Is there a Riemannian metric on the configuration space of $n$ distinct points with "nice" geodesics? 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).
 A: This is just a long comment, and a pretty speculative one at that. However it might perhaps be of interest to you since:

*

*there is a natural connection to physics,

*the construction only works in three dimensions,

*the construction is equivariant with respect to the action of the permutations.

I have in mind the (conjectured) map described by Atiyah in [1] which maps configurations of points to the complex flag manifold:
$$
  C_n(\mathbb{R}^3) \to U(n) / T^n.
$$
Since the flag manifold is homogeneous, this map would provide a metric on $C_n(\mathbb{R}^3)$ if we could spot a natural metric on the fibres. I don't know if this is possible but for $n=2$, the fibres of the map are pairs of distinct points defining the same direction (first point looking at second point) and so are naturally parameterised by their midpoint $m$ and distance apart $t$. It's a bit of a stretch but if we give this fibre the metric of $dm^2 + (dt/t)^2$ then we get something you might regard as "nice".
[1] Atiyah, M., "Configurations of Points", R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001), no. 1784, 1375-1387.
A: The answer is no.  This relies on two things:


*

*A uniquely geodesic proper metric space is contractible; see here for a proof.

*$C_n$ is not contractible.  Indeed, it has many nontrivial homology groups (there is a huge literature on this).
