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Bulletin of the London Mathematical Society Advance Access originally published online on February 6, 2009
Bulletin of the London Mathematical Society 2009 41(1):155-163; doi:10.1112/blms/bdn115
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© 2009 London Mathematical Society

Global smooth fibrations of R3 by oriented lines

Marcos Salvai

FaMAF-CIEM
Ciudad Universitaria
5000 Córdoba
Argentina

Received 23 July 2007. Revision received 29 May 2008.

A smooth fibration of R3 by oriented lines is given by a smooth unit vector field V on R3 all of whose integral curves are straight lines. Such a fibration is said to be nondegenerate if dV vanishes only in the direction of V. Let L be the space of oriented lines of R3 endowed with its canonical pseudo-Riemannian neutral metric. We characterize the nondegenerate smooth fibrations of R3 by oriented lines as the closed (in the relative topology) definite connected surfaces in L. In particular, local conditions on L imply the existence of a global fibration. Besides, for any such fibration the base space is diffeomorphic to the open disc and the directions of the fibers form an open convex set of the two-sphere. We characterize as well, in a similar way, the smooth (possibly degenerate) fibrations.

2000 Mathematics Subject Classification 53C12 (primary), 53C40, 53C50 (secondary).

The author was partially supported by FONCyT, Antorchas, CIEM (CONICET) and SECyT (UNC).


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