Riemannian submersions which preserve the eigenforms of the Laplacian (Q1922053)

Let \(\pi: Z\to Y\) be a Riemannian submersion of closed Riemannian manifolds. Let \(E_p (\lambda, Z)\) and \(E_p (\lambda, Y)\) be the eigenspaces of the Laplacians \(\Delta^Z_p\) and \(\Delta^Y_p\) on \(p\)-forms over \(Z\) and \(Y\), respectively. We say that the pullback \(\pi^*\) preserves the eigenforms of the Laplacian \(\Delta^Y_p\) if for all \(\lambda\), there is \(\mu (\lambda)\) so \(\pi^* E_p (\lambda, Y) \subset E_p (\mu(\lambda),Z)\). The authors generalize previous work by Goldberg-Ishihara and Watson to prove: Theorem 1. The following conditions are equivalent: a) The fibers of \(\pi\) are minimal. b) We have \(\Delta^Z_0 \pi^*= \pi^* \Delta^Y_0\). c) Pullback preserves the eigenfunctions of the Laplacian \(\Delta^Y_0\). Theorem 2. The following conditions are equivalent: a) The fibers of \(\pi\) are minimal and the horizontal distribution of \(\pi\) is integrable. b) We have \(\Delta^Z_p \pi^*= \pi^* \Delta^Y_p\) for all \(0\leq p\leq \dim (Y)\). c) There exists \(p\) with \(1\leq p\leq \dim(Y)\) so that pullback preserves the eigenforms of the Laplacian \(\Delta^Y_p\).