TITLE: Helix translates as geodesics
QUESTION [13 upvotes]: I believe one can fill $\mathbb{R}^3$ with
horizontal translates of the helix
$(\cos t, \sin t, t) \;,\; t \in \mathbb{R}$,
so that every point of $\mathbb{R}^3$
lies in exactly one helix.
I am wondering if it is possible to assign a metric to $\mathbb{R}^3$
so that these helices are all geodesics?
(I am imagining these as world lines of particles stationary in the plane.)

REPLY [6 votes]: I'm pretty sure that Nil geometry works, but I don't know a reference. I seem to remember that one may think of Nil geometry as fibering over the plane, and that geodesics connecting different points in the same fiber had projections to circles. I think then these give helices. 
A more general criterion was given by Dennis Sullivan for when a foliation may be realized as the geodesics of a Riemannian metric.