Ambient restriction formula for the spherical Laplacian ($P_{S^d}(F{|}_{S^d})=F_{rr}+d{F}_r-{\Delta }_{\mathrm{Euc}}F$)

Let $F$ be a smooth function near the unit sphere $S^d$, and write ${\Delta }_{\mathrm{Euc}}F={\sum }_{a=1}^{d+1}{\partial }_a^{2}F$ for the ordinary ambient second-derivative trace. For the positive Laplace-Beltrami operator $P_{S^d}=-\mathrm{div}\nabla$,evaluated at radius one. The negative of this formula is the restriction identity for the nonpositive Laplace-Beltrami operator convention.

Choose an orthonormal basis $e_i$ of $T_p{S}^d$. The great circles ${\gamma }_i(t)=p\cos t+e_i\sin t$ giveThe negative sum is $P_{S^d}(F|)$; supplementing the tangent basis by the radial vector $p$ gives the displayed formula. If $F$ is a degree-$\ell$ homogeneous polynomial that is a harmonic polynomial, then $F_r=\ell F$, $F_{rr}=\ell (\ell -1)F$ and ${\Delta }_{\mathrm{Euc}}F=0$. Its restriction therefore has positive-convention eigenvalue $\ell (\ell +d-1)$, as in the spectrum of the Laplacian on a sphere.