Eigenvalues of the formvalued Laplacian for Riemannian submersions

Let ¥ð : Z ¡æ Y be a Riemannian submersion of closed manifolds. Let  ¥Õ_p be an eigen p-form of the Laplacian on Y with eigenvalue ¥ë which pulls back to an eigen p-form of the Laplacian on Z with eigenvalue ¥ì. We are interested in when the eigenvalue can change.  We show that ¥ë¡Â¥ì, so the eigenvalue can only increase; and we give some examples where ¥ë£¼¥ì, so the eigenvalue changes. If the horizontal distribution is integrable and if Y is simply connected, then ¥ë£½¥ì, so the eigenvalue does not change.