 Eigenfunction pullback by a Riemannian submersion

For a surjective Riemannian submersion $\pi :M\to N$, let $H={\sum }_{\alpha }({\nabla }_{V_{\alpha }}{V}_{\alpha }{)}^{\mathrm{hor}}$ be the unnormalised mean curvature vector of its fibres, using an orthonormal vertical frame. With the positive Laplace-Beltrami operator, tracing the Riemannian Hessian in horizontal and vertical directions gives ${\Delta }_M(f\circ \pi )=({\Delta }_{N}f)\circ \pi +d(f\circ \pi )(H)$. Horizontal traces agree with those on the base; a vertical Hessian term is $-d(f\circ \pi )({\nabla }_{V_{\alpha }}{V}_{\alpha })$. Thus minimal fibres, in particular totally geodesic submanifolds as fibres, make pullback an injection from each base eigenspace into the corresponding total-space eigenspace. Its image is exactly the eigenfunctions constant on each whole fibre.