# Some Properties of Curvature Tensors and Foliations of Locally Conformal Almost Kähler Manifolds

Authors: Ntokozo Sibonelo Khuzwayo; Fortuné Massamba
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

We investigate a class of locally conformal almost Kähler structures and prove that, under some conditions, this class is a subclass of almost Kähler structures. We show that a locally conformal almost Kähler manifold admits a canonical foliation whose leaves are hypersurfaces with the mean curvature vector field proportional to the Lee vector field. The geodesibility of the leaves is also characterized, and their minimality coincides with the incompressibility of the Lee vector field along the leaves.


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## Body

## 1. Introduction

The study of manifolds whose metric is locally conformal to an almost Kähler metric is considered as one of the most interesting studies in the field of differential geometry (see [1] for details and references therein). This is because of its richness in the theory that is applicable in physics, algebraic geometry, symplectic geometry, etc. To our knowledge, locally conformal (almost) Kähler structures were first studied by Libermann [2] in the 1950s. In 1966, Gray [3] also contributed to the study by considering (almost) Hermitian manifolds whose metric is conformal to a local (almost) Kähler metric. However, globally conformal (almost) Kähler manifolds share the same topological properties with locally conformal (almost) Kähler manifolds [4]. It is therefore provocative to consider those almost Hermitian manifolds whose metric is locally conformal to an almost Kähler metric. The difference between locally conformal Kähler manifolds and locally conformal almost Kähler manifolds is the condition of integrability of an almost complex structure. This is equivalent to an almost complex structure being parallel with respect to a globally defined connection or the vanishing of a Nijenhuis tensor. Therefore, the geometric properties which do not depend on the almost complex structure will apply to both of these manifolds.Libermann defined a locally conformal (almost) Kähler metric as a metricg at which in the neighborhood of each point of an almost Hermitian manifold, it is locally conformal to an (almost) Kähler metric.In this paper, we investigate some properties of curvature tensors and foliations of locally conformal almost Kähler manifolds. For the foliations, we pay attention to the ones that arise naturally when the Lee form is nowhere vanishing. The paper is organized as follows. In Section2, we recall the definition of locally conformal almost Kähler structures supported by an example. In Section 3, we deal with curvature tensors. The latter generalizes those found by Olszak in [5]. Under some conditions, we prove that a class of locally conformal almost Kähler structures is a subclass of almost Kähler structures. Section 4 is devoted to the canonical foliations that arise for the nonvanishing Lee form. We prove that these foliations are Riemannian if and only if the Lee vector field is autoparallel. We also prove that the locally conformal almost Kähler manifolds contain leaves with mean curvature vector field proportional to the Lee vector field, and their geodesibility coincides with the killing condition of the Lee vector field. The latter is incompressible along the leaves if and only if the leaves are minimal.


## 2. Locally Conformal Almost Kähler Metrics

LetM be a 2n-dimensional almost Hermitian manifold with the metric g and the almost complex structure J satisfying(1)J2=−I,gJX,JY=gX,Y,for any vector fields X and Y tangent to M, where I stands for the identity transformation of tangent bundle TM. Then, for any vector fields X and Y, the tensor(2)ΩX,Y=gX,JYdefines the fundamental 2-form of M which is nondegenerate and gives an almost symplectic structure on M. If Ω is closed, i.e., dΩ=0, then M,J,g is called an almost Kähler manifold [6].Now, letM,J,g be a 2n-dimensional almost Hermitian manifold. Such a manifold is said to be a locally conformal almost Kähler manifold [4] if there is an open covering Utt∈I of M and a family ftt∈I of C∞-functions ft:Ut⟶ℝ such that, for any t∈I, the metric form(3)gt=exp−ftg|Utis an almost Kähler metric.If the structuresJ,gt defined in (3) are Kähler, then M,J,g is called locally conformal Kähler. Moreover, a locally conformal almost Kähler manifold M is almost Kähler if and only if dft=0.The Lee form is important because it characterizes locally conformal almost Kähler manifolds. Locally conformal almost Kähler manifolds were characterized by Vaisman in [4]. This is stated as follows: an almost Hermitian manifold M,J,g is a locally conformal almost Kähler manifold if and only if there exists 1-form ω such that(4)dΩ=ω∧Ω,dω=0.Example 1.
We consider the 4-dimensional manifoldM4=p∈ℝ4|x1≠0,x2>0, where p=x1,x2,y1,y2 are the standard coordinates in ℝ4. The vector fields,(5)Xi=x2∂∂xi,Yi=1x23∂∂yi,i=1,2,are linearly independent at each point of M. Let g be the Riemannian metric on M defined by gXi,Xj=gYi,Yj=δij, where δij is the Kronecker symbol, gXi,Yj=0. That is, the form of the metric becomes(6)g=1x22dx12+dx22+x26dy12+dy22.
LetJ be the 1,1-tensor field defined by JX1=Y1, JX2=−Y2, JY2=X2, and JY1=−X1. Thus, J,g defines an almost Hermitian structure on M4. The nonzero component of fundamental 2-form J is(7)Ω∂∂x1,∂∂y1=−1x22andΩ∂∂x2,∂∂y2=1x22,and we have(8)Ω=1x22−dx1∧dy1+dx2∧dy2.
Its differential gives(9)dΩ=2x23dx1∧dy1∧dx2.
By letting(10)ω=−1x2dx2,we have(11)dΩ=2ω∧Ω.
It is easy to see thatdω=0, and the dual vector field B is given by(12)B=−1x22X2.
Let us consider the open neighborhoodU of M given by U=p∈M4|x2>0, and there exists a differentiable function f on U such that ω=df, where f=−lnx2. By the aforementioned characterization given in (4), M4,J,g is a locally conformal almost Kähler manifold.
Next, we wish to study the relationship of the Levi-Civita connections induced by the locally conformal Kähler metricsgt and g.
Throughout this paper,ΓΞ will denote the ℱM-module of differentiable sections of a vector bundle Ξ.
Let∇ and ∇t be the Levi-Civita connections associated with the metrics g and gt, respectively. As is well known, they are connected by(13)∇XtY=∇XY−12ωXY+ωYX−gX,YB,for all X,Y∈ΓTM.


