  We present some unexpected examples related to the Kobayashi pseudodistance:
For an unramified covering, the vanishing of the Kobayashi pseudodistance on
the base does not imply the vanishing on the total space. The vanishing of the
Kobayashi pseudodistance does not imply the vanishing of the Kobayashi Royden
pseudo metric. Given a locally holomorphic trivial fiber bundle, the Kobayashi
pseudodistance may vanish identically on the total space even if the fiber is
hyperbolic.
