= Solution
Write
$$
H_{\alpha\beta}
=\partial_\alpha\partial_\beta\omega\big|_p,
\qquad
H=\eta^{\alpha\beta}H_{\alpha\beta}.
$$
The <Levi-Civita connection> of
$$
g_{\alpha\beta}=e^{2\omega}\eta_{\alpha\beta}
$$
is
$$
\Gamma^\rho_{\alpha\beta}
=\delta^\rho_\alpha\partial_\beta\omega
+\delta^\rho_\beta\partial_\alpha\omega
-\eta_{\alpha\beta}\partial^\rho\omega.
$$
The assumed first-derivative condition makes $\Gamma^\rho_{\alpha\beta}(p)=0$. Differentiating once more gives, at $p$,
$$
\begin{aligned}
R_{\alpha\beta\gamma\delta}
=e^{2\omega}\bigl(
&\eta_{\alpha\delta}H_{\beta\gamma}
-\eta_{\alpha\gamma}H_{\beta\delta}\\
&-\eta_{\beta\delta}H_{\alpha\gamma}
+\eta_{\beta\gamma}H_{\alpha\delta}
\bigr).
\end{aligned}
$$
Its contractions are
$$
R_{\beta\delta}
=-2H_{\beta\delta}-\eta_{\beta\delta}H,
\qquad
R=-6e^{-2\omega}H.
$$
Substitution into the defining expression for $C_{\alpha\beta\gamma\delta}$ cancels every Hessian term, leaving
$$
\boxed{C_{\alpha\beta\gamma\delta}(p)=0}.
$$
Since $p$ was arbitrary, this is the <Weyl tensor of a conformally flat metric>.
Back to article page
