I'm studying Riemannian Geometry from different sources and I have a problem trying to solve one of the exercises from Petersen's Riemannian Geometry:
Show, that any Riemannian Manifold $(M, g)$ admits a conformal change $(M,\lambda^2g)$, where $\lambda: M \rightarrow (0,\infty)$, such that $(M,\lambda^2g)$ is complete.
After being stuck for a certain amount of time, I found out the article by Nomizu, Katsumi, and Hideki Ozeki, "The existence of complete Riemannian metrics." But I don't really understand how the relative compactness is used here to prove the completeness.
Could somebody give me some hints to understand the proof from the article? Or is there maybe an easier way to solve this exercise?
To show a completeness, we have a claim that any closed and bounded set $C$ wrt $\lambda^2g$ is compact.
(1) $C$ is bounded wrt $g$ : If $d_\lambda$ is a distance wrt $\lambda^2 g$, then since $C$ is bounded so $B_R^\lambda (x_0)$ contains $C$. Hence for some $r$, $B_r(x_0)$ contains $C$.
(2) $C$ is closed wrt $g$ : Assume that $x_n\in C$ and $d(x_n,x)\rightarrow 0$ So $d_\lambda (x_n,x)\rightarrow 0$ Since $C$ is closed wrt $g_\lambda$ so $x\in C$.
(3) Hence $C$ is compact wrt $g$. For any sequence $x_n\in C$, we have convergence subsequence $y_n$ s.t. $d(y_n,y)\rightarrow 0,\ y\in C$ That is $d_\lambda (y_n,y)\rightarrow 0$ Since $C$ is closed wrt $\lambda^2 g$, so $y\in C$.