  Our aim is to give an explicit description of the Arens-Michael envelope for
the universal enveloping algebra of a finite-dimensional nilpotent complex Lie
algebra. It turns out that the Arens-Michael envelope belongs to a class of
completions introduced by R.~Goodman in 70s. To find a precise form of this
algebra we preliminary characterize the set of holomorphic functions of
exponential type on a simply connected nilpotent complex Lie group. This
approach leads to unexpected connections to Riemannian geometry and the theory
of order and type for entire functions.
  As a corollary, it is shown that the Arens-Michael envelope considered above
is a homological epimorphism. So we get a positive answer to a question
investigated earlier by Dosi and Pirkovskii.
