Symmetry breaking and mass generation by space-time topology

Scalar-field theory with a $\lambda {\phi }^4$ self-interaction is studied in space-times with a non-Minkowskian topology. An application of the effective potential to symmetry breaking is discussed. In addition, expressions for the vacuum energy density and topological mass to order $\lambda$ are obtained for a theory which is massless at the tree-graph level. One of the cases which is considered is the scalar-field version of the Casimir effect. In this case we also obtain the one-loop vacuum energy density for the massive scalar field and discuss its renormalization.