Complete revolution surface I was reading the Delaunay theorem in which is mentioned the term "complete surface of revolution", and I has wondering what does it mean? Is it a surface of revolution with no edges? Or it has another meaning? Thank you
 A: After searching a bit for the precise theorem you're mentioning, I have confirmed Ivo Terek's comment is indeed the right answer. "Complete" in this context means "geodesically complete": i.e, every geodesic of the surface is defined on all of $\mathbb{R}$. Equivalently (by the Hopf-Rinow theorem), the surface is complete as a metric space (with the metric induced by the Riemannian distance function, which is determined by the Riemannian metric). Yet another characterization would be saying the exponential map of your surface is defined on the whole tangent bundle (or saying that the exponential map at any point is defined on the whole tangent space of that point). Informally, being complete means the surface is whole, i.e it's not a piece of any other surface.
For concrete examples, you can take the cylinder (which is a surface of revolution which is complete because its geodesics are circles and lines), the catenoid (which is complete by Hopf-Rinow because it's closed - as it's diffeomorphic to $\mathbb{S}^1 \times C$, where $C$ is the catenary curve - and $\mathbb{R}^3$ is complete). The pseudo-sphere is an example of a surface of revolution which is not complete.
