= Ambient restriction formula for the spherical Laplacian
{title2=$\Delta_{S^d}(F|_{S^d})=\widetilde\Delta F-F_{rr}-dF_r$}
For a smooth ambient function near the unit <sphere>, the <chain rule> along the <great circles> $\gamma_i(t)=p\cos t+e_i\sin t$ gives $(F\circ\gamma_i)''(0)=D^2F(e_i,e_i)-DF(p)$. Sum over an <orthonormal basis> of the <tangent space> and supplement it by the radial vector $p$. The <geodesic trace formula for the Laplace-Beltrami operator> then gives the displayed formula, evaluated at radius one. In particular a degree-$\ell$ <harmonic polynomial> restricts to a <spherical harmonic> with positive <eigenvalue> $\ell(\ell+d-1)$.
Back to article page
