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 ω is shown to possess two canonical foliations F and Fc, the first of which is given by the Pfaff equation ω=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 Ω+=(Λ+∖Λ0)/Gλ (similar to the Boothby metric on a complex Hopf manifold) and prove a CR extension result for CR functions on the leafs of F when M=Ω+ (where Λ+∖Λ0⊂Csn is −|z1|2−⋯−|zs|2+|zs+1|2+⋯+|zn|2>0). We study the geometry of the second fundamental form of the leaves of F and Fc. 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].