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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.

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I do not know of a reference, but maybe the problem can be reduced to studying the problem on the sphere $S^2$ and on the torus $T$ and then looking for solutions with certain symmetries. For instance, if $f$ is a function on $RP^2$ then it comes from a function $g$ on $S^2$ such that $g(p) = g(-p)$. Solve $\nabla^{2} u = f$ on $S^2$, then I think that $u' = (u(p) + u(-p))/2$ is a solution of your problem on $RP^2$.

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