Basic-function Laplacian identity (${\Delta }_M{\pi }^{\ast }f={\pi }^{\ast }{\Delta }_{N}f$)

For a Riemannian submersion with totally geodesic submanifolds as fibers, the positive Laplace-Beltrami operator commutes with pullback of functions. Horizontal terms in the Riemannian Hessian are pulled back from the base, and vertical terms vanish. Minimal fibers suffice because only the trace of the vertical second fundamental form enters.