Indefinite locally conformal Kähler manifolds

We study the basic properties of an indefinite locally conformal Kähler (l.c.K.) manifold. Any indefinite l.c.K. Manifold $M$ with a parallel Lee form $\omega$ is shown to possess two canonical foliations $\mathcal F$ and $\mathcal{F}_c$ , the first of which is given by the Pfaff equation $\omega = 0$ and the second is spanned by the Lee and the anti-Lee vectors of $M$. We build an indefinite l.c.K. metric on the noncompact complex manifold $\Omega_+ = (\Lambda_+ \setminus \Lambda_0)/G_\lambda$ (similar to the Boothby metric on a complex Hopf manifold) and prove a CR extension result for CR functions on the leafs of $\mathcal F$ when $M = \Omega_+$ (where $\Lambda_+ \setminus \Lambda_0 \subset \mathbb{C}_s^n$ is $- |z_1|^2 - \cdots - |z_s|^2 + |z_{s+1}|^2 + \cdots !z_n|^2 >0$). We study the geometry of the second fundamental form of the leaves of $\mathcal F$ and $\mathcal{F}_c$. In the degenerate cases (corresponding to a lightlike Lee vector) we use the technique of screen distributions and (lightlike) transversal bundles developed by A. Bejancu et al. [K.L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364, Kluwer Academic, Dordrecht, 1996].