# Some New Results on Trans-Sasakian Manifolds

Authors: Lei Wang; Yan Zhao
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 manifoldM2n+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(1)∇XϕY=αgX,Yξ−ηYX+βgϕX,Yξ−ηYϕX,for all vector fields XandY, where both α and β are smooth functions. The classical Sasakian, Kenmotsu, and cosymplectic manifolds (see [1]) are all its trivial cases.In general, a trans-Sasakian manifold of typeα,β is said to be proper (see [4–6]) 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. [9–12], 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-spaceℝ4 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

LetM2n+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], M2n+1 is called an almost-contact metric manifold if(2)ϕ2=−Id+η⊗ξ,ηξ=1,η∘ϕ=0,gϕX,ϕY=gX,Y−ηXηY,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=−2dη⊗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. This is not necessarily true for higher dimension.A normal almost-contact metric manifold is said to be anα-Sasakian manifold if dη=αΦ and dΦ=0, where α is a nonzero constant. An α-Sasakian manifold reduces to a Sasakian manifold when α=1. A normal almost-contact metric manifold is called a β-Kenmotsu manifold if it satisfies dη=0 and dΦ=2βη∧Φ, where β is a nonzero constant. A β-Kenmotsu manifold becomes a Kenmotsu manifold when β=1. A normal almost-contact metric manifold is said to be a cosymplectic manifold if it satisfies dη=0 and dΦ=0. Obviously, the set of all α-Sasakian manifolds (resp., β-Kenmotsu) is a proper subset of that of all trans-Sasakian manifolds of type α,0 (resp., 0,β).PuttingY=ξ into (1) and using (2), we have(3)∇Xξ=−αϕX+βX−ηXξ,for any vector field X. In this paper, all manifolds are assumed to be connected.


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

LetMnc be a complete and simply connected complex space form which is complex analytically isometric to the following:(i)
A complex projective spaceℂPnc if c>0(ii)
A complex Euclidean spaceℂn if c=0(iii)
A complex hyperbolic spaceℂHnc if c<0Here,c is the constant holomorphic sectional curvature.LetM be a real hypersurface immersed in a complex space form Mnc and N be a unit normal vector field of M. We denote by ∇¯ the Levi-Civita connection of the metric g¯ of Mnc 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. Then, the Gauss and Weingarten formulas are given, respectively, as follows:(4)∇¯XY=∇XY+gAX,YN,∇¯XN=−AX,for any vector fields XandY, where A denotes the shape operator of M in Mnc. For any vector field X, we put(5)JX=ϕX+ηXN,JN=−ξ.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:=ηAξ, 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 Mnc and using (4) and (5), we have(6)∇XϕY=ηYAX−gAX,Yξ,for any vector fields XandYX,Y. Let R be the Riemannian curvature tensor of M. As Mnc is of constant holomorphic sectional curvature c, the Gauss equation of M in Mnc is given by(7)RX,YZ=c4gY,ZX−gX,ZY+gϕY,ZϕX−gϕX,ZϕY−2gϕX,YϕZ+gAY,ZAX−gAX,ZAY,for any vector fields XandYX,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 [25–27]).Theorem 1.
LetM2n−1 be a trans-Sasakian manifold. Then, M2n−1 is realized as a real hypersurface in a complex space form Mnc, n≥2, if and only if the following statements are valid:(1)
IfMnc=ℂPnc, M2n−1 is locally congruent to a geodesic hypersphere.(2)
IfMnc=ℂHnc, M2n−1 is locally congruent to(i)
a horosphere(ii)
a geodesic hypersphere(iii)
a tube around a totally geodesicℂHn−1(3)
IfMnc=ℂn, M2n−1 is locally congruent to(i)
a hyperplaneℝ2n−1(ii)
a sphereS2n−1(iii)
a cylinder over a plane curveγ×ℝ2n−2Proof.
If a real hypersurfaceM2n−1 in a complex space form Mnc is trans-Sasakian, by definition, from (1) and (6), we get(8)αgX,Yξ−ηYX+βgϕX,Yξ−ηYϕX=ηYAX−gAX,Yξ,for any vector fields XandYX,Y. In the above equality, setting Y=ξ gives(9)AX=ηAXξ−βϕX+αηXξ−X,for any vector field X. Obviously, it follows that Aξ=ηAξξ and hence, M2n−1 is Hopf. Using this in the previous equality, we get(10)AX=α+ηAξηXξ−βϕX−αX,for any vector field X. Recall that the shape operator is self-adjoint; it follows directly that β=0, and hence,(11)AX=α+ηAξηXξ−αX,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 isℂPnc or ℂHnc, following [28, 29], we observe that a totally η-umbilical real hypersurface satisfying (11) is locally congruent to the following:(i)
A geodesic sphere of radiusr in ℂPn with α=−cr/2cotcr/2, where 0<r<π/c(ii)
A horosphere inℂHn with α=−−c/2(iii)
A geodesic sphere of radiusr in ℂHn with α=−−c/2coth−c/2, where 0<r<∞(iv)
A tube of radiusr around a totally geodesic complex hyperplane ℂHn−1c in ℂHnc with α=−−c/2tanh−c/2, where 0<r<∞
When the ambient space isℂn, from Gauss equations (7) and (11), we see that the hypersurface is pseudo-Einstein, i.e.,(12)QX=α2n−3α−ηAξX−2n−3αα+ηAξηXξ,for any vector field X, where Q 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 Theorem1, a new characterization for the property of trans-Sasakian 3-manifolds is given. We remark that α in two cases in the proof of Theorem 1 is both constant.Corollary 1.
A trans-Sasakian 3-manifold is anα-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 typeα,β realized as a hypersurface in ℝ4 is isometric to the Sasakian manifold S3α2 provided that the hypersurface is compact. Such a situation occurs in our Theorem 1 in view of (12) and (11) for n=2 (for more details, see ([14], Theorem 2)).


