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).

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