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

Elisabetta Barletta, Sorin Dragomir and Francesco Esposito
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Elisabetta Barletta: Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, 85100 Potenza, Italy
Sorin Dragomir: Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, 85100 Potenza, Italy
Francesco Esposito: Dipartimento di Matematica e Fisica Ennio De Giorgi, Università del Salento, 73100 Lecce, Italy

Mathematics, 2021, vol. 9, issue 4, 1-15

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 ? = 0 , provided that ? ? = 0 and c = ? ? ? ? 0 ( ? 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 2 s , 0 < s < n , is locally biholomorphically homothetic to an indefinite complex Hopf manifold C H s n ( ? ) , 0 < ? < 1 , equipped with the indefinite Boothby metric g s , n .

Keywords: indefinite locally conformal Kähler manifold; indefinite Hopf manifold; indefinite Boothby metric; indefinite Vaisman manifold; Lee vector field; Lee form; canonical foliation; indefinite Sasakian structure (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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