Global smooth fibrations of R^3 by oriented lines

MARCOS SALVAI

La Falda, Córdoba

Congreso; Segundo Congreso Latinoamericano de Grupos de Lie en Geometría; 2008

A smooth fibration of R^3 by oriented lines is given by a smooth unit vector field V on R^3 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 R^3 endowed with its canonical pseudo-Riemannian neutral 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 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) global fibrations.