(* Neumann eigenvalues for the flat cylinder *)
NDEigenvalues[{-Laplacian[u[x,y],{x,y}], PeriodicBoundaryCondition[u[x,y],x==0,Function[p,p+{Pi,0}]]},u[x,y],{x,y}∈Rectangle[{0,0},{Pi,Pi}],10]
(* eigenvalues of a flat torus *)
NDEigenvalues[{-Laplacian[u[x,y],{x,y}], PeriodicBoundaryCondition[u[x,y],x==0,Function[p,p+{Pi,0}]],
PeriodicBoundaryCondition[u[x,y],y==0,Function[p,p+{0,Pi}]]},u[x,y],{x,y}∈Rectangle[{0,0},{Pi,Pi}],10]
(* eigenvalues of Möbius strip *)
NDEigenvalues[{-Δ[u], PeriodicBoundaryCondition[u[x,y],x==0,Function[p,{Pi,Pi}-p]]},u[x,y],{x,y}∈Rectangle[{0,0},{Pi,Pi}],10]