Counterexamples related to the Kobayashi Pseudodistance

Abstract

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.

11 pages, LaTeX. Some constructions have been made more explicit; references added