# Remarks on Almost Cosymplectic 3-Manifolds with RICCI Operators

Authors: Quanxiang Pan; Yajie Wang
Journal: Journal of Mathematics
 (2020)

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## Abstract

LetM be an almost cosymplectic 3-h-a-manifold. In this paper, we prove that the Ricci operator of M is transversely Killing if and only if M is locally isometric to a product space of an open interval and a surface of constant Gauss curvature, or a unimodular Lie group equipped with a left invariant almost cosymplectic structure. Some corollaries of this result and some examples illustrating main results are given.


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## Body

## 1. Introduction

An almost cosymplectic manifold can be regarded as an odd-dimensional analogy of almost Kähler manifolds from topological points of view (see [1]) and was first introduced by Goldberg and Yano in [2]. In some recent literature, such kind of manifolds was also referred to as almost coKähler manifolds (see [3–9]); for some differences between these two types of manifolds, we refer the reader to a survey [1]. In this paper, we aim to investigate symmetry classification problems of almost cosymplectic manifolds and we adopt those notations and formulas introduced by Blair [10].The curvature properties of cosymplectic manifolds were first studied by Blair who in [11] proved that a cosymplectic manifold of constant sectional curvature is locally flat. Olszak, in [12], extended this result and proved that an almost cosymplectic manifold of nonzero constant sectional curvature does not exist in dimensions ≥5. Later, such a result was generalized to dimension three by Olszak in [13]. Perrone, in [14], extended Olszak’s results and proved that a locally symmetric almost cosymplectic manifold of dimension three is cosymplectic and is locally isometric to a Riemannian product of a one-dimensional manifold and a Kähler surface of constant sectional curvature. Wang, in [5], studied locally symmetric almost cosymplectic manifolds and in particular, in [7], studied locally symmetric almost cosymplectic 5-manifolds. In recent years, many classification results on almost cosymplectic manifolds of dimension three emerged. For example, Cho, in [15], studied Reeb flow symmetry (that is, the Ricci tensor is invariant along the Reeb flow) on almost cosymplectic 3-manifolds. Moreover, semisymmetry, local ϕ-symmetry, curvature, and ball homogeneities on almost cosymplectic 3-manifolds were considered in [3, 4, 8, 9], respectively. Some other symmetry properties in terms of the Ricci operators on such manifolds were also studied in [6, 16]. For many symmetry properties on cosymplectic 3-manifolds, we refer the reader to [17–19].The notion of a Killing tensor fieldT of type 1,1 was introduced by Blair in [20], which is defined by(1)∇XTX=0,where X and ∇ denote an arbitrary vector field and the Levi-Civita connection on a Riemannian manifold M,g, respectively. Relation (1) is much weaker than parallelism of Blair. In [20], Blair applied (1) for the structure tensor field to give a characterization for an almost contact metric manifold to be cosymplectic. In this paper, applying relation (1) for the Ricci operator, we obtain some local classification results of almost cosymplectic 3-manifolds and they are extensions of those in [6, 14].


## 2. Almost Cosymplectic Manifolds

By an almost contact metric manifold, we mean a Riemannian manifoldM,g of dimension 2n+1 on which there exists a quadruple ϕ,ξ,η,g satisfying(2)ϕ2=−id+η⊗ξ,η∘ϕ=0,ηξ=1,gϕX,ϕY=gX,Y−ηXηY,for any vector fields XandY, where ϕ is a 1,1-type tensor field, ξ is a vector field called the Reeb vector field, and η is a global 1-form called the almost contact 1-form and g is a Riemannian metric (see Blair [10]). By an almost cosymplectic manifold we mean an almost contact metric manifold on which there holds dη=0 and dΦ=0, where Φ is the fundamental 2-form defined by ΦX,Y=gX,ϕY (see [10, 12]). We consider the product M2n+1×ℝ of an almost contact metric manifold M2n+1 and ℝ and define on it an almost complex structure J by(3)JX,fddt=ϕX−fξ,ηXddt,where X denotes a vector field tangent to M2n+1, t is the coordinate of ℝ, and f is a C∞-function on M2n+1×ℝ. The almost contact metric manifold is said to be normal if J is integrable, or equivalently,(4)ϕ,ϕ=−2dη⊗ξ,where ϕ,ϕ denotes the Nijenhuis tensor of ϕ. A normal almost cosymplectic manifold is said to be a cosymplectic manifold (cf. [10]). An almost cosymplectic manifold is a cosymplectic manifold if and only if(5)∇ϕ=0.LetM2n+1 be an almost cosymplectic manifold. We consider three tensor fields l=R⋅,ξξ, h=1/2ℒξϕ, and h′=h∘ϕ on M2n+1, where R is the Riemannian curvature tensor of g and ℒ is the Lie differentiation. From [10], we know that the three 1,1-type tensor fields l, h′, and h are symmetric and satisfy(6)hξ=0,lξ=0,trh=0,trh′=0,hϕ+ϕh=0,and(7)∇ξ=h′.


