The sperctral geometry of Riemannian submersions for manifolds with boundary

We study the spectral geometry of a Riemannian submersion ¥ð : Z ¡æ Y where Z and Y are compact Riemannian manifolds with smooth boundaries and where ¥ð: ¡ÓZ ¡æ ¡ÓY is also a Riemannian submersion. We impose suitable boundary conditions and give necessary and sufficient conditions that ¥ð^* preserve all the eigenforms of the Laplacian.