Eigenvalues of the form valued Laplacian for Riemannian submersions

Abstract:

Let $\pi :Z\rightarrow Y$ be a Riemannian submersion of closed manifolds. Let $\Phi _{p}$ 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 eigenvalue $\mu$. We are interested in when the eigenvalue can change. We show that $\lambda \le \mu$, so the eigenvalue can only increase; and we give some examples where $\lambda <\mu$, so the eigenvalue changes. If the horizontal distribution is integrable and if $Y$ is simply connected, then $\lambda =\mu$, so the eigenvalue does not change.