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

Elisabetta Barletta,Francesco Esposito,Sorin Dragomir

doi:10.3390/math9040333

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, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

Highlights

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Any indefinite Vaisman manifold admits two canonical foliations F and G, the first of which is given by the Pfaffian equation ω = 0, while the second is tangent to the distribution spanned by the Lee and anti-Lee fields i.e., T (G) = R B ⊕ R A
Our study is confined to the semi-Riemannian case (c 6= 0), and in that case knowledge of the first-order geometric structure of F together with the semi-Riemannian de Rham decomposition theorem leads to the local description of the metric structure of any geodesically complete, conformally flat, indefinite Vaisman manifold

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Let M be a complex n-dimensional indefinite Hermitian manifold, of index 0 ≤ ν < 2n, with the complex structure J and the semi-Riemannian metric g (ν is necessarily even, i.e., ν = 2s). ( M, J, g) is an indefinite Kähler manifold if ∇ J = 0, where ∇ is the LeviCivita connection of ( M, g). Indefinite Kähler manifolds were studied by M. 1976 work pointed out (cf [4]) the counterexample of a complex Hopf manifold C H0n (λ) with the Boothby metric g0,n and disproved Betti number of the (compact, complex) manifold C H0n (λ) is 1, so that C H0n (λ) admits no globally defined Kähler metrics) Aubin’s “finding”. Regarding the geometry of l.c.K. structures, it was not until the work by K.L. Duggal et al

Objectives

Conclusion