 Solution

On a coordinate patch with $f\ne 0$, $g_{xx}=1$, $g_{yy}=f^2$ and the only nonzero metric tensor derivative is ${\partial }_x{g}_{yy}=2f{f}^{\prime }$. ThusAll remaining Christoffel symbols vanish. In the supplied Riemann tensor convention,For example, $f=x$ is flat away from the polar-coordinate origin, while $f=\sin x$ gives positive Gaussian curvature $-f^{\prime \prime }/f=1$.