Global smooth fibrations of R^3 by oriented lines

SALVAI, MARCOS

Santiago de Compostela

Otro; VIII International Colloquium on Differential Geometry; 2008

A smooth fibration of R^3 by oriented lines is given by a smoothunit vector field V on R3 all of whose integral curves arestraight lines. Such a fibration is said to be nondegenerate if dVvanishes only in the direction of V. Let L be the space of oriented lines of R^3 endowed with its canonical pseudo-Riemannianneutral metric. We characterize the nondegenerate smooth fibrations of  R^3 by oriented lines as the definite closed (in the relative topology)connected surfaces in L. In particular, local conditions on L imply the existence of a global fibration. Besides, for anysuch fibration, the base space is diffeomorphic to the open disc and thedirections of the fibers form an open convex set of the two-sphere. Wecharacterize as well, in a similar way, the smooth (possibly degenerate)global fibrations.