Bose-Einstein condensation as symmetry breaking in compact curved spacetimes

We examine Bose-Einstein condensation as a form of symmetry breaking in the specific model of the Einstein static universe. We show that symmetry breaking never occurs in the sense that the chemical potential μ never reaches its critical value. This leads us to some statements about spaces of finite volume in general. In an appendix we clarify the relationship between the standard statistical mechanical approaches and the field theory method using ζ functions.