Lee Form Research Articles

On Generalized Kenmotsu Manifolds as Hypersurfaces of Vaisman-Gray Manifolds

In this paper, we conclude that the hypersurfaces of Vaisman-Gray manifolds have generalized Kenmotsu structures under some conditions for the Lee form, Kirichenko's tensors and the second fundamental form of the immersion of the hypersurface into the manifold of Vaisman-Gray class. Moreover, the components of the second fundamental form are determined when the foregoing hypersurfaces have generalized Kenmotsu structures or any special kind of it or Kenmotsu structures, such that some of these components are vanish. Also, some components of Lee form and some components of some Kirichenko's tensors in the Vaisman--Gray class are equal to zero. On the other hand, the minimality of totally umbilical, totally geodesic hypersurfaces of Vaisman--Gray manifolds with generalized Kenmotsu structures are investigated. In addition, we deduced that the hypersurface of Vaisman--Gray manifold that have generalized Kenmotsu structure is totally geodesic if and only if it is totally umbilical and some components of Lee form are constants.

Kähler Magnetic Curves in Conformally Euclidean Schwarzschild Space

In this paper, we study the magnetic curves on a Kähler manifold which is conformally equivalent to Euclidean Schwarzschild space. We show that Euclidean Schwarzschild space is locally conformally Kähler and transform it into a Kähler space by applying a conformal factor coming from its Lee form. We solve Lorentz equation to find analytical expressions for magnetic curves which are compatible with the almost complex structure of the proposed Kähler manifold. We also calculate the energy of magnetic curves.

Adapted metrics on locally conformally product manifolds

We show that the Gauduchon metric g 0 g_0 of a compact locally conformally product manifold ( M , c , D ) (M,c,D) of dimension greater than 2 2 is adapted, in the sense that the Lee form of D D with respect to g 0 g_0 vanishes on the D D -flat distribution of M M . We also characterize adapted metrics as critical points of a natural functional defined on the conformal class.

Locally conformally product structures

A locally conformally product (LCP) structure on compact manifold [Formula: see text] is a conformal structure [Formula: see text] together with a closed, non-exact and non-flat Weyl connection [Formula: see text] with reducible holonomy. Equivalently, an LCP structure on [Formula: see text] is defined by a reducible, non-flat, incomplete Riemannian metric [Formula: see text] on the universal cover [Formula: see text] of [Formula: see text], with respect to which the fundamental group [Formula: see text] acts by similarities. It was recently proved by Kourganoff that in this case [Formula: see text] is isometric to the Riemannian product of the flat space [Formula: see text] and an incomplete irreducible Riemannian manifold [Formula: see text]. In this paper, we show that for every LCP manifold [Formula: see text], there exists a metric [Formula: see text] such that the Lee form of [Formula: see text] with respect to [Formula: see text] vanishes on vectors tangent to the distribution on [Formula: see text] defined by the flat factor [Formula: see text], and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev–Nikolayevsky, Kourganoff and Oeljeklaus–Toma (OT-manifolds).

Geometry of Harmonic Nearly Trans-Sasakian Manifolds

This paper considers a class of nearly trans-Sasakian manifolds. The local structure of nearly trans-Sasakian structures with a closed contact form and a closed Lee form is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds with a closed contact form locally conformal to the closely cosymplectic manifolds. A wide class of harmonic nearly trans-Sasakian manifolds has been identified (i.e., nearly trans-Sasakian manifolds with a harmonic contact form) and an exhaustive description of the manifolds of this class is obtained. Also, examples of harmonic nearly trans-Sasakian manifolds are given.

Second Chern-Einstein metrics on four-dimensional almost-Hermitian manifolds

Abstract We study four-dimensional second Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis, the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe four-dimensional compact second Chern-Einstein locally conformally symplectic manifolds, and we give some examples of such manifolds. Finally, we study the second Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second Chern-Einstein metric with a parallel non-zero Lee form.

