Show that the Riemannian manifold $\Bbb R^2$ equipped with a scaled version of the flat Euclidean metric $g^E$ given by $g_{(x,y)} = e^yg^E_{(x,y)}$ is not a complete Riemannian manifold.
It is sufficient to prove that this manifold is not complete as a metric space, but I don't see what is the issue. At every point $(x,0)$ in the $x$-axis the metric $g$ is the same as the Euclidean metric, and when we start to go above the $x$-axis to points $(x,y)$ the metric is a scaled version of the standard Euclidean metric so the length's of the vectors become longer.
Similarly if we go below the $x$-axis to points $(x,-y)$ the vectors become shorter due to $e^{-y}$ decreasing them.
I now need to find a Cauchy sequence in this space with the property that it does not converge. I can't think of how I can utilize this scaling factor to generate a Cauchy sequence that wouldn't converge. I have the intuition that this scaling factor might let me be able to consider sequences $(x_n,y_n)$ that would not be Cauchy in the standard flat Euclidean space, but I don't know how to come up with one. Could someone give me a hint on what can we do with this scaled metric?
Consider the sequence $\{x_n\}_{n\in \Bbb N} = \{(0,-n)\}_{n\in \Bbb N}$. The path $$\gamma(t) = (1-t)x_n + tx_m = (0,-n + t(n-m)),$$ for $t\in [0,1]$ and $m > n$, joins $x_n$ to $x_m$. It satisfies $\gamma'(t) = (0,n-m)$. Hence, for $m > n$, \begin{align} d(x_n,x_m) &\leqslant \ell(\gamma) \\ &= \int_0^1 |\gamma'(t)|_g \mathrm{d}t \\ &= \int_0^1 \sqrt{g_{\gamma(t)}(\gamma'(t),\gamma'(t))}\mathrm{d}t \\ &= \int_0^1 e^{-\frac{n + t(m-n)}{2}}(m-n) \mathrm{d}t \\ &= e^{-\frac{n}{2}}(m-n)\int_0^1 e^{-t\frac{m-n}{2}} \mathrm{d}t \\ &= e^{-\frac{n}{2}}(m-n)\frac{2}{m-n}\left(1-e^{-\frac{m-n}{2}}\right) \\ &= 2(e^{-\frac{n}{2}}-e^{-\frac{m}{2}}). \end{align} Can you go on from there?