  In this paper we evaluate the Wightman functions associated with a massive
quantum scalar field in de Sitter and anti-de Sitter spacetimes in the presence
of a cosmic string. Having these functions we calculate the corresponding
renormalized vacuum expectation values of the field squared and present the
behavior of the contributions induced by the cosmic string as function of the
proper distance to it for different values of the parameter which codify the
presence of this linear topological defect.
