Christoffel symbols of a radially conformal polar metric ($d{s}^2=e^{2\chi (r)}(d{r}^2+r^{2}d{\theta }^2)$)

For the displayed metric tensor, its inverse has components $g^{rr}=e^{-2\chi }$ and $g^{\theta \theta }=e^{-2\chi }/r^2$. The nonzero second-kind Christoffel symbols are ${\Gamma }^r{}_{rr}={\chi }^{\prime }$, ${\Gamma }^r{}_{\theta \theta }=-r-r^2{\chi }^{\prime }$ and ${\Gamma }^{\theta }{}_{r\theta }={\Gamma }^{\theta }{}_{\theta r}=1/r+{\chi }^{\prime }$. They follow by differentiating the two diagonal metric entries and raising the first index of the first-kind connection coefficients. Polar coordinates require treating $r=0$ separately.