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

Mathematics

Francesco Esposito¹  Sorin Dragomir²  Elisabetta Barletta² 

[1] Dipartimento di Matematica e Fisica Ennio De Giorgi, Universita` del Salento, 73100 Lecce, Italy;Dipartimento di Matematica, Informatica ed Economia, Universita` degli Studi della Basilicata, 85100 Potenza, Italy;

关键词: indefinite locally conformal Kähler manifold;    indefinite Hopf manifold;    indefinite Boothby metric;    indefinite Vaisman manifold;    Lee vector field;    Lee form;   

DOI  :  10.3390/math9040333

来源: DOAJ

【 摘 要 】

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 $c=\parallel \omega \parallel \ne 0$ ($\omega$ 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. 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 $\mathbb{C}{H}_s^n\left(\lambda \right)$, $0<\lambda <1$, equipped with the indefinite Boothby metric $g_{s,n}$.

【 授权许可】

Unknown