Some New Results on Trans-Sasakian Manifolds

Wang, Lei; Zhao, Yan

doi:https://doi.org/10.1155/2022/1276797

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Research Article | Open Access

Volume 2022 | Article ID 1276797 | https://doi.org/10.1155/2022/1276797

Some New Results on Trans-Sasakian Manifolds

Lei Wang¹and Yan Zhao¹

Academic Editor: Antonio Masiello

Received28 Jan 2022

Accepted16 Feb 2022

Published15 Mar 2022

Abstract

In this paper, we classify trans-Sasakian manifolds which are realized as real hypersurfaces in a complex space form. We also investigate trans-Sasakian manifolds whose Reeb vector fields are harmonic-Killing. The above results bring some new characterizations for the property of trans-Sasakian 3-manifolds.

1. Introduction

In differential geometry of almost-contact Riemannian manifolds, the so-called trans-Sasakian manifolds play important roles when studying topology as well as geometry of almost-contact structures. Here, an almost-contact metric manifold of dimension , , together with its almost-contact metric structure , is said to be a trans-Sasakian manifold (it is, often referred to, of type ) (see [1–3]) if it satisfiesfor all vector fields , where both and are smooth functions. The classical Sasakian, Kenmotsu, and cosymplectic manifolds (see [1]) are all its trivial cases.

In general, a trans-Sasakian manifold of type is said to be proper (see [4–6]) when either or vanishes identically. Marrero in [7] proved that a trans-Sasakian manifold of dimension greater than 3 must be proper. However, such a property holds not necessarily true for general trans-Sasakian manifolds of dimension three. In the past decade, to determine on what geometric conditions a connected, compact, or complete trans-Sasakian three-manifold is proper has been proposed by Deshmukh in [8] and later considered by many authors (see recent results by De et al. [9–12], Deshmukh et al. [8, 13–19], Wang and Wang and Liu [20, 21], Wang [4, 22, 23], Zhao [5, 6] and Ma and Pei [24].

It is interesting to point out that trans-Sasakian three-manifolds isometrically immersed in the Euclidean four-space have been studied in [14]. In the present paper, extending Deshmukh’s above results, we consider a trans-Sasakian manifold of an arbitrary dimension immersed in a complex space form realized as a real hypersurface. As an immediate corollary, we also present a new characterization for the property of trans-Sasakian three-manifolds without compactness restriction. On the other hand, Zhao [6] provided a characterization for the property by considering the Reeb vector field of a trans-Sasakian three-manifold being affine Killing. In the present paper, we generalize such a result by weakening the above restriction; namely, we need only to suppose that the Reeb vector field is harmonic-Killing (see its definition in Section 4).

2. Trans-Sasakian Manifolds

Let be a smooth Riemannian manifold of dimension on which there exist a -type, -type, and -type tensor fields , , and , respectively. According to Blair [1], is called an almost-contact metric manifold iffor any vector fields and . is said to be the Reeb or structure vector field. An almost-contact metric manifold is said to be normal if , where denotes the Nijenhuis tensor of . An almost-contact metric manifold is said to be trans-Sasakian if it satisfies equality (1). A three-dimensional almost-contact metric manifold is trans-Sasakian if and only if it is normal. This is not necessarily true for higher dimension.

A normal almost-contact metric manifold is said to be an -Sasakian manifold if and , where is a nonzero constant. An -Sasakian manifold reduces to a Sasakian manifold when . A normal almost-contact metric manifold is called a -Kenmotsu manifold if it satisfies and , where is a nonzero constant. A -Kenmotsu manifold becomes a Kenmotsu manifold when . A normal almost-contact metric manifold is said to be a cosymplectic manifold if it satisfies and . Obviously, the set of all -Sasakian manifolds (resp., -Kenmotsu) is a proper subset of that of all trans-Sasakian manifolds of type (resp., ).

Putting into (1) and using (2), we havefor any vector field . In this paper, all manifolds are assumed to be connected.

3. Trans-Sasakian Manifolds as Real Hypersurfaces in Complex Space Forms

Let be a complete and simply connected complex space form which is complex analytically isometric to the following:(i)A complex projective space if (ii)A complex Euclidean space if (iii)A complex hyperbolic space if

Here, is the constant holomorphic sectional curvature.

Let be a real hypersurface immersed in a complex space form and be a unit normal vector field of . We denote by the Levi-Civita connection of the metric of and the complex structure. Let and be the induced metric from the ambient space and the Levi-Civita connection of the metric , respectively. Then, the Gauss and Weingarten formulas are given, respectively, as follows:for any vector fields , where denotes the shape operator of in . For any vector field , we put

One can check that (2) holds and hence, on real hypersurfaces, there exist natural almost-contact metric structures. If the structure vector field is principal, that is, at each point, where , then is called a Hopf hypersurface and is called Hopf principal curvature.

Moreover, applying the parallelism of the complex structure (i.e., ) of and using (4) and (5), we havefor any vector fields . Let be the Riemannian curvature tensor of . As is of constant holomorphic sectional curvature , the Gauss equation of in is given byfor any vector fields .

Because an almost-contact metric structure exists on a real hypersurface, then it is very interesting to ask what almost-contact metric structure can be if it is realized as a real hypersurface in complex space forms? Some authors have studied contact, Sasakian, and generalized Sasakian space form structures on real hypersurfaces (see [25–27]).