## 3. Main Results

It is known that an almost cosymplectic manifold of dimension three is cosymplectic if and only ifh=0 identically. Thus, we have to discuss our main theorems from two cases and first we consider the non-Kenmotsu case. Let M3 be a three-dimensional almost cosymplectic manifold. Following Perrone [21], let U1 be the open subset of M3 on which h≠0 and U2 the open subset defined by U2=p∈M3:h=0in a  neighborhood  ofp. Therefore, U1∪U2 is an open and dense subset of M3. For any point p∈U1∪U2, we find a local orthonormal basis ξ,e1,e2=ϕe1 of three distinct unit eigenvector fields of h in certain neighborhood of p. On U1, we assume that he1=λe1 and hence he2=−λe2, where λ is assumed to be a positive function. Note that λ is continuous on M3 and smooth on U1∪U2.Lemma 1.
(see [21]). On U1, we have(8)∇ξe1=ae2,∇ξe2=−ae1,∇e1ξ=−λe2,∇e2ξ=−λe1,∇e1e1=12λe2λ+σe1e2,∇e2e2=12λe1λ+σe2e1,∇e2e1=λξ−12λe1λ+σe2e2,∇e1e2=λξ−12λe2λ+σe1e1,where f is a smooth function, σ is the 1-form defined by σ⋅=S⋅,ξ, and S is the Ricci tensor.Applying Lemma1, we obtain the Ricci operator Q associated to the Ricci tensor given by (see [21])(9)Q=αid+βη⊗ξ+ϕ∇ξh−σϕ2⊗ξ+σe1η⊗e1+σe2η⊗e2,where α=1/2r+trh2, β=−1/2r+3trh2, and r denotes the scalar curvature. Now, from (9), we write the Ricci operator as follows:(10)Qξ=−2λ2ξ+σe1e1+σe2e2,Qe1=σe1ξ+12r+2λ2−4λae1+ξλe2,Qe2=σe2ξ+ξλe1+12r+2λ2+4λae2,with respect to the local orthonormal basis ξ,e1,e2. Moreover, according to Lemma 1, by a simple calculation, we obtain(11)∇ξh=1/λξλh−2ah′.In view of equation (11), Wang, in [6, 9], considered a special almost coysmplectic 3-manifold determined by the condition ∇ξh=−2ah′ with a∈ℝ. In this paper, we say that an almost cosymplectic 3-manifold satisfying the previous condition is an almost cosymplectic 3-h-a-manifold. In particular, when a vanishes, an almost cosymplectic 3-h-a-manifold reduces to an almost cosymplectic 3-h-manifold (see [4]). Notice that on a cosymplectic manifold such a condition is meaningless because of h=0. There are many nontrivial examples of almost cosymplectic 3-manifolds satisfying such a condition even when a is a smooth function (see [4, 6]).As seen in Section1, the Ricci operator of an almost cosymplectic manifold is said to be transversely Killing if there holds(12)∇XQX=0,for any vector field X orthogonal to the Reeb vector field ξ. Obviously, (12) is much weaker than Ricci parallelism (∇Q=0) and local symmetry (∇R=0).Now, we are ready to give our first main result.Theorem 1.
The Ricci operator of a noncosymplectic almost cosymplectic 3-h-a-manifold is transversely Killing if and only if the manifold is locally isometric to the group E2 of rigid motions of Euclidean 2-plane, the group E1,1 of rigid motions of Minkowski 2-plane, or the Heisenberg group Nil3 equipped with a left invariant almost cosymplectic structure.Proof.
