Punctured circular cone
= Punctured circular cone
The punctured circular cone
$$
S=\{(r\cos\theta,r\sin\theta,ar):r>0\},
\qquad a>0,
$$
is <geodesically incomplete> because a generator reaches the missing vertex in finite time. It is an <inextendible embedded surface>: any proper connected embedded extension would have to add the vertex, but its tangent planes have different limits as $\theta$ varies. Nevertheless its <Gaussian curvature of a cone away from its vertex> is identically zero.