The spectral geometry of the canonical Riemannian submersion of a compact Lie group

Abstract

Let G be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m ( x , y ) = x y be the multiplication operator. We show the associated fibration m : G × G → G is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show that the pull-backs of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions for the pull-back of a form on the base to be harmonic on the total space.