure of a nearly trans-Sasakian manifold. If is parallel, then one has where and is a tensor field of type . Also, is totally geodesic if

Proof. From (4) and (13), we have easily found the relation Since is parallel then (35) reduces to Applying both sides, we obtain In view of (37), we have where .
In a similar way we have Using (37)–(39), we have the result (33).
Next, from (30) and (33), we have If nearly trans-Sasakian with contact structure, then which implies

As a consequence, we have the following.

Corollary 5. Let be a noninvariant hypersurface with -structure of a nearly cosymplectic manifold. If is parallel, then one has

Theorem 6. Let be a noninvariant hypersurface with -structure of a nearly trans-Sasakian manifold. If a vector field is parallel, then one has Consequently, if is totally geodesic and for all , then eigenvalue of is .

Proof. Consider Therefore From (4), we have From (47), we have Then from (17) we have Using (46), (48), and (49), we get Now equating tangential part, we have If is parallel then (51) implies If is totally geodesic then ; that is, .
Therefore If , then , which implies .
Hence eigenvalue of is .

Theorem 7. If is parallel vector field in noninvariant hypersurface with -structure of a nearly cosymplectic manifold admitting contact structure, then

Proof. We know that, for a nearly cosymplectic manifold , From (28) and (55), we have Equating tangential part, we have Since nearly cosymplectic manifold admits contact structure, then .
If is parallel, then   .
Since , therefore   , which gives   , which implies is totally geodesic.
Proposition 2 also enables us to deduce the above result by substituting , for a nearly cosymplectic manifold admitting contact structure (i.e., ) provided is parallel.

Conflict of Interests

The authors declare that there is no conflict of interests regarding to the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous reviewer for his valuable comments and suggestions to improve the quality of the paper. They also thank Dr. Rajendra Prasad for his help.