Tolman wormhole ($d{s}^2=(1+\lambda /r^2{)}^2{\delta }_{ij}d{x}^{i}d{x}^j$)

For $\lambda >0$, this four-dimensional Euclidean metric has two asymptotically flat ends. Inversion $\rho =\lambda /r$ puts the metric near the $r=0$ end into the same form with $\rho \to \infty$. On $r>0$, its Ricci scalar vanishes because $\Delta (r^{-2})=0$, but its Ricci tensor is $R_{ij}=4\lambda ({\delta }_{ij}-4x_i{x}_j/r^2)/(r^2+\lambda {)}^2$. It therefore does not satisfy the Vacuum Einstein equations unless $\lambda =0$. For negative $\lambda$ the conformal factor vanishes at a finite radius, so that metric is not a smooth wormhole across that sphere.