Abstract: | For a solvable Lie group G the surjectivity of the exponential function expG is equivalent to the connectedness of the near-Cartan subgroups and to the connectedness of the centralizers in a Cartan subgroup of all nilpotent elements in its Lie algebra g. Furthermore, these conditions are satisfied if and only if for all elements g ? G there is an x ? g with g = expG x in which expG is regular. |