$(\lambda ,\lambda )$-eigenfunctions on compact manifolds

In this note we study $(\lambda ,\mu )$-eigenfamilies on compact Riemannian manifolds when $\lambda =\mu$. We show that any compact manifold admitting a $(\lambda ,\lambda )$-eigenfunction is a mapping torus and that any $(\lambda ,\lambda )$-eigenfamily is one dimensional. Additionally, we consider generalised eigenfamilies, which can have higher dimension, and relate these to harmonic Riemannian submersions to a torus.