Spectral-Geometric Deformations of Function Algebras on Manifolds

TL;DR

This paper introduces a novel intrinsic spectral deformation of smooth function algebras on compact Riemannian manifolds, connecting classical deformation frameworks with spectral and geometric methods.

Contribution

It develops a spectral-based deformation method using Laplace eigenvalues, extending classical group action deformations to a broader geometric setting.

Findings

Deformation constructed via Laplace spectral decomposition.

Compatibility conditions for complex conjugation and Sobolev extensions established.

Connections made with classical deformation frameworks like Rieffel and Connes-Landi.

Abstract

We introduce an intrinsic deformation of the algebra of smooth functions on a compact Riemannian manifold using only the Laplace spectral decomposition. The construction twists the canonical multiplication-projection channels by unimodular phases, producing a well-defined bilinear product on the finite spectral core with values in $L^2(M)$. We give a simple condition for compatibility with complex conjugation and isolate a Sobolev boundedness hypothesis under which the product extends to a Sobolev algebra and admits iteration; in that setting, associativity is equivalent to an explicit identity for the twisted spectral channels. We analyze gauge and coboundary aspects for scalar twists and obtain rigidity statements in the action-free regime. We also compare with classical strict deformation frameworks arising from actions of locally compact abelian groups -- Rieffel's deformation for $\mathbb{R}^d$-actions, Connes-Landi's torus isospectral deformations, and Kasprzak's cocycle deformation via Landstad theory -- showing that, when the relevant abelian group action has a discrete spectral decomposition (in particular, in the compact abelian/periodic case where the algebra decomposes into homogeneous subspaces indexed by characters of the acting group), their deformed products are recovered uniformly as refined instances of our channel twist. Finally, we formulate a grading-based obstruction and classification for graded scalar twists.