I would appreciate a reference describing the analysis of PDEs on the Klein bottle and the real projective plane. As an example, is there a reference discussing the existence and uniqueness of the solution of the Poisson equation $\nabla^2 u = f$ on either of these? I would prefer to avoid embedding in a higher dimensional space, if possible.
More specifically, I am interested in a `flat rectangular region having the topology of' a Klein bottle or of the real projective plane. I lifted this language from the answer to a question about wave equations on a Mobius strip (see below) because it is clearer than my original question.