On an almost cosymplectic 3-h-a-manifold, from the definition, we have ∇ξh=−2ah′, and thus from (11), we may use ξλ=0 in (10) to obtain(13)Qξ=−2λ2ξ+σe1e1+σe2e2,Qe1=σe1ξ+12r+2λ2−4λae1,Qe2=σe2ξ+12r+2λ2+4λae2.According to (13) and Lemma 1, by a standard calculation, we obtain(14)∇e1Qe1=e1σe1−12λσe2e2λ+σe1ξ+12e1r+2λ2−4aλe1−λσe1+2ae2λ+σe1e2,(15)∇e2Qe2=e2σe2−12λσe1e1λ+σe2ξ−λσe2−2ae1λ+σe2e1+12e2r+2λ2+4aλe2.Because the Ricci operator is transversely Killing, from (12), we have ∇e1Qe1=0 and ∇e2Qe2=0, which are compared with (14) and (15), respectively, implying(16)e1σe1−12λσe2e2λ+σe1=0,e1r+2λ2−4aλ=0,λσe1+2ae2λ+σe1=0,and(17)e2σe2−12λσe1e1λ+σe2=0,λσe2−2ae1λ+σe2=0,e2r+2λ2+4aλ=0,respectively. On the contrary, applying Lemma 1, we also have(18)ξ,e1=a+λe2,e2,ξ=a−λe1,e1,e2=12λe1λ+σe2e2−12λe2λ+σe1e1.Putting (18) into the well-known Jacobi identity, we obtain(19)e1λ−a+ξe2λ+σe12λ+a−λ2λe1λ+σe2=0,e2λ+a+ξe1λ+σe22λ−a+λ2λe2λ+σe1=0.Recall that, in this situation,a∈ℝ and ξλ=0 and the abovementioned relations reduce to(20)e1λ+12λξe2λ+σe1+a−λ2λe1λ+σe2=0,e2λ+12λξe1λ+σe2−a+λ2λe2λ+σe1=0.Taking into account the last term of (16) (or the second term of (17)), we have to divide our discussions into the following two cases.Case 1.
a
=
0. Because λ is assumed to be a positive function, using a=0 in the last term of (16) and the second term of (17), we obtain(21)σe1=σe2=0.
Asξλ=0 and a=0, from the first two terms of (18), we obtain ξe1λ=λe2λ and ξe2λ=λe1λ. Putting these two equations into (20), with the aid of (21) and a=0, we obtain(22)e1λ=e2λ=0,where we have used that λ is a positive function. In view of ξλ=0, together with the abovementioned relations, we see that λ is a positive constant. Now (18) reduces to(23)ξ,e1=λe2,e2,ξ=−λe1,e1,e2=0.
According to Milnor [22] and the abovementioned relations, it is easily seen that the manifold is locally isometric to the group E1,1 of rigid motions of the Minkowski 2-space equipped with a left invariant almost cosymplectic structure.Case 2.
a
≠
0. In this case, replacing X by e1+e2 in (12), with the aid of ∇e1Qe1=0 and ∇e1Qe1=0, we obtain(24)∇e1Qe2+∇e2Qe1=0.
On the contrary, with the aid ofξλ=0, applying (13) and Lemma 1, we obtain(25)∇e1Qe2=e1σe2+12λr+4aλ+6λ2+12λσe1e2λ+σe1ξ−λσe1+2ae2λ+σe1e1−2λσe2−12e1r+2λ2+4aλe2,∇e2Qe1=e2σe1+12λr−4aλ+6λ2+12λσe2e1λ+σe2ξ−2λσe1−12e2r+2λ2−4aλe1−λσe2−2ae1λ+σe2e2,which are compared with (24), implying that(26)e1σe2+e2σe1+λr+6λ2+12λσe2e1λ+σe2+12λσe1e2λ+σe1=0,(27)3λσe1+2ae2λ+σe1−12e2r+2λ2−4aλ=0,(28)3λσe2−2ae1λ+σe2−12e1r+2λ2+4aλ=0.
Putting the last term of (17) and the second term of (16) into (27) and (28), respectively, we obtain(29)3λ+2aσe1+6ae2λ=0,and(30)3λ−2aσe2−6ae1λ=0,respectively. The subtraction of (29) from the third term of (16) multiplied by 3 implies σe1=0, where we have used the assumption a≠0. Using this relation back in (29), we obtain(31)e2λ=0.
Similarly, in view ofa≠0, the subtraction of (30) from the second term of (17) multiplied by 3 gives σe2=0, which is applied back in (30), implying(32)e1λ=0.
