= Incomplete surface z equals r to three halves
{title2=$z=r^{3/2}$}
The <surface of revolution>
$$
S=\{(r\cos\theta,r\sin\theta,r^{3/2}):r>0\}
$$
is <geodesically incomplete>: a meridian reaches the missing origin in finite length. Its Gaussian curvature, computed using the <curvatures of a parametrized surface of revolution>, is
$$
K(r)=\frac{9}{8r(1+9r/4)^2},
$$
which tends to infinity as $r\downarrow0$. Every finite-time endpoint of a constant-speed geodesic must approach the origin, the only finite point in the ambient closure missing from $S$, so this surface satisfies the curvature-blowup criterion.
Back to article page
