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.)