EIGENVALUES OF THE FORM VALUED LAPLACIAN FOR RIEMANNIAN SUBMERSIONS

Let pi : Z --> Y be a Riemannian submersion of closed manifolds. Let P hi(p), be an eigen p-form of the Laplacian on Y with eigenvalue lambda which pulls back to an eigen p-form of the Laplacian on Z with eigenv alue mu. We are interested in when the eigenvalue can change. We show that lambda less than or equal to mu, so the eigenvalue can only incre ase; and we give some examples where lambda < mu, so the eigenvalue ch anges. If the horizontal distribution is integrable and if Y is simply connected, then lambda = mu, so the eigenvalue does not change.