## 3. Curvature Relations of Locally Conformal Almost Kähler Metrics

LetM,J,g be a 2n-dimensional almost Hermitian manifold. Here, we keep the formalism of local transformations and other formulas defined in the previous section. For the Riemann curvature R of a metric g, we use the following convention:(14)RX,Y,Z,W=gRX,YZ,W,where(15)RX,YZ=∇X∇YZ−∇Y∇XZ−∇X,YZ,for any vector field X, Y, and Z on M.LetEi1≤i≤2n be the orthonormal basis with respect to g. The Ricci curvature tensor ρ and the scalar curvature τ are given by(16)ρX,Y=∑i=12nREi,X,Y,Ei,τ=∑i=12nρEi,Ei.Now, we consider the Ricci∗-curvature tensor ρ∗ and the scalar ∗-curvature τ∗ defined by(17)ρ∗X,Y=∑i=12nREi,X,JY,JEi,τ∗=∑i=12nρ∗Ei,Ei.Similarly, the curvatures corresponding to the metricgt will be denoted by Rt, ρt, τt, ρt∗, and τt∗, respectively.Lemma 1.
LetM,J,g be a locally conformal almost Kähler manifold. Then, the curvature tensors Rt and R with respect to the metrics gt and g, respectively, are related as(18)RtX,YZ=RX,YZ+12∇YωZ+12ωYωZX−12∇XωZ+12ωXωZY+12gY,Z∇XB+12ωXB−12gX,Z∇YB+12ωYB−B24gY,ZX−gX,ZY,where B2=gB,B.Proof.
Using the convention in (15) for the curvature tensors Rt and R and relation (13) and for any vector fields X, Y, and Z tangent to M, the expressions(19)∇Xt∇YtZ=∇X∇YZ−12ωX∇YZ−12ω∇YZ+12gX,∇YZB−12XωYZ−12ωY∇XZ+14ωXωYZ+14ωYωZX−14ωYgX,ZB−12XωZY−12ωZ∇XY+14ωXωZY+14ωYωZX−14ωZgX,YB+12XgY,ZB+12gY,Z∇XB−14B2gY,ZX.
It is worth noting that(20)∇X,YtZ=∇X,YZ−12ωX,YZ−12ωZX,Y+12gX,Y,ZB.
Putting pieces (19) and (20) together, one obtains relation (18). This completes the proof.
Next, from the above lemma, we define0,2-tensor field P by(21)PX,Y=∇XωY+12ωXωY−14B2gX,Y,and this trace is given by(22)traceP=divB−121−nB2.Lemma 2.
The0,2-tensor field P is symmetric.Proof.
For any vector fieldsX and Y tangent to M and since ω is closed, we have(23)PX,Y=∇YωX+12ωXωY−14B2gX,Y=YωX−ω∇YX+12ωXωY−14B2gX,Y=YωX−ωY,X−ω∇XY+12ωXωY−14B2gX,Y=∇XωY+12ωXωY−14B2gX,Y,which completes the proof.
The Lie derivativeg with respect to the vector field B gives, for any vector fields X and Y,(24)LBgX,Y=XgB,Y−gB,X,Y−gX,B,Y=∇XωY+∇YωX=2∇XωY.
The last equality of (24) follows from the fact that smooth 1-form ω is closed.Lemma 3.
The dual vector fieldB of ω preserves the metric g if and only if the Lee form ω is ∇-parallel.The Riemannian curvatures are related by, for anyX, Y, Z, and W tangent to M,(25)expftRtX,Y,Z,W=RX,Y,Z,W+12gX,WPY,Z−gY,WPX,Z+12gY,ZPX,W−gX,ZPY,W.LetEi be the orthonormal basis with respect to g. Then, we have(26)gEi,Ej=1ifi=j,0ifi≠j.LetEit=expft1/2Ei, for any i=1,2,…,2n. Therefore, we have the following.Lemma 4.
The frameEit1≤i≤2n is the orthonormal basis with respect to gt.The following identities generalize the ones given in [9, p.216].Lemma 5.
The Ricci curvature tensorsρt and ρ with respect to gt and g, respectively, are related by(27)ρtX,Y=ρX,Y+n−1PX,Y+12gX,YtraceP.Proof.
Using Lemma4 and for any vector fields X and Y tangent to M, one has(28)ρtX,Y=∑i=12nRtEit,X,Y,Eit=∑i=12nexpftRtEi,X,Y,Ei=∑i=12nREi,X,Y,Ei+12∑i=12ngEi,EiPX,Y−∑i=12ngX,EiPEi,Y+12∑i=12ngX,YPEi,Ei−∑i=12ngEi,YPX,Ei=ρX,Y+n−1PX,Y+12gX,YtraceP,which completes the proof.
