 Incomplete surface z equals r to three halves

The surface of revolutionis geodesically incomplete: a meridian reaches the missing origin in finite length. Its Gaussian curvature, computed using the curvatures of a parametrized surface of revolution, iswhich tends to infinity as $r\downarrow 0$. 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.