Canonical Foliation of an Indefinite Locally Conformal Kähler Manifolds

We investigate the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l. c. K.) manifold M, described by the Pfaffian equation ω = ۰, under the condition ∇ω = ۰ and c = k ω k ۶= ۰ (where ω represents the Lee form of M). If M is conformally flat, then each leaf of F is proven to be a completely geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/۴, carrying an indefinite c-Sasakian structure. As a consequence of this result, along with a semi-Riemannian version of the de Rham decomposition theorem, any geodesically complete, conformally flat, indefinite Vaisman manifold of index ۲s, where ۰ < s < n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CH n s (λ), where ۰ < λ < ۱, equipped with the indefinite Boothby metric g s, n.

Keywords:

Canonical foliation , Lee form , Indefinite locally conformal Kähler manifold , Indefinite Hopf manifold