Also, corresponding Ricci∗-curvatures are related by(29)ρt∗X,Y=ρ∗X,Y+12PX,Y+PJX,JY.Corollary 1.
The scalar curvaturesτt and τ are related by(30)exp−ftτt=τ+2n−1divB−121−nB2.Proof.
Using Lemma 4 and the scalar curvatureτt, we have(31)τt=∑i=12nρtEit,Eit=expft∑i=12nρtEi,Ei.
Then, applying equation (26) to (31), we get(32)exp−ftτt=∑i=12nρEi,Ei+n−1∑i=12nPEi,Ei+ntraceP=τ+2n−1traceP=τ+2n−1divB−121−nB2.
Therefore,(33)exp−ftτt=τ+2n−1divB−121−nB2,which completes the proof.
Now, if we consider a relation between scalar∗-curvature τt∗ and τ∗, we get the following.Corollary 2.
The scalar∗-curvatures τt∗ and τ∗ are related by(34)exp−ftτt∗=τ∗+divB+n−1B2.Proof.
The scalar∗-curvature τt∗ is given by(35)τt∗=∑i=12nρt∗Eit,Eit=expft∑i=12nρt∗Ei,Ei.
Now, applying relation (29) to (35), we compute(36)exp−ftτt∗=∑i=12nρt∗Ei,Ei=∑i=12nρ∗X,Y+12∑i=12nPEi,Ei+PJEi,jEi=τ∗+divB+n−1B2.
Hence,(37)exp−ftτt∗=τ∗+divB+n−1B2,as required.
Gray in [7] considered some curvature identities for Hermitian and almost Hermitian manifolds. Let ℒ be the class of almost Hermitian manifolds as defined in [7]. Then, the manifold under consideration is an element of class ℒ. Now, consider as in [7] the curvature operator Rt of a locally conformal almost Kähler manifold M:(38)1RtX,Y,Z,W=RtX,Y,JZ,JW,2RtX,Y,Z,W−RtJX,JY,Z,W=RtJX,Y,JZ,W,+RtJX,Y,Z,JW,3RtX,Y,Z,W=RtJX,JY,JZ,JW,for any X, Y, Z, and W tangent to M. Item (1) is called a Kähler identity if M is a locally conformal Kähler manifold (see [7] for more details and references therein).
We will focus, throughout the rest of this note, on item (1). Using further notations as in [7], we denoted by ℒi the subclass of manifolds whose curvature operator Rt satisfies identity (i). Here, (i) may be either item (1), (2), or (3) above. As in [7], it is easy to see that(39)ℒ1⊆ℒ2⊆ℒ3⊆ℒ.
Therefore, we have the following result.Lemma 6.
If a locally conformal almost Kähler manifold is in classℒ1, then the equality holds(40)τ∗−τ=2n−1traceP.Proof.
The proof follows from a straightforward calculation using the fact that, for any vector fieldsX and Y tangent to M, we have(41)ρtX,Y=∑i=12nexpftRtEi,X,Y,Ei=∑i=12nexpftRtEi,X,JY,JEi=ρ∗tX,Y,which leads to(42)ρ∗−ρX,Y=n−32PX,Y+12gX,YtraceP−12PJX,JY.
This completes the proof.
Relation (41) leads to(43)τt=∑i=12n∑j=12nRtEj,Ei,Ei,Ej=∑i=12n∑j=12nRtEj,Ei,JEi,JEj=τt∗.Theorem 1.
LetM,J,g be a 2n-dimensional compact locally conformal almost Kähler manifold with n>1 and contained in ℒ1. If(44)τ∗=τ,then M,J,g is an almost Kähler manifold.Proof.
By Lemma 6, we haveτ∗−τ=2n−1traceP, with traceP=divB−1/21−nB2. Taking into account this, integrating relation (40), and using Green’s theorem, we have 0=∫Mτ∗−τ=n−12∫MB2. Hence, under our assumption, we obtain B=0. Therefore, ω=0 identically on M. Hence, M.J,g is an almost Kähler manifold.
As an example for this theorem, we have compact flat locally almost Kähler manifolds. Compact flat manifolds have been detailed in [8] and references therein.


