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

Let G be a compact Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy be the multiplication operator. The associated fibration m:GxG->G is a Riemannian submersion with totally geodesic fibers. The associated spectral geometry of the submersion is studied. Eigen functions on G pull back to eigen funtions on GxG with the same eigenvalue. Eigen p-forms for p>0 on the base pull back to eigen p-forms on GxG with finite Fourier series; there are examples where the number of eigenvalues in the Fourier series of the pull back on GxG is arbitrarily large. If w is an harmonic p-form on the base, necessary and sufficient conditions are given to ensure the pull back of w is harmonic on GxG.

This is joint work with Corey Dunn (Cal State San Bernadino USA) and JeongHyeong Park (SungKyungKwan University Korea).