In view ofξλ=0, we see that λ is a positive constant. Moreover, from (26), we see that the scalar curvature of the manifold is a negative constant. In this case, from (18), we also have(33)ξ,e1=a+λe2,e2,ξ=a−λe1,e1,e2=0.
Finally, according to Milnor [22] and (33), we observe that the manifold is locally isometric to a unimodular Lie group G endowed with a left invariant noncosymplectic almost cosymplectic structure because(34)traceei⟶ei,ej=0,for anye1,e2,e3=ξ.
More specifically,G is the group E2 of rigid motions of Euclidean 2-plane if a>λ or a<−λ; the group E1,1 of rigid motions of Minkowski 2-plane if −λ<a<λ; or the Heisenberg group Nil3 if a=λ or a=−λ. We refer the reader to [14] for the constructions of almost cosymplectic structures on the abovementioned unimodular Lie groups. The converse is easy to check. This completes the proof.
From proof of Theorem1, we haveCorollary 1.
The Ricci operator of a noncosymplectic almost cosymplectic 3-h-manifold is transversely Killing if and only if the manifold is locally isometric to the group E1,1 of rigid motions of Minkowski 2-plane equipped with a left invariant almost cosymplectic structure.In Theorem1, the condition “the Ricci operator is transversely Killing” is essential because we haveExample 1.
LetG be a three-dimensional nonunimodular Lie group equipped with a left invariant metric g whose Lie algebra is given by(35)e1,e2=αe2,e2,e3=0,e1,e3=βe2,where e1,e2,e3 is an orthonomal basis with respect to g and α,β∈ℝ∗. On G, there exists an almost cosymplectic structure (see [14]). According to [4], we have(36)∇e1Qe1=0,∇e2Qe2=12αβ2e1.
It is easily seen that the left-invariant Ricci operator ofG is not transversely Killing even if ∇ξh=βh.
Next, we give the cosymplectic version of Theorem1 as follows.Theorem 2.
The Ricci operator of a cosymplectic 3-manifold is transversely Killing if and only if the manifold is locally isometric to a Riemannian product of an open interval and a surface of constant Gauss curvature.Proof.
On a cosymplectic 3-manifold, applyingh=0 on (7), we see that ξ is parallel, and hence we get RX,Yξ=0 and(37)Qξ=0.On a Riemannian 3-manifold, the curvature tensorR is given by(38)RX,YZ=gY,ZQX−gX,ZQY+gQY,ZX−gQX,ZY−r2gY,ZX−gX,ZY,for any vector fields X,Y,andZ, where r is the scalar curvature. Replacing Z by ξ in (38), with the aid of (37) and RX,Yξ=0, we obtain(39)QX=r2X−r2ηXξ,for any vector field X. According to (7) and (37), a simple calculation shows ∇ξQξ=0. If the Ricci operator is transversely Killing, from (12), we obtain ∇eQe=0 and ∇ϕeQϕe=0, where e,ϕe is an arbitrary local basis of the contact distribution ξ⊥. Thus, applying the previous three relations in the well-known formula divQ=1/2gradr, we obtain that the scalar curvature r is a constant. Therefore, taking the derivative of (39), we obtain(40)∇Q=0.That is, the manifold is locally symmetric, and hence following [14], we see that the manifold is locally isometric to a product space of an open interval and a surface of constant Gauss curvature. The converse is easy to check.An almost cosymplectic 3-manifold is said to be an almost cosymplectic 3-h-manifold if the Reeb vector field ξ is principal direction of the Ricci operator, or equivalently, ξ is harmonic (see [4]).Lemma 2.
