  Wigner's theorem characterizes isometries of the set of all rank one
projections on a Hilbert space. In metric geometry nonexpansive maps and
noncontractive maps are well studied generalizations of isometries. We show
that under certain conditions Wigner symmetries can be characterized as
nonexpansive or noncontractive maps on the set of all projections of rank one.
The assumptions required for such characterizations are injectivity or
surjectivity and they differ in the finite and the infinite-dimensional case.
Motivated by a recently obtained optimal version of Uhlhorn's generalization of
Wigner's theorem, we also give a description of nonexpansive maps which satisfy
a condition that is much weaker than surjectivity. Such maps do not need to be
Wigner symmetries. The optimality of all presented results is shown by
counterexamples.
