Dynamical Systems On Generalised Klein Bottles

The paper defines K(k1,k2,B) as a quotient of R^{k1+k2} by the free action of G = Z^{k1} ⋊_φ Z^{k2} with (a,b)·(x,y) = (φ(b)x + a, y + b), proves the action is free and a covering action, and identifies the quotient with the cube [0,1]^{k1+k2} with the stated face identifications (translations in x, 2-periodicity in y, and a flip of those x_i for which B_{i,j}=1 when y_j increases by 1) . For scalar fields, the paper factors the quotient via T = R^{k1+k2}/(Z^{k1}×(2Z)^{k2}) and a free (Z_2)^{k2}-action on T, characterizing descended functions as the kernel of an averaging operator L and deriving the Fourier coefficient constraint (L⋆c)(λ,ζ) = c(λ,ζ) − 2^{-k2}∑_β (−1)^{ζ·β} c(φ(β)^Tλ,ζ) . In the standard Klein case K(1,1), this yields parity constraints giving cosine modes for even ζ and sine modes for odd ζ . For vector fields, the paper shows descent is equivalent to the equivariance V(β·z)=A(β)V(z) with A(β)=φ(β)⊕I, and again uses averaging to get Fourier constraints; in K(1,1) the X-component takes sine (even ζ) and cosine (odd ζ), while Y follows the scalar-field parities . The candidate solution reproduces precisely these constructions: the same group action and fundamental domain, the same torus cover and (Z_2)^{k2}-averaging projector for scalars and vectors, the same Fourier constraints with φ(β)^T acting on λ, and the same K(1,1) parity bases. Minor differences are expository (e.g., noting irreducibility of B is not needed for manifoldness, which aligns with the paper’s use of irreducibility only to avoid product decompositions).