= Solution
Work on a regular patch with a smooth nonzero conformal factor. Its sign is constant there, so replace it by $|\Omega|$ if necessary and assume $\Omega>0$. Put $u=\log\Omega$, and let all partial derivatives, index raising on $u_i,\Omega_i$, and the operators $\Delta,|\nabla\cdot|^2$ in the following formulas use the flat Cartesian metric $\delta$. The <orthonormal coframe> $E^i=\Omega dx^i$ has
$$
dE^i=\Omega^{-2}\Omega_k E^k\wedge E^i.
$$
The proposed <connection 1 forms> reduce to
$$
\boxed{\omega^i{}_j=\Omega^{-2}(\Omega_jE^i-\Omega_iE^j)=u_jdx^i-u_idx^j.}
$$
Lowering the first frame index with $\delta$ makes these antisymmetric, which establishes <metric compatibility>. Moreover,
$$
\omega^i{}_j\wedge E^j=\Omega^{-2}\Omega_jE^i\wedge E^j=-dE^i.
$$
Thus <Cartan's first structure equation> has zero torsion. Uniqueness of the metric-compatible <torsion-free connection> identifies these forms as the <Levi-Civita connection>; this also explains why merely guessing a skew connection would not have sufficed.
To calculate the <Riemann curvature tensor>, use <Cartan's second structure equation>. Direct differentiation and multiplication give
$$
\begin{aligned}
d\omega^i{}_j&=u_{jk}dx^k\wedge dx^i-u_{ik}dx^k\wedge dx^j,\\
\omega^i{}_k\wedge\omega^k{}_j
&=u_j u_kdx^i\wedge dx^k-|\nabla u|^2dx^i\wedge dx^j+u_i u_kdx^k\wedge dx^j.
\end{aligned}
$$
Introduce the <Schouten tensor> in Cartesian coordinate components,
$$
S_{ij}=-u_{ij}+u_i u_j-\frac12|\nabla u|^2\delta_{ij}
=-\frac{\Omega_{ij}}\Omega+\frac{2\Omega_i\Omega_j}{\Omega^2}-\frac{|\nabla\Omega|^2}{2\Omega^2}\delta_{ij}.
$$
The preceding calculation groups into the <curvature 2-forms>
$$
\mathcal R^i{}_j=S_{jl}\,dx^i\wedge dx^l+S_{ik}\,dx^k\wedge dx^j.
$$
Keep the convention $R_{ijkl}=g(\partial_i,R(\partial_k,\partial_l)\partial_j)$, with $R(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$, as in the first solution. Converting frame indices to coordinate indices gives the <curvature of a conformally Euclidean metric>:
$$
\boxed{R_{ijkl}=\Omega^2(\delta_{ik}S_{jl}+\delta_{jl}S_{ik}-\delta_{il}S_{jk}-\delta_{jk}S_{il}).}
$$
These are coordinate components. Components in the dual <orthonormal frame> are $\Omega^{-4}R_{ijkl}$, hence the same expression with the overall factor $\Omega^{-2}$. This distinction is important when contracting the <Riemann curvature tensor>.
Contract with $g^{ik}=\Omega^{-2}\delta^{ik}$. In four dimensions the <Ricci tensor> becomes $R_{jl}=2S_{jl}+\delta_{jl}\sum_kS_{kk}$, and therefore
$$
\boxed{R_{ij}=-2u_{ij}+2u_i u_j-\delta_{ij}\bigl(\Delta u+2|\nabla u|^2\bigr)
=-\frac{2\Omega_{ij}}\Omega+\frac{4\Omega_i\Omega_j}{\Omega^2}-\delta_{ij}\left(\frac{\Delta\Omega}\Omega+\frac{|\nabla\Omega|^2}{\Omega^2}\right).}
$$
Taking the remaining metric trace yields the <Ricci scalar>
$$
\boxed{R=-6\Omega^{-2}\bigl(\Delta u+|\nabla u|^2\bigr)=-6\Omega^{-3}\Delta\Omega.}
$$
The cancellation of the gradient-squared terms in the last expression is special to dimension four.
For the <Tolman wormhole>, compute on $r>0$:
$$
\Omega_i=-\frac{2\lambda x_i}{r^4},\qquad
\Omega_{ij}=-\frac{2\lambda\delta_{ij}}{r^4}+\frac{8\lambda x_ix_j}{r^6},\qquad
\Delta\Omega=0,\qquad |\nabla\Omega|^2=\frac{4\lambda^2}{r^6}.
$$
Substitution into the <Ricci tensor> and <Ricci scalar> gives
$$
\boxed{R=0,\qquad R_{ij}=\frac{4\lambda}{(r^2+\lambda)^2}\left(\delta_{ij}-\frac{4x_ix_j}{r^2}\right).}
$$
For example, the coordinate radial eigenvalue is $-12\lambda/(r^2+\lambda)^2$, while each of the three tangential eigenvalues is $4\lambda/(r^2+\lambda)^2$. Their sum vanishes, but none vanishes when $\lambda\ne0$. The <Vacuum Einstein equations> with zero <cosmological constant> require $R_{ij}-\tfrac12Rg_{ij}=0$ and hence $R_{ij}=0$. \b[The metric is scalar-flat but not a vacuum solution for any nonzero $\lambda$; only $\lambda=0$ gives flat vacuum space.] A nonzero cosmological constant does not rescue it: its trace would force $R=4\Lambda$, so $R=0$ would first force $\Lambda=0$.
The smooth wormhole interpretation uses $\lambda>0$; $r=0$ is another asymptotically flat end, not an included origin, as inversion $\rho=\lambda/r$ shows. For $\lambda<0$, the conformal factor vanishes at $r=\sqrt{-\lambda}$, where the metric is degenerate. The local non-vacuum calculation remains valid on each regular region away from that sphere, but such a metric is not a smooth wormhole across it.
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