When working out the question in Do Carmo, I encounter a problem to show a given Riemannian manifold is incomplete. The given metric is $$g_{ij}=\frac{\delta_{ij}}{(1+\frac{K}{4}\sum x_i^2)^2}$$ defined on the Euclidean space $\mathbb{R}^n$. Where $K>0$ is a constant.
I have no idea of the approaches to this question. It seems to be a HUGE work to calculate the expression of the geodesics on this Riemannian space. Furthermore, I am wandering whether there is a proof of the completeness in Euclidean space with a arbitrary given metric. Any suggestion or hint will be appreciated.
I have just got the answer for this particular metric, which is obviously isometric to the incomplete metric on sphere $\mathbb{S}^n\setminus{N}$ removing North Pole $N$ induced by stereographic projection with positive constant curvature $K$.