Some properties of curvature tensors and foliations of locally conformal almost K\"ahler manifolds

We investigate a class of locally conformal almost K\"ahler structures and prove that, under some conditions, this class is a subclass of almost K\"ahler structures. We show that locally conformal almost K\"ahler manifolds admits canonical foliation whose leaves are locally conformal almost K\"ahler hypersurfaces. The geodesibility of the leaves is also characterized, and their minimality coincides with the incompressibility of the Lee vector field along the leaves.


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 [3] 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 P. Libermann [7] in the 1950s. In 1966, A Gray [4] 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 [11]. 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 metric g 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 Section 2, 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 [9]. Under some conditions, we prove that a class of locally conformal almost Kähler structures is a subclass of almost Kähler structures. The Section 4 is devoted to the canonical foliations that arise for non-vanishing Lee form. We prove that these foliations are Riemannian if and only if the Lee vector field is auto-parallel. We also prove that the locally conformal almost Kähler manifolds contain leaves which are locally conformal almost Kähler hypersurfaces 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.

LOCALLY CONFORMAL ALMOST KÄHLER METRICS
Let M be a 2n-dimensional almost Hermitian manifold with the metric g and the almost complex structure J satisfying J 2 = −I, g(JX, JY ) = g(X, Y ), for any vector fields X and Y tangent to M , where I stands for the identity transformation of tangent bundle T M . Then for any vector fields X and Y , the tensor defines the fundamental 2-form of M which is non-degenerate and gives an almost symplectic structure on M . Now, let (M, J, g) be a 2n-dimensional almost Hermitian manifold. Such manifold is said to be locally conformal almost Kähler manifold [11] if there is an open covering {U t } t∈I of M and a family {f t } t∈I of C ∞ -functions f t : U t → R such that, for any t ∈ I, the metric form is an almost Kähler metric.
If the structures (J, g t ) defined in (2.2) 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 df t = 0.
The Lee form is important because it characterizes locally conformal almost Kähler manifolds. Locally confornal almost Kähler manifolds were characterized by Vaisman in [11]. This is stated follows: An almost Hermitian manifold (M, J, g) is a locally conformal almost Kähler manifold if and only if there exists a 1-form ω such that dΩ = ω ∧ Ω and dω = 0. (2.3) Example 2.1. We consider the 4-dimensional manifold where p = (x 1 , x 2 , y 1 , y 2 ) are the standard coordinates in R 4 . The vector fields, are linearly independent at each point of M . Let g be the Riemannian metric on That is, the form of the metric becomes Let J be the (1, 1)-tensor field defined by, Thus, (J, g) defines an almost Hermitian structure on M 4 . The non-zero component of the fundamental 2-form J is and we have Its differential gives By letting we have, It is easy to see that dω = 0 and the dual vector field B is given by . By the characterization given in (2.3) above-mentioned, (M 4 , J, g) is a locally conformal almost Kähler manifold.
Next, wish to study the relationship of the Levi-Civita connections induced by the locally conformal Kähler metric g t and g.
Throughout this paper, Γ(Ξ) will denote the F(M )-module of differentiable sections of a vector bundle Ξ.
Let ∇ and ∇ t be the Levi-Civita connections associated with the metrics g and g t , respectively. As is well-known, they are connected by

