Let $C\_n = C\_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^3$. Is there a Riemannian metric $g$ on $C\_n$ such that given any two configurations in $C\_n$, there is a unique geodesic joining them?


In addition, it would be nice if $g$ was also geodesically complete, and if $g$ came from natural considerations in Physics (for instance if it is the kinetic term of some naturally occuring Lagrangian etc.).


Edit 1: I accepted Andy Putman's answer below, because it does answer negatively my question (thank you!). However, could someone please indicate whether or not there exists a complete Riemannian metric $g$ on $C\_n$? Is it more appropriate to create another post perhaps? I just found out that Nomizu and Ozeki proved that any connected smooth (second countable) manifold admits a complete Riemannian metric. This is nice. However, is there a known explicit such complete Riemannian metric $g$ on $C\_n$? If two of the points say are going towards each other and seem about to collide, there has to be a repulsive force that forbids collision (in physical terms).



Is there a Riemannian metric on the configuration space of $n$ distinct points with "nice" geodesics?
The answer is no. This relies on two things:


1. A uniquely geodesic proper metric space is contractible; see [here](https://math.stackexchange.com/questions/479022/is-a-uniquely-geodesic-space-contractible-i) for a proof.
2. $C\_n$ is not contractible. Indeed, it has many nontrivial homology groups (there is a huge literature on this).


