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.

Twisted product CR-submanifolds in a locally conformal Kähler space forms

Certain twisted product CR-submanifolds in a K?hler manifold and some inequalities of the second fundamental form of these submanifolds are presented ([14]). Then the length of the second fundamental form of a twisted product CR-submanifold in a locally conformal K?hler manifold is considered (2013), ([15]). In this paper, we consider the relation of the mean curvature and the length of the second fundamental form in two twisted product CR-submanifolds in a locally conformal K?hler space forms.

The Concircular Curvature Tensor Of The Locally Conformal Kahler Manifold

In this research, we are calculated components conharmonic curvature tensor in some aspects Hermeation manifolding in particular of the Locally Conformal Kahler manifold. And we prove that this tensor possesses the classical symmetry properties of the Riemannian curvature. They also, establish relationships between the components of the tensor in this manifold http://dx.doi.org/10.25130/tjps.24.2019.137

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 conformal symplectic nilmanifolds with no locally conformal Kähler metrics

We report on a question, posed by L. Ornea and M. Verbitsky in [32], about examples of compact locally conformal symplectic manifolds without locally conformal Kähler metrics. We construct such an example on a compact 4-dimensional nilmanifold, not the product of a compact 3-manifold and a circle.

Example of a six-dimensional LCK solvmanifold

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.