If on an almost cosymplectic 3-h-manifold M, the Ricci operator is transversely Killing, then we have ξλ=0 and da∧η=0.Proof.
Ifξ is harmonic, we have Qξ=Sξ,ξξ, which is applied in Lemma 1 implying σe1=0 and σe2=0. Using this, now (10) becomes(41)Qξ=−2λ2ξ,Qe1=12r+2λ2−4λae1+ξλe2,Qe2=ξλe1+12r+2λ2+4λae2.Taking into account Lemma1 and (41), we obtain(42)∇e1Qe1=λξλξ+e1ξλ−2ae2λe2+12e1r+2λ2−4λa−1λξλe2λe1,and(43)∇e2Qe2=λξλξ+e2ξλ+2ae1λe1+12e2r+2λ2+4λa−1λξλe1λe2.If the Ricci operator is transversely Killing, from (12), we have ∇e1Qe1=0 and ∇e2Qe2=0, which are compared with (42) and (43), implying(44)ξλ=0,ae2λ=0,e1r+2λ2−4aλ=0,and(45)ξλ=0,ae1λ=0,e2r+2λ2+4aλ=0,respectively, where we have applied that λ is a positive function. If a vanishes, the proof is completed already. Next we consider the case that a≠0 on some open subset. In this case, from (44) and (45), we know that λ is a positive constant. Moreover, from Lemma 1 and (41), we obtain(46)∇ξQξ=0.Applying (46), ∇e1Qe1=0, and ∇e2Qe2=0 in the formula divQ=1/2gradr, we see that r is also a constant. Thus, according to (44) and (45), we obtain(47)e1a=e2a=0.It follows directly thatda=ξaη and hence by the definition of almost cosymplectic structure we get da∧η=0. This completes the proof.Theorem 3.
Let M be a noncosymplectic almost cosymplectic 3-h-manifold such that a is an invariant along Reeb flow. Then, the Ricci operator is transversely Killing if and only if the manifold is locally isometric to the group E2 of rigid motions of Euclidean 2-plane, the group E1,1 of rigid motions of Minkowski 2-plane, or the Heisenberg group Nil3 equipped with a left invariant almost cosymplectic structure.Proof.
On a noncosymplectic almost cosymplectic 3-h-manifold M, if a is an invariant along the Reeb flow, from Lemma 2, we see that a is a constant. Thus, together with ξλ=0, we see that M is in fact an almost cosymplectic 3-h-a-manifold. Finally, the proof follows immediately from Theorem 1.Note that the condition “a is invariant along the Reeb flow and (12) is true” is essential. We show an almost cosymplectic 3-h-manifold on which previous condition does not hold (see [4, 23]).Example 2.
LetM3 be an open subset of ℝ3 defined by M3:=x,y,z∈ℝ3:z>0. On M3 there is an almost cosymplectic structure defined by(48)ξ=∂∂z,η=dz,ϕ∂∂x=z2∂∂y,ϕ∂∂y=−1z2∂∂x,ϕ∂∂z=0,g=z2dx⊗dx+1z2dy⊗dy+dz⊗dz.
An orthonormal basis for the tangent space at each point ofM is given by ξ,e1=1/z∂/∂x,e2=z∂/∂y. According to [4], we have(49)Qξ=−2z2ξ,∇e1Qe1=−3z2ξ,∇e1Qe2=∇e2Qe1=0,∇e2Qe2=1z2ξ.
From this, we see that the Reeb vector field is harmonic but the Ricci operator is not transversely Killing.



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