non complete metric on $\mathbb{R}^2$

Question

I'm looking for a riemannian metric on $\mathbb{R}^2$ which is not complete.

In this forum post second one someone gave an example of a metric $d$ on $\mathbb R$ such that $(\mathbb R,d)$ is not complete. I tried to generalize this result on $\mathbb R^2$ and find the riemannian metric $g$ of $d$, but it didn't work.

Does someone have another idea how to find one or how to use the result in the forum post to construct one.

Answer 1

Map $\mathbb R^2$ to the sphere minus a point, using stereographic projection, and use the metric induced by the standard metric on the sphere. Then sequences that go off to infinity (in any direction) are Cauchy sequences, since on the sphere they converge to the projection point; but there's no corresponding point in $\mathbb R^2$ for them to converge to.

Answer 2

$f:(\{ x| |x|<1\},\|\ \|_2) \rightarrow \mathbb{R}^2$ by $ f(x)=\frac{x}{1-|x|}$. Here $f$ gives a metric on $\mathbb{R}^2$.