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 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, 0sn, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0λ1, equipped with the indefinite Boothby metric gs,n.

Keywords

• indefinite locally conformal Kähler manifold
• indefinite Hopf manifold
• indefinite Boothby metric
• indefinite Vaisman manifold
• Lee vector field
• Lee form