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

Abstract

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kahler (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 2 s , 0 < s < n , is locally biholomorphically homothetic to an indefinite complex Hopf manifold C H s n ( λ ) , 0 < λ < 1 , equipped with the indefinite Boothby metric g s , n .