%0 Journal Article
%T Geodesics on a supermanifold and projective equivalence of super connections
%A Thomas Leuther
%A Fabian Radoux
%A Gijs Tuynman
%J Mathematics
%D 2012
%I arXiv
%R 10.1016/j.geomphys.2013.01.005
%X We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field associated with the connection. Our (super) geodesics possess the same properties as the in the classical case: there exists a unique (super) geodesic satisfying a given initial condition and when the connection is metric, our supergeodesics coincide with the trajectories of a free particle with unit mass. Moreover, using our definition, we are able to establish Weyl's characterization of projective equivalence in the super context: two torsion-free (super) connections define the same geodesics (up to reparametrizations) if and only if their difference tensor can be expressed by means of a (smooth, even, super) 1-form.
%U http://arxiv.org/abs/1202.1077v2