Theorem 1. Let be a trans-Sasakian manifold. Then, is realized as a real hypersurface in a complex space form , , if and only if the following statements are valid:(1)If , is locally congruent to a geodesic hypersphere.(2)If , is locally congruent to(i)a horosphere(ii)a geodesic hypersphere(iii)a tube around a totally geodesic (3)If , is locally congruent to(i)a hyperplane (ii)a sphere (iii)a cylinder over a plane curve

Proof. If a real hypersurface in a complex space form is trans-Sasakian, by definition, from (1) and (6), we getfor any vector fields . In the above equality, setting givesfor any vector field . Obviously, it follows that and hence, is Hopf. Using this in the previous equality, we getfor any vector field . Recall that the shape operator is self-adjoint; it follows directly that , and hence,for any vector field . Now, the hypersurface is totally -umbilical. Conversely, the application of the above equality in (6) implies that the hypersurface is always a trans-Sasakian manifold of type . Next, we divide our discussions into two cases.
When the ambient space is or , following [28, 29], we observe that a totally -umbilical real hypersurface satisfying (11) is locally congruent to the following:(i)A geodesic sphere of radius in with , where (ii)A horosphere in with (iii)A geodesic sphere of radius in with , where (iv)A tube of radius around a totally geodesic complex hyperplane in with , where When the ambient space is , from Gauss equations (7) and (11), we see that the hypersurface is pseudo-Einstein, i.e.,for any vector field , where denotes the Ricci operator. The remaining proof follows immediately from Proof of Theorem 1 in [27] (see also [30]).
The converse is easy to check.
In view of Theorem 1, a new characterization for the property of trans-Sasakian 3-manifolds is given. We remark that in two cases in the proof of Theorem 1 is both constant.

Corollary 1. A trans-Sasakian 3-manifold is an -Sasakian manifold if it is realized as a real hypersurface in the complex space form.

As pointed out in the Introduction section, a trans-Sasakian 3-manifold of type realized as a hypersurface in is isometric to the Sasakian manifold provided that the hypersurface is compact. Such a situation occurs in our Theorem 1 in view of (12) and (11) for (for more details, see ([14], Theorem 2)).

4. Harmonic-Killing Reeb Vector Field

From [31], a vector field on a Riemannian manifold is called affine Killing ifwhere denotes the Levi-Civita connection of the metric (see also [32]). According to [33, 34], a vector field on a Riemannian manifold is called harmonic-Killing if each local parameter group of infinitesimal transformations associated to is a group of harmonic maps. For any harmonic-Killing vector field , from Theorem 2.1 in [33], we have

By considering the Reeb vector field of trans-Sasakian three-manifolds being affine Killing, Zhao [6] studied the property of trans-Sasakian three-manifolds. In this section, we consider a weaker condition on trans-Sasakian manifolds of arbitrary dimensions.

Lemma 1. If the Reeb vector field of trans-Sasakian manifolds of type is harmonic-Killing, then we have

Moreover, if , we have .

Proof. Recall that on any differentiable manifold, there holds (see Yano ([35], pp. 23))for any vector fields . Notice that in our case, the Riemannian metric is parallel and it follows thatfor any vector fields . Cyclicly interchanging the roles of , and in the above equality, we obtainfor any vector fields . The addition of (17) with (18) gives an equality; subtracting this equality from (19), with the aid of the symmetry of , we havefor any vector fields .
From (3), we havefor any vector fields . By a direct calculation, taking the covariant derivative of the above equality, with the aid of (3), we havefor any vector fields .
We consider a local orthonormal frame of the tangent space at each point. By a direct calculation, from (22), we havefor any vector field .
If the Reeb vector field of a trans-Sasakian manifold is harmonic-Killing, from (14) and (20), we havefor any vector field , which is simplified by using (23) and (24) yieldingwhere denotes the gradient operator. Obviously, taking the inner product of the above equality with implies . Moreover, if , substituting into (26) gives .

Lemma 2. (see [36]). If on a Riemannian manifold there exists a Killing vector field of constant length satisfyingfor a nonzero constant and any vector fields , then is homothetic to a Sasakian manifold.

Based on the above two lemmas, one of our main results is given.

Theorem 2. If the Reeb vector field of a compact and simply connected trans-Sasakian three-manifold of type is harmonic-Killing, then the manifold is homothetic to a Sasakian 3-manifold.

Proof. Taking into account (15), we havewhere we have used (3). As the manifold is assumed to be compact, applying Stokes’ theorem on the above equality yields . Moreover, now from (21), we observe that is Killing of constant length one. We also claim that is a constant and such an assertion is the same with the proof of Theorem 3.1 in [18]. If the constant , the manifold is cosymplectic. However, this is impossible. In fact, if , with the help of (3), we see thatfor any vector fields . Then, is closed. Since the manifold is assumed to be simply connected, then is exact; i.e., there exists a smooth function on the manifold such that . Consequently, and there exists a point on the manifold on which vanishes, where we have used the compactness of the manifold. However, as seen in Section 2, is always a unit vector field, contradicting the above statement. Thus, we conclude that is a nonzero constant. Finally, by (1), it is easy to check that (27) is valid. In fact, now the manifold is isometric to a three-sphere (see [19]). This completes the proof.
Theorem 2 is an extension of Corollary 3.7.1 in [6].
In Lemma 1, we have obtained a property, i.e., . In fact, such an equality is just one of the requirements when defining a local conformal cosymplectic manifold in the sense of Olszak (for more details, see [37]).

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors have read and approved the final manuscript.

Acknowledgments

This work was supported by the Doctoral Foundation of Henan University of Technology (No. 2018BS061).

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Copyright © 2022 Lei Wang and Yan Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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