I've read several books with this content:
Let $g = dr^2 + f^2(r)d\theta^2$ be a smooth metric on $\Bbb R^2$ expressed in polar coordinates. This metric is complete and the volume can be finite even though this manifold is not-compact.
I thought that $\Bbb R^2$ always have infinite volume.
Question: Why authors emphasize that this metric is complete while we know that $\Bbb R^2$ is complete topological space. Is there any non-complete metric on $\Bbb R^2$?
There exists a diffeomorphism from $\Bbb R^2$ to the open unit disc. One such function, given in polar form, is $(r,\theta)\mapsto(\tanh r,\theta)$. You can replace $\tanh$ with your favorite sigmoid function. Give the open unit disc the standard metric, and then give $\Bbb R^2$ the metric it inherits from the diffeomorphism. This metric is not complete.
This doesn't contradict the author's statement, because metrics of the form $dr^2+f^2(r )d\theta^2$ do not change the lengths of rays from the origin, and so the metric I described above is not of this form.