We study the semi-Riemannian geometry of the foliation $mathcal F$ of an indefinite locally conformal Kähler (l.c.K.) manifold $M$, given by the Pfaffian equation $omega = 0$, provided that $nabla omega = 0$ and $c = | omega | neq 0$ ($omega$ is the Lee form of $M$). If $M$ is conformally flat then every leaf of $mathcal F$ 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 (in the sense of T. Takahasi). As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem (due to H. Wu) any geodesically complete, conformally flat, indefinite Vaisman manifold of index $2s$, $0 < s < n$, is locally biholomorphically homothetic to an indefinite complex Hopf manifold ${mathbb C}H^n_s (lambda )$, $0 < lambda < 1$, equipped with the indefinite Boothby metric $g_{s, n}$.
On the canonical foliation of an indefinite locally conformal Kähler manifold with a parallel Lee form
Elisabetta Barletta;Sorin Dragomir
;Francesco Esposito
2021-01-01
Abstract
We study the semi-Riemannian geometry of the foliation $mathcal F$ of an indefinite locally conformal Kähler (l.c.K.) manifold $M$, given by the Pfaffian equation $omega = 0$, provided that $nabla omega = 0$ and $c = | omega | neq 0$ ($omega$ is the Lee form of $M$). If $M$ is conformally flat then every leaf of $mathcal F$ 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 (in the sense of T. Takahasi). As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem (due to H. Wu) any geodesically complete, conformally flat, indefinite Vaisman manifold of index $2s$, $0 < s < n$, is locally biholomorphically homothetic to an indefinite complex Hopf manifold ${mathbb C}H^n_s (lambda )$, $0 < lambda < 1$, equipped with the indefinite Boothby metric $g_{s, n}$.File | Dimensione | Formato | |
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On the canonical foliation of an indefinite locally conformal Kahler manifold with a parallel Lee form.pdf
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