The Möbius and Klein Transforms: Spectral Analysis on Non-Orientable Manifolds

The classical Fourier transform is the harmonic analysis of the circle (and torus) – an orientable manifold. In analogy with the Fourier transform, we define two named, closed-form, unitary transform pairs on the two simplest non-orientable manifolds, the Möbius strip (the Möbius transform, MT) and the Klein bottle (the Klein transform, KT). The guiding principle is that the topological twist is carried not by a phase factor pasted onto the basis functions but by the constraint defining the function space – antiperiodicity for the Möbius strip, deck-transformation invariance for the Klein bottle – from which the orthogonal basis is derived. We give the continuous and discrete transform pairs, prove orthogonality, completeness and the Parseval identity, show that the discrete versions are implementable via the FFT, and extend the construction to differential forms and vector and tensor fields by introducing twisted sectors. Numerical checks agree at machine precision. The construction carries over to quantum mechanics on the twisted space, to spinor fields via Pin lifts (the P^2=-1 class forcing a quarter-integer lattice), and to a flat three-dimensional quotient, where one parity rule fixes every tensor slice and the curl splits rather than switches sector. The Casimir energy of a vector field is the sum over its components' sectors, not a multiple of the scalar value — in two dimensions in closed form, in three numerically with a closed-form sector gap: the sector rule lowers the magnitude in both two and three dimensions, and more steeply in two (Section 9.7). The four Pin structures differ even in sign. The bounds are stated explicitly: there is no global convolution theorem, and the discrete one-dimensional transform is numerically the half-shift GDFT — what is added is the reading of that lattice as bundle structure, not a new algorithm. Two closing sections link the framework to the Laplace transform, where the denominator 1+e^-sT of an antiperiodic image puts the candidate poles on the half-integer MKT lattice — a connection rather than new mathematics — and to space-reflecting time translations, where an odd spatial profile enforces a subharmonic response: the known selection rule of temporal glide symmetry.