= Ambient restriction formula for the spherical Laplacian
{title2=$P_{S^d}(F|_{S^d})=F_{rr}+dF_r-\Delta_{\mathrm{Euc}}F$}
Let $F$ be a smooth <function> near the unit <sphere> $S^d$, and write $\Delta_{\mathrm{Euc}}F=\sum_{a=1}^{d+1}\partial_a^2F$ for the ordinary ambient second-derivative <trace>. For the <positive Laplace-Beltrami operator> $P_{S^d}=-\operatorname{div}\nabla$,
$$
P_{S^d}(F|_{S^d})=
F_{rr}+dF_r-\Delta_{\mathrm{Euc}}F,
$$
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 $T_pS^d$. The <great circles> $\gamma_i(t)=p\cos t+e_i\sin t$ give
$$
(F\circ\gamma_i)''(0)=D^2F(e_i,e_i)-DF(p).
$$
The 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>.
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