Conformal transformation of metric on $\mathbb{R}^n$

Question

Let us define the following metric on $\mathbb{R}^n$: $$ g|_v(X, Y) := e^{-|v|^2} \langle X, Y\rangle,$$ where the brackets denote the standard scalar product.

How does the resulting manifold look like? If $n=2$, can it be isometrically embedded in $3$-space?

It has finite volume, but is it geodesically complete?

Answer 1

It is not geodesically complete.

Any ray that begins at the origin is clearly the image of a geodesic. Since each one of those rays has a finite length with respect to $g$, geodesics are not defined at infinite time.