Consequences of Hopf-Rinow Theorem

Question

So the Hopf-Rinow theorem tells us that if we have a connected Riemannian manifold then it is equivalent saying that $M$ is geodesically complete and $(M,d)$ is a complete metric space, where $d(p,q)$ is the infimum of the lenght of all curves connecting the 2 points $p$ and $q$. My question is suppose we have $M=\mathbb{R}^n$ or for the matter any complete metric space with respect to some distance $\rho$, is there any way that we can relate the distance $d$ that is induced by the metric to the distance $\rho$ that makes the space into a complete metric space, so we might be able to conclude that it is geodesically complete? Also these 2 distances $d$ and $\rho$ can give us very different topologies on $M$ right ?

Answer 1

The issue is that a Riemannian manifold $(M,g)$ gives rise to a metric space $(M,d_g)$. Now, if you have an arbitrary distance function $d$ on a manifold $M$, compatible with its topology, it is not necessarily the case that $d = d_g$ for some Riemannian metric $g$ (e.g., one necessary condition is that $d\colon M\times M \to \Bbb R$ is smooth in a neighborhood of the diagonal $\Delta \subseteq M\times M$).

So in general, if you start with an arbitrary distance function, there is no need for this to be related to any Riemannian metric at all. Also, there are examples of two distance functions inducing the same topology on a given set with one of them complete but the other not, and we can have two complete distance functions inducing different topologies.