On the geometry of nearly trans-Sasakian manifolds

The geometry of nearly trans-Sasakian manifolds is researched in this paper. The complete group of structural equations and the components of the Lee vector on the space of the associated $G$-structure are obtained for such manifolds. Conditions are found under which a nearly trans-Sasakian structure is a trans-Sasakian, a cosymplectic, a closely cosymplectic, a Sasakian structure or a Kenmotsu structure. The conditions are obtained when the nearly trans-Sasakian structure is a special generalized Kenmotsu structure of the second kind. A complete classification of nearly trans-Sasakian manifolds is obtained, i.e. it is proved that a nearly trans-Sasakian manifold is either a trans-Sasakian manifold or has a closed contact form. It is proved that the nearly trans-Sasakian structure with a nonclosed contact form is homothetic to the Sasakian structure. The criterion of ownership of a nearly trans-Sasakian structure is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds with a closed contact form, which are locally conformal to the closely cosymplectic manifolds. Examples of such manifolds are given. The necessary and sufficient conditions for the complete integrability of the first fundamental distribution of a nearly trans-Sasakian manifold are obtained. It is proved that a nearly K\"ahler structure on the leaves of the first fundamental distribution of a nearly trans-Sasakian manifold is induced.

Scalar curvatures in almost Hermitian geometry and some applications

On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the Kähler metric, the balanced metric, the locally conformal Kähler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier (2020) and a conjecture by Angella et al. (2018).

Correction to: A note on Euler number of locally conformally Kähler manifolds

Our aim is to clarify the Definition 2.8 in the original article. Let (M, J, g) be a compact 2n-dimensional LCK manifold with the Lee form.

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/4, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

Small Gauge Transformations and Universal Geometry in Heterotic Theories

The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is 'holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in α and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a choice of holomorphic structure and Lee form on a 'universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.

Variational Problems in Conformal Geometry

We study the Euler-Lagrange equation for several natural functionals defined on a conformal class of almost Hermitian metrics, whose expression involves the Lee form $\theta$ of the metric. We show that the Gauduchon metrics are the unique extremal metrics of the functional corresponding to the norm of the codifferential of the Lee form. We prove that on compact complex surfaces, in every conformal class there exists a unique metric, up to multiplication by a constant, which is extremal for the functional given by the $L^2$-norm of $dJ\theta$, where $J$ denotes the complex structure. These extremal metrics are not the Gauduchon metrics in general, hence we extend their definition to any dimension and show that they give unique representatives, up to constant multiples, of any conformal class of almost Hermitian metrics.

A note on Euler number of locally conformally Kähler manifolds

Let $$M^{2n}$$ be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call $$(M,J,\theta )$$ a d(bounded) locally conformally Kahler manifold if the lifted Lee form $${\tilde{\theta }}$$ on the universal covering space of M is d(bounded). We show that if $$M^{2n}$$ is homeomorphic to a d(bounded) LCK manifold, then its Euler number satisfies the inequality $$(-1)^{n}\chi (M^{2n})\ge $$ (resp. >) 0.

The exterior derivative of the Lee form of almost Hermitian manifolds

The exterior derivative dθ of the Lee form θ of almost Hermitian manifolds is studied. If ω is the Kähler two-form, it is proved that the Rω-component of dθ is always zero. Expressions for the other components, in [λ01,1] and in [[λ2,0]], of dθ are also obtained. They are given in terms of the intrinsic torsion. Likewise, it is described some interrelations between the Lee form and U(n)-components of the Riemannian curvature tensor.

Infinitesimal Transformations of Locally Conformal Kähler Manifolds

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric.

Locally conformally Kähler solvmanifolds: a survey

AbstractA Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.

On locally conformal symplectic manifolds of the first kind

We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the corresponding nilpotent Lie groups. Under certain topological restrictions related with the compactness of the canonical foliation, we prove a structure theorem for locally conformal symplectic manifolds of the first kind. In the non compact case, we show that they are the product of a real line with a compact contact manifold and, in the compact case, we obtain that they are mapping tori of compact contact manifolds by strict contactomorphisms. Motivated by the aforementioned examples, we also study left-invariant locally conformal symplectic structures on Lie groups. In particular, we obtain a complete description of these structures (with non-zero Lee form) on connected simply connected nilpotent Lie groups in terms of locally conformal symplectic extensions and symplectic double extensions of symplectic nilpotent Lie groups. In order to obtain this description, we study locally conformal symplectic structures of the first kind on Lie algebras.

Example of a six-dimensional LCK solvmanifold

Abstract The purpose of this paper is to prove that there exists a lattice on a certain solvable Lie group and construct a six-dimensional locally conformal Kähler solvmanifold with non-parallel Lee form.

Structure theorem for Vaisman completely solvable solvmanifolds

Locally conformal Kähler manifold is said to be a Vaisman manifold if the Lee form is parallel with respect to the Riemannian metric. In this paper, we have the structure theorem for Vaisman completely solvable solvmanifolds.

On Pluricanonical Locally Conformally Kähler Manifolds

We prove that compact pluricanonical locally conformally Kähler manifolds have parallel Lee form.