METRICS
Let (M, J, g) be a 2n-dimensional almost Hermitian manifold. Here we keep the formalism of local transformations and others formulas defined in the previous Section. For the Riemann curvature R of a metric g, we use the following convention for any vector field X, Y and Z on M . Let {E i } 1≤i≤2n be the orthonormal basis with respect to g. The Ricci curvature tensor ρ and the scalar curvature τ are given by Now we consider the Ricci * -curvature tensor ρ * and the scalar * -curvature τ * defined by Similarly, the curvatures corresponding to the metric g t will be denoted by R t , ρ t , τ t , ρ t * and τ t * , respectively.
Lemma 3.1. Let (M, J, g) be a locally conformal almost Kähler manifold. Then the curvature tensors R t and R with respect to the metrics g t and g, respectively, are related as Proof. Using the convention in (3.2) for the curvature tensors R t and R and the relation (2.4), and for any vector fields X, Y and Z tangent to M , the expressions It is worth noting that Putting the pieces (3.6) and (3.7) together, one obtains the relation (3.5). This completes the proof.
Next, from the above Lemma, we define (0, 2)-tensor field P by and this trace is given by Lemma 3.2. The (0, 2)-tensor field P is symmetric.
Proof. For any vector fields X and Y tangent to M and since ω is closed, we have which completes the proof.
The Lie derivative g with respect to the vector field B gives, for any vector fields X and Y , (3.10) The last equality of (3.10) follows from the fact that the smooth 1-form ω is closed. Let {E i } be the orthonormal basis with respect to g. Then, we have 2 E i , for any i = 1, 2, · · · , 2n. Therefore, we have: The following identities generalize the ones given in [9, p.216].
Lemma 3.5. The Ricci curvature tensors ρ t and ρ with respect to g t and g, respectively, are related by Proof. Using the Lemma 3.4 and for any vector fields X and Y tangent to M , one has which completes the proof.
Also, corresponding Ricci * -curvatures are related by Corollary 3.6. The scalar curvatures τ t and τ are related by (3.14) Proof. Using the Lemma 3.4, the scalar curvature τ t , we have Then, applying Equation (3.12) into (3.15), we get Therefore, which completes the proof.
Now if we consider a relation between the scalar * -curvature τ t * and τ * , we get: Corollary 3.7. The scalar * -curvatures τ t * and τ * are related by Proof. The scalar * -curvature τ t * is given by Now applying the relation (3.13) into (3.17), we compute Hence, as required.
Gray in [5] considered some curvature identities for Hermitian and almost Hermitian manifolds. Let L be the class of almost Hermitian manifolds as defined in [5]. Then the manifold under consideration is an element of the class L. Now consider as in [5] the curvature operator R t of a locally conformal almost Kähler manifold M : for any X, Y , Z and W tangent to M . The item (1) is called called Kähler identity if M is locally conformal Kähler manifold (see [5] for more details and reference therein).
We will focus, throughout the rest of this note, on the item (1). Using further notations as in [5], we denoted by L i the subclass of manifolds whose curvature operator R t satisfies identity (i). Here (i) may be either the item (1), (2) or (3) above. As in [5], it is easy to see that Therefore we have the following result.
Proof. The proof follows from a straightforward calculation using the fact that, for any vector fields X and Y tangent to M , we have This completes the proof.
As an example for this Theorem, we have compact flat locally almost Kähler manifolds. For compact flat manifolds have been detailed in [2] and reference therein.

LEE FORM AND CANONICAL FOLIATIONS
Let (M, 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 [6] for more details and reference therein).
Let D := ker ω be the distribution on M and D ⊥ be the distribution spanned the vector field B.Then, we have the following decomposition where ⊕ denotes the orthogonal direct sum. By the decomposition (4.1), any X ∈ Γ(T M ) is written as where Q and Q ⊥ are the projection morphisms of T M into D and D ⊥ , respectively. Here, it is easy to see that Q ⊥ X = 1 ||B|| 2 ω(X)B and Let F 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 [1] and [10] for more details): for any X, Y , Z ∈ Γ(T M ). A foliation F is said to be Riemannian on (M, J, g) if the Riemannian metric g on M is bundle-like for F. Let F ⊥ 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 ⊥ . Therefore Fixing a local orthonormal frame {e 1 , · · · , e 2n−1 } in T M ′ , one has, The last assertion follows and this completes the proof.
The first assertion of this theorem is similar the one found by Massamba and Maloko Mavambou in [8,Theorem 3.8]. Therefore we have the following results.