The Spectral Geometry of Reimannian Submersions

Peter B. Gilkey, John V. Leahy, Jeong Hyeong Park

We study the spectral geometry of a Riemannian submersion $\pi : Z -> Y$. We give necessary and sufficient conditions that $\pi$ preserve the eigenforms of the Laplacian. We show that if the pull-back of an eigenform is an eigenform, then the eigenvalue can only increase. If $G$ is a compact, connected Lie group with $H^1(G;R)\neq 0$, we give examples of principal $G$ bundles over homogeneous manifolds where the pull-back of an eigenform from the base is an eigenform on the total space with different eigenvalue.