  The present note deals with the dynamics of metric connections with vectorial
torsion, as already described by E. Cartan in 1925. We show that the geodesics
of metric connections with vectorial torsion defined by gradient vector fields
coincide with the Levi-Civita geodesics of a conformally equivalent metric. By
pullback, this yields a systematic way of constructing invariants of motion for
such connections from isometries of the conformally equivalent metric, and we
explain in as much this result generalizes the Mercator projection which maps
sphere loxodromes to straight lines in the plane. An example shows that
Beltrami's theorem fails for this class of connections. We then study the
system of differential equations describing geodesics in the plane for vector
fields which are not gradients, and show among others that the Hopf-Rinow
theorem does also not hold in general.
