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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.) |
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12
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Consider the following reparametrization of $\mathbb R^3$: $$(x,y,z)\mapsto (x-\cos z,y-\sin z,z).$$ Note that the horizontal translations of your helix go to the vertical lines. So the pullback of the canonical metric on $\mathbb R^3$ is the metric you want. |
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5
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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. |
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