= Curvature of a two-dimensional warped metric
{title2=$ds^2=dx^2+f(x)^2dy^2,\quad K=-f''/f$}
For positive twice <differentiable> $f$, the only nonzero <Christoffel symbols> are $\Gamma^x_{yy}=-ff'$ and $\Gamma^y_{xy}=\Gamma^y_{yx}=f'/f$. In the convention $R^d{}_{abc}=\partial_b\Gamma^d_{ac}-\partial_c\Gamma^d_{ab}+\Gamma^e_{ac}\Gamma^d_{eb}-\Gamma^e_{ab}\Gamma^d_{ec}$, substitution gives $R^x{}_{yxy}=-ff''$. Dividing $R_{xyxy}$ by $g_{xx}g_{yy}-g_{xy}^2=f^2$ gives the <Gaussian curvature> $K=-f''/f$. 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.
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