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

YAMILE GODOY; MARCOS SALVAI

San Pablo

Otro; 16th School of Differential Geometry; 2010

Universidad de San Pablo

For $kappa =0,1,-1$, let $M_{kappa }$ be the three dimensional simply connected manifold of constant sectional curvature $kappa $. Let $mathcal{L% }_{kappa }$ be the manifold of all (unparametrized) oriented geodesics of $% M_{kappa }$, endowed with its canonical pseudo-Riemannian metric of signature $left( 2,2 ight) $ and K"{a}hler structure $J$. A smooth curve in $mathcal{L}_{kappa }$ determines a ruled surface in $M_{kappa }$. We characterize the ruled surfaces of $M_{kappa }$ associated with the magnetic geodesics of $mathcal{L}_{kappa }$, that is, those curves $sigma $ in $mathcal{L}_{kappa }$ satisfying $ abla _{dot{sigma}}dot{sigma}=J% dot{sigma}$. More precisely: a time-like or a space-like magnetic geodesic describes the ruled surface in $M_{kappa }$ 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.