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

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.

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 metric 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 [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

Let be a -dimensional almost Hermitian manifold with the metric and the almost complex structure satisfyingfor any vector fields and tangent to , where stands for the identity transformation of tangent bundle . Then, for any vector fields and , the tensordefines the fundamental 2-form of which is nondegenerate and gives an almost symplectic structure on . If is closed, i.e., , then is called an almost Kähler manifold [6].

Now, let be a -dimensional almost Hermitian manifold. Such a manifold is said to be a locally conformal almost Kähler manifold [4] if there is an open covering of and a family of -functions such that, for any , the metric formis an almost Kähler metric.

If the structures defined in (3) are Kähler, then is called locally conformal Kähler. Moreover, a locally conformal almost Kähler manifold is almost Kähler if and only if .

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 is a locally conformal almost Kähler manifold if and only if there exists 1-form such that

Example 1. We consider the 4-dimensional manifold , where are the standard coordinates in . The vector fields,are linearly independent at each point of . Let be the Riemannian metric on defined by , where is the Kronecker symbol, . That is, the form of the metric becomesLet be the -tensor field defined by , , , and . Thus, defines an almost Hermitian structure on . The nonzero component of fundamental 2-form isand we haveIts differential givesBy lettingwe haveIt is easy to see that , and the dual vector field is given byLet us consider the open neighborhood of given by , and there exists a differentiable function on such that , where . By the aforementioned characterization given in (4), 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 metrics and .
Throughout this paper, will denote the -module of differentiable sections of a vector bundle .
Let and be the Levi-Civita connections associated with the metrics and , respectively. As is well known, they are connected byfor all .

3. Curvature Relations of Locally Conformal Almost Kähler Metrics

Let be a -dimensional almost Hermitian manifold. Here, we keep the formalism of local transformations and other formulas defined in the previous section. For the Riemann curvature of a metric , we use the following convention:wherefor any vector field , , and on .

Let be the orthonormal basis with respect to . The Ricci curvature tensor and the scalar curvature are given by

Now, we consider the Ricci -curvature tensor and the scalar -curvature defined by

Similarly, the curvatures corresponding to the metric will be denoted by , , , , and , respectively.

Lemma 1. Let be a locally conformal almost Kähler manifold. Then, the curvature tensors and with respect to the metrics and , respectively, are related aswhere .

Proof. Using the convention in (15) for the curvature tensors and and relation (13) and for any vector fields , , and tangent to , the expressionsIt is worth noting thatPutting pieces (19) and (20) together, one obtains relation (18). This completes the proof.
Next, from the above lemma, we define -tensor field byand this trace is given by

Lemma 2. The -tensor field is symmetric.

Proof. For any vector fields and tangent to and since is closed, we havewhich completes the proof.
The Lie derivative with respect to the vector field gives, for any vector fields and ,The last equality of (24) follows from the fact that smooth 1-form is closed.

Lemma 3. The dual vector field of preserves the metric if and only if the Lee form is -parallel.

The Riemannian curvatures are related by, for any , , , and tangent to ,

Let be the orthonormal basis with respect to . Then, we have

Let , for any . Therefore, we have the following.

Lemma 4. The frame is the orthonormal basis with respect to .

The following identities generalize the ones given in [9, p.216].

Lemma 5. The Ricci curvature tensors and with respect to and , respectively, are related by

Proof. Using Lemma 4 and for any vector fields and tangent to , one haswhich completes the proof.
Also, corresponding Ricci -curvatures are related by

Corollary 1. The scalar curvatures and are related by

Proof. Using Lemma 4 and the scalar curvature , we haveThen, applying equation (26) to (31), we getTherefore,which completes the proof.
Now, if we consider a relation between scalar -curvature and , we get the following.

Corollary 2. The scalar -curvatures and are related by

Proof. The scalar -curvature is given byNow, applying relation (29) to (35), we computeHence,as required.
Gray in [7] considered some curvature identities for Hermitian and almost Hermitian manifolds. Let be the class of almost Hermitian manifolds as defined in [7]. Then, the manifold under consideration is an element of class . Now, consider as in [7] the curvature operator of a locally conformal almost Kähler manifold :for any , , , and tangent to . Item (1) is called a Kähler identity if 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 the subclass of manifolds whose curvature operator satisfies identity (i). Here, (i) may be either item (1), (2), or (3) above. As in [7], it is easy to see thatTherefore, we have the following result.

Lemma 6. If a locally conformal almost Kähler manifold is in class , then the equality holds

Proof. The proof follows from a straightforward calculation using the fact that, for any vector fields and tangent to , we havewhich leads toThis completes the proof.
Relation (41) leads to

Theorem 1. Let be a -dimensional compact locally conformal almost Kähler manifold with and contained in . Ifthen is an almost Kähler manifold.

Proof. By Lemma 6, we have , with . Taking into account this, integrating relation (40), and using Green’s theorem, we have Hence, under our assumption, we obtain . Therefore, identically on . Hence, 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

Let be a locally conformal almost Kähler manifold, and assume that the Lee form is never vanishing on . Then, defines on an integrable distribution, and hence a foliation , on (see [9] for more details and references therein).

Let be the distribution on and be the distribution spanned by the vector field . Then, we have the following decomposition:where denotes the orthogonal direct sum. By decomposition (45), any is written aswhere and are the projection morphisms of into and , respectively. Here, it is easy to see that , and

Let be a foliation on a locally conformal almost Kähler manifold of codimension 1. The metric is said to be bundle-like for the foliation if the induced metric on the transversal distribution is parallel with respect to the intrinsic connection on . This is true if and only if the Levi-Civita connection of satisfies (see [10, 11] for more details)for any , , . A foliation is said to be Riemannian on if the Riemannian metric on is bundle-like for .

Let be the orthogonal complementary foliation generated by . Now, we provide necessary and sufficient conditions for the metric on an locally conformal almost Kähler manifold to be bundle-like for foliations and .

Theorem 2. Let be a locally conformal almost Kähler manifold, and let be a foliation on of codimension 1. Then, the following assertions are equivalent:(i)The foliation is Riemannian.(ii)The Lee vector field is autoparallel with respect to , that is,

Proof. For any , , , we have and , and the left-hand side of (48) givesfor which the equivalence follows.
Let be a leaf of the distribution . Since is a submanifold of and for any , , we havewhere and are the Levi-Civita connection and the second fundamental form of , respectively. Here, is the shape operator with respect to . On the contrary, we have ; hence,for any . Therefore, Weingarten formula (52) becomes

Proposition 1. Let be a locally conformal almost Kähler manifold. Then, the mean curvature vector field of the leaves of the integrable distribution defined in (45) is given byMoreover, these leaves are totally geodesic hypersurfaces of if and only if the dual vector field of preserves their metrics.

Proof. Let be a leaf of the integrable distribution . Using (51) and (54), the second fundamental form of givesfor any , . Fixing a local orthonormal frame in , one hasThe last assertion follows, and this completes the proof.
Therefore, we have the following result.

Corollary 4. Let be a locally conformal almost Kähler manifold. Then, the leaves of the distribution in (45) are minimal if and only if the dual vector field is incompressible along .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was based on the research supported wholly/in part by the National Research Foundation of South Africa (Grant nos. 95931 and 106072).