Global smooth foliations of R^3 by oriented lines

SALVAI, MARCOS

BULLETIN OF THE LONDON MATHEMATICAL SOCIETY

Oxford University Press

Año: 2009 vol. 41 p. 155 - 163

0024-6093

A smooth fibration of R^3 by oriented lines is given by a smooth unitvector field V on R^3 all of whose integral curves are straight lines. Such afibration is said to be nondegenerate if dV vanishes only in the directionof V . Let L be the space of oriented lines of R^3 endowed with its canon-ical pseudo-Riemannian neutral metric. We characterize the nondegeneratesmooth fibrations of R3 by oriented lines as the definite closed (in the relativetopology) connected surfaces in L. In particular, local conditions on L implythe existence of a global fibration. Besides, for any such fibration, the basespace is diffeomorphic to the open disc and the directions of the fibers forman open convex set of the two-sphere. We characterize as well, in a similarway, the smooth (possibly degenerate) global fibrations.The spirit of the article is similar to that of the characterization given byH. Gluck and F. Warner (Duke Math. J., 1983) of the oriented great circlefibrations of S^3: They determine which subsets of the manifold of orientedcircles C= S^2 x S^2 are base spaces of such fibrations. In the Euclideansituation, by contrast, apart from the noncompactness of the ambient space,one has the difficulties arising from the fact that L is not Riemannian andfrom the possibility of degeneracy (existence of infinitesimally parallel fibers,not present in the spherical case).