Spectral geometry, homogeneous spaces, and differential forms with finite Fourier series

TL;DR

This paper establishes a characterization of differential forms with finite Fourier series on Riemannian manifolds under group actions, linking their properties via G-equivariant Riemannian submersions.

Contribution

It provides a necessary and sufficient condition for differential forms to have finite Fourier series under G-equivariant Riemannian submersions.

Findings

Finite Fourier series on N correspond to finite Fourier series on M via pullback.

The result connects spectral properties of differential forms with group actions and submersions.

Advances understanding of spectral geometry in homogeneous spaces.

Abstract

Let G be a compact Lie group acting transitively on Riemannian manifolds M and N. Let p be a G equivariant Riemannian submersion from M to N. We show that a smooth differential form on N has finite Fourier series if and only if the pull back has finite Fourier series on M