 Basic-function Laplacian identity (source code)

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= Basic-function Laplacian identity
{title2=$\Delta_M\pi^*f=\pi^*\Delta_Nf$}

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.

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