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.Comment: 20 page
