 Curvature of a two-dimensional warped metric

For positive twice differentiable $f$, the only nonzero Christoffel symbols are ${\Gamma }_{yy}^x=-f{f}^{\prime }$ and ${\Gamma }_{xy}^y={\Gamma }_{yx}^y=f^{\prime }/f$. In the convention $R^d{}_{abc}={\partial }_b{\Gamma }_{ac}^d-{\partial }_c{\Gamma }_{ab}^d+{\Gamma }_{ac}^e{\Gamma }_{eb}^d-{\Gamma }_{ab}^e{\Gamma }_{ec}^d$, substitution gives $R^x{}_{yxy}=-f{f}^{\prime \prime }$. Dividing $R_{xyxy}$ by $g_{xx}{g}_{yy}-g_{xy}^2=f^2$ gives the Gaussian curvature $K=-f^{\prime \prime }/f$. Thus $f=x$ gives the flat metric in polar coordinates away from its axis, $f=\sin x$ gives unit positive curvature, and $f=\sinh x$ gives unit negative curvature.