Generalized Fourier Transforms for Momentum-Space Construction on Riemannian Manifolds

TL;DR

This paper generalizes Fourier analysis to Riemannian manifolds using spectral decomposition, resolving spectral degeneracies with symmetry-adapted operators, and establishing a framework for harmonic analysis on curved spaces.

Contribution

It introduces a constructive method for defining a Generalized Fourier Transform on Riemannian manifolds, incorporating symmetry-adapted operators to resolve spectral degeneracies.

Findings

Proves a generalized Parseval-Plancherel theorem for GFT.

Provides an algorithm for generating symmetry-adapted commuting operators.

Classifies momentum label spaces based on geometric symmetry constraints.

Abstract

We extend Fourier analysis to curved spaces by defining a Generalized Fourier Transform (GFT) on any Riemannian manifold $\Sigma$ via spectral decomposition. Under minimal requirements that the transform is an isometric isomorphism and has a kernel diagonalizing the Laplace-Beltrami operator, we prove that the GFT satisfies a generalized Parseval-Plancherel theorem. To resolve the spectral degeneracy that obscures "momentum space" in such settings, we require the degenerate sector to be resolved by a local, symmetry-adapted maximal Abelian commuting set (a fiberwise MASA), constructed from geometric differential operators, most notably from Killing data when such symmetries are available. We provide a constructive algorithm for generating these commuting operators and show that the resulting momentum label spaces $\mathcal{F}$ (discrete, continuous, or mixed) reflect geometric symmetry constraints. We introduce a dual classification: (i) by MASA completeness and Stackel separability, and (ii) by the topology of $\mathcal{F}$. Finally, we distinguish unitary changes induced by true isometries (which preserve the GFT structure) from changes of coordinate-adapted degeneracy resolution/separation schemes, which may induce inequivalent $k$-space labelings (e.g. Cartesian vs spherical constructions in $\mathbb{R}^{3}$) while remaining unitarily equivalent on $\mathcal{L}^{2}\left[\Sigma\right]$. This symmetry-adapted harmonic analysis is intended as a foundation for curved-space mode decompositions; dynamical applications are developed in the subsequent work.