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 functions 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 a 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 Bernardino USA) and
JeongHyeong Park (SungKyungKwan University Korea)