The magnetic flow on the manifold of oriented geodesics of a three dimensional space form

YAMILE GODOY; MARCOS SALVAI

San Pablo, Brasil

Otro; XVI School of Differential Geometry; 2010

Universidad de San Pablo

For k=0,1,-1, let M_{k} be the three dimensional simply connected manifold of constant sectional curvature k. Let L_{k} be the manifold of all (unparametrized) oriented geodesics of M_{k}, endowed with its canonical pseudo-Riemannian metric of signature (2,2) and Kähler structure J. A smooth curve in L_{k} determines a ruled surface in M_{k}. We characterize the ruled surfaces of M_{k} associated with the magnetic geodesics of L_{k}. More precisely: a time-like or a space-like magnetic geodesic describes the ruled surface in M_{k} given by the binormal vector field along a helix with non-zero torsion. Null magnetic geodesics describe cones, cylinders or, in the hyperbolic case, also cones with vertices at infinity.