Some New Results on Trans-Sasakian Manifolds

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 M2n+1 of dimension 2n + 1, n≥ 1, together with its almost-contact metric structure (φ, ξ, η, g), is said to be a trans-Sasakian manifold (it is, often referred to, of type (α, β)) (see [1–3]) if it satisfies


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 M 2n+1 of dimension 2n + 1, n ≥ 1, together with its almost-contact metric structure (ϕ, ξ, η, g), is said to be a trans-Sasakian manifold (it is, often referred to, of type (α, β)) (see [1][2][3]) if it satisfies for all vector fields X and Y, where both α and β are smooth functions. e classical Sasakian, Kenmotsu, and cosymplectic manifolds (see [1]) are all its trivial cases.
In general, a trans-Sasakian manifold of type (α, β) is said to be proper (see [4][5][6]) when either α or β 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][10][11][12], Deshmukh et al. [8,[13][14][15][16][17][18][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 threemanifolds isometrically immersed in the Euclidean fourspace R 4 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).

Trans-Sasakian Manifolds
Let (M 2n+1 , g) be a smooth Riemannian manifold of dimension 2n + 1 on which there exist a (1, 1)-type, (1, 0)-type, and (0, 1)-type tensor fields ϕ, ξ, and η, respectively. According to Blair [1], M 2n+1 is called an almostcontact metric manifold if for any vector fields X and Y. ξ is said to be the Reeb or structure vector field. An almost-contact metric manifold is said to be normal if [ϕ, ϕ] � − 2 dη ⊗ ξ, where [ϕ, ϕ] denotes the Nijenhuis tensor of ϕ. 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. is is not necessarily true for higher dimension.
Putting Y � ξ into (1) and using (2), we have for any vector field X. In this paper, all manifolds are assumed to be connected.

Trans-Sasakian Manifolds as Real Hypersurfaces in Complex Space Forms
Let M n (c) be a complete and simply connected complex space form which is complex analytically isometric to the following: Here, c is the constant holomorphic sectional curvature. Let M be a real hypersurface immersed in a complex space form M n (c) and N be a unit normal vector field of M. We denote by ∇ the Levi-Civita connection of the metric g of M n (c) and J the complex structure. Let g and ∇ be the induced metric from the ambient space and the Levi-Civita connection of the metric g, respectively. en, the Gauss and Weingarten formulas are given, respectively, as follows: for any vector fields X and Y, where A denotes the shape operator of M in M n (c). For any vector field X, 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 ξ is principal, that is, Aξ � δξ at each point, where δ: � η(Aξ), then M is called a Hopf hypersurface and δ is called Hopf principal curvature.
Moreover, applying the parallelism of the complex structure (i.e., ∇ J � 0) of M n (c) and using (4) and (5), we have for any vector fields X and YX, Y. Let R be the Riemannian curvature tensor of M. As M n (c) is of constant holomorphic sectional curvature c, the Gauss equation of M in M n (c) is given by for any vector fields X and YX, Y.
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][26][27]).  (1) and (6), we get for any vector fields X and YX, Y. In the above equality, setting Y � ξ gives for any vector field X. Obviously, it follows that Aξ � η(Aξ)ξ and hence, M 2n− 1 is Hopf. Using this in the previous equality, we get for any vector field X. Recall that the shape operator is selfadjoint; it follows directly that β � 0, and hence, for any vector field X. Now, the hypersurface is totally η-umbilical. Conversely, the application of the above equality in (6) implies that the hypersurface is always a trans-Sasakian manifold of type (α, 0). Next, we divide our discussions into two cases. When the ambient space is CP n (c) or CH n (c), following [28,29], we observe that a totally η-umbilical real hypersurface satisfying (11) is locally congruent to the following: When the ambient space is C n , from Gauss equations (7) and (11), we see that the hypersurface is pseudo-Einstein, i.e., for any vector field X, where Q denotes the Ricci operator. e remaining proof follows immediately from Proof of eorem 1 in [27] (see also [30]). e converse is easy to check. In view of eorem 1, a new characterization for the property of trans-Sasakian 3-manifolds is given. We remark that α in two cases in the proof of eorem 1 is both constant. □

Corollary 1. A trans-Sasakian 3-manifold is an α-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 (α, β) realized as a hypersurface in R 4 is isometric to the Sasakian manifold S 3 (α 2 ) provided that the hypersurface is compact. Such a situation occurs in our eorem 1 in view of (12) and (11) for n � 2 (for more details, see ( [14], eorem 2)).

Harmonic-Killing Reeb Vector Field
From [31], a vector field V on a Riemannian manifold (M, g) is called affine Killing if where ∇ denotes the Levi-Civita connection of the metric g (see also [32]). According to [33,34], a vector field V on a Riemannian manifold is called harmonic-Killing if each local parameter group of infinitesimal transformations associated to V is a group of harmonic maps. For any harmonic-Killing vector field V, from eorem 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.
Proof. Recall that on any differentiable manifold, there holds (see Yano ([35], pp. 23)) for any vector fields X, Y, and Z. Notice that in our case, the Riemannian metric g is parallel and it follows that for any vector fields X, Y, and ZX, Y, Z. Cyclicly interchanging the roles of X, Y, and Z in the above equality, we obtain for any vector fields X, Y, and ZX, Y, Z. e addition of (17) with (18) gives an equality; subtracting this equality from (19), with the aid of the symmetry of L ξ ∇, we have Journal of Mathematics 3 for any vector fields X, Y, and ZX, Y, Z. From (3), we have for any vector fields X, Y, and ZX, Y, Z. By a direct calculation, taking the covariant derivative of the above equality, with the aid of (3), we have for any vector fields X, Y, and ZX, Y, Z.
We consider a local orthonormal frame e 1 , . . . , e 2n+1 of the tangent space at each point. By a direct calculation, from for any vector field Z.
□ Lemma 2 (see [36]). If on a Riemannian manifold M there exists a Killing vector field ξ of constant length satisfying for a nonzero constant k and any vector fields X and Y, then M 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 (α, β) is harmonic-Killing, then the manifold is homothetic to a Sasakian 3-manifold.
Proof. Taking into account (15), we have where we have used (3). As the manifold is assumed to be compact, applying Stokes' theorem on the above equality yields β � 0. Moreover, now from (21), we observe that ξ is Killing of constant length one. We also claim that α is a constant and such an assertion is the same with the proof of eorem 3.1 in [18]. If the constant α � 0, the manifold is cosymplectic. However, this is impossible. In fact, if α � 0, with the help of (3), we see that for any vector fields X and Y. en, η is closed. Since the manifold is assumed to be simply connected, then η is exact; i.e., there exists a smooth function f on the manifold such that η � df. Consequently, ξ � Df and there exists a point on the manifold on which Df vanishes, where we have used the compactness of the manifold. However, as seen in Section 2, ξ is always a unit vector field, contradicting the above statement. us, we conclude that α 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 S 3 (α 2 ) (see [19]). is completes the proof. eorem 2 is an extension of Corollary 3.7.1 in [6]. In Lemma 1, we have obtained a property, i.e., dβ ∧η � 0. 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
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.