A generalized Fourier transform is defined on any Riemannian manifold that satisfies a Parseval-Plancherel theorem and constructs unique momentum-space labels by resolving degeneracy with fiberwise maximal Abelian commuting sets from geometric symmetries.
Then: (1) ˆUφ =φ ∗ is unitary on L2 (Σ,dµ Σ ), (2) ˆUφ =φ ∗ commute with the Laplace-Beltrami operator: φ ∗ △ = △ φ ∗
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Generalized Fourier Transforms for Momentum-Space Construction on Riemannian Manifolds
A generalized Fourier transform is defined on any Riemannian manifold that satisfies a Parseval-Plancherel theorem and constructs unique momentum-space labels by resolving degeneracy with fiberwise maximal Abelian commuting sets from geometric symmetries.