## 4. Harmonic-Killing Reeb Vector Field

From [31], a vector field V on a Riemannian manifold M,g is called affine Killing if(13)ℒV∇=0,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 Theorem 2.1 in [33], we have(14)trℒV∇=0.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 manifoldsM2n+1 of type α,β is harmonic-Killing, then we have(15)ξβ=−2β2.Moreover, ifn>1, we have dβ=−2β2η.Proof.
Recall that on any differentiable manifold, there holds (see Yano ([35], pp. 23))(16)Lξ∇Xg−∇XLξg−∇ξ,XgY,Z=−gLξ∇X,Y,Z−gLξ∇X,Z,Y,for any vector fields X,Y,andZ. Notice that in our case, the Riemannian metric g is parallel and it follows that(17)∇XLξgY,Z=gLξ∇X,Y,Z+gLξ∇X,Z,Y,for any vector fields X,Y,andZX,Y,Z. Cyclicly interchanging the roles of X,Y, and Z in the above equality, we obtain(18)∇YLξgZ,X=gLξ∇Y,Z,X+gLξ∇Y,X,Z,(19)∇ZLξgX,Y=gLξ∇Z,X,Y+gLξ∇Z,Y,X,for any vector fields X,Y,andZX,Y,Z. The addition of (17) with (18) gives an equality; subtracting this equality from (19), with the aid of the symmetry of Lξ∇, we have(20)2gLξ∇X,Y,Z=∇XLξgY,Z+∇YLξgZ,X−∇ZLξgX,Y,for any vector fields X,Y,andZX,Y,Z.
From (3), we have(21)ℒξgY,Z=2βgY,Z−ηYηZ,for any vector fields X,Y,andZX,Y,Z. By a direct calculation, taking the covariant derivative of the above equality, with the aid of (3), we have(22)∇XLξgY,Z=2XβgY,Z−ηYηZ+2αβgϕX,YηZ+2αβgϕX,ZηY−2β2gX,YηZ−2β2gX,ZηY+4β2ηXηYηZ,for any vector fields X,Y,andZX,Y,Z.
We consider a local orthonormal framee1,…,e2n+1 of the tangent space at each point. By a direct calculation, from (22), we have(23)∑i=12n+1∇eiLξgei,Z=2Zβ−2ξβηZ−4nβ2ηZ,(24)∑i=12n+1∇ZLξgei,ei=4nZβ,for any vector field Z.
If the Reeb vector field of a trans-Sasakian manifold is harmonic-Killing, from (14) and (20), we have(25)∑i=12n+1∇eiLξgei,Z+∑i=12n+1∇eiLξgZ,ei=∑i=12n+1∇ZLξgei,ei,for any vector field Z, which is simplified by using (23) and (24) yielding(26)n−1Dβ=−ξβ+2nβ2ξ,where D denotes the gradient operator. Obviously, taking the inner product of the above equality with ξ implies ξβ=−2β2. Moreover, if n>1, substituting ξβ=−2β2 into (26) gives Dβ=−2β2ξ.Lemma 2. (see [36]).
If on a Riemannian manifoldM there exists a Killing vector field ξ of constant length satisfying(27)k2∇X∇Yξ−∇∇XYξ=gY,ξX−gX,Yξ,for a nonzero constant k and any vector fields XandY, 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 typeα,β is harmonic-Killing, then the manifold is homothetic to a Sasakian 3-manifold.Proof.
Taking into account (15), we have(28)divβ3ξ=3β2ξβ+β3divξ=−4β4,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 Theorem 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(29)dηX,Y=g∇Xξ,Y−g∇Yξ,X=0,for any vector fields XandY. Then, 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. Thus, 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 S3α2 (see [19]). This completes the proof.
Theorem2 is an extension of Corollary 3.7.1 in [6].
In Lemma1, 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]).



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