= Eigenfunction pullback by a Riemannian submersion
{title2=$\Delta_M\pi^*f=\pi^*\Delta_Nf+df(d\pi H)$}
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 $\Delta_M(f\circ\pi)=(\Delta_Nf)\circ\pi+d(f\circ\pi)(H)$. 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.
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