Parabolic geodesics as parallel curves in parabolic geometries

Marc Herzlich (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

We give a simple characterization of the parabolic geodesics introduced by Cap, Slovak and Zadnik for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manifold and a Weyl structure along the curve, so that the curve is a geodesic for its companion Weyl structure and the Weyl structure is parallel along the curve and in the direction of the tangent vector of the curve.

Mots clés: Parabolic geometries; parabolic geodesics

Domaines: Géométrie différentielle [math.DG]

Marc Herzlich. Parabolic geodesics as parallel curves in parabolic geometries. International Journal of Mathematics, 2013, 24 (9), pp.1350067. ⟨10.1142/S0129167X13500675⟩. ⟨hal-00718588⟩