## 4. Lee Form and Canonical Foliations

LetM,J,g be a locally conformal almost Kähler manifold, and assume that the Lee form ω is never vanishing on M. Then, ω=0 defines on M an integrable distribution, and hence a foliation ℱ, on M (see [9] for more details and references therein).LetD:=kerω be the distribution on M and D⊥ be the distribution spanned by the vector field B. Then, we have the following decomposition:(45)TM=D⊕D⊥,where ⊕ denotes the orthogonal direct sum. By decomposition (45), any X∈ΓTM is written as(46)X=QX+Q⊥X,where Q and Q⊥ are the projection morphisms of TM into D and D⊥, respectively. Here, it is easy to see that Q⊥X=1/B2ωXB, and(47)X=QX+1B2ωXB.Letℱ be a foliation on a locally conformal almost Kähler manifold M,J,g of codimension 1. The metric g is said to be bundle-like for the foliation ℱ if the induced metric on the transversal distribution D⊥ is parallel with respect to the intrinsic connection on D⊥. This is true if and only if the Levi-Civita connection ∇ of M,J,g satisfies (see [10, 11] for more details)(48)g∇Q⊥YQX,Q⊥Z+g∇Q⊥ZQX,Q⊥Y=0,for any X, Y, Z∈ΓTM. A foliation ℱ is said to be Riemannian on M,J,g if the Riemannian metric g on M is bundle-like for ℱ.Letℱ⊥ be the orthogonal complementary foliation generated by B. Now, we provide necessary and sufficient conditions for the metric on an locally conformal almost Kähler manifold to be bundle-like for foliations ℱ and ℱ⊥.Theorem 2.
LetM,J,g be a locally conformal almost Kähler manifold, and let ℱ be a foliation on M of codimension 1. Then, the following assertions are equivalent:(i)
The foliationℱ is Riemannian.(ii)
The Lee vector fieldB is autoparallel with respect to ∇, that is,(49)∇BB=BlnBB.Proof.
For anyX, Y, Z∈ΓTM, we have Q⊥Y=1/B2ωYB and Q⊥Z=1/B2ωZB, and the left-hand side of (48) gives(50)g∇Q⊥YQX,Q⊥Z+g∇Q⊥ZQX,Q⊥Y=2B2ωYωZω∇BQX,for which the equivalence follows.
LetM′ be a leaf of the distribution D. Since M′ is a submanifold of M and for any X, Y∈ΓTM′, we have(51)∇XY=∇X′Y+αX,Y,(52)∇XB=−ABX+∇X′⊥B,where ∇′ and α are the Levi-Civita connection and the second fundamental form of M′, respectively. Here, AB is the shape operator with respect to B. On the contrary, we have g∇XB,B=XωB−g∇XB,B; hence,(53)g∇X′⊥B,B=12XωB,for any X∈ΓTM′. Therefore, Weingarten formula (52) becomes(54)∇XB=−ABX+12XωBB.Proposition 1.
LetM,J,g be a locally conformal almost Kähler manifold. Then, the mean curvature vector field H′ of the leaves of the integrable distribution D defined in (45) is given by(55)H′=12n−1div|M′BB.
Moreover, these leaves are totally geodesic hypersurfaces ofM if and only if the dual vector field B of ω preserves their metrics.Proof.
LetM′ be a leaf of the integrable distribution D. Using (51) and (54), the second fundamental form of M′ gives(56)αX,Y=gABX,YB=g∇XB,YB,for any X, Y∈ΓTM′. Fixing a local orthonormal frame e1,…,e2n−1 in TM′, one has(57)H=12n−1∑i=12n−1αei,ei=12n−1div|M′BB.
The last assertion follows, and this completes the proof.
Therefore, we have the following result.Corollary 4.
LetM,J,g be a locally conformal almost Kähler manifold. Then, the leaves M′ of the distribution D in (45) are minimal if and only if the dual vector field B is incompressible along M′.



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