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Is there a unique geodesics between any two points in the NIL (resp. SOL) geometry? If so, is there a nice way of parametrizing them? For example geodesics in $S^3$ can be parametrized using the embedding in $\mathbb{R}^4$ and $\sin , \cos$ functions. Geodesics in hyperbolic space can be parametrized using the hyperboloid model and the functions sinh,cosh.

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Look at [Kaplan, On the geometry of groups of Heisenberg type. Bull. London Math. Soc. 15 (1983), no. 1, 35--42], as well as the papers referring to it in mathscinet. –  Igor Belegradek Sep 29 '10 at 13:31
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up vote 14 down vote accepted

The geodesics between points are not unique in both cases. Moreover the following is true: if $M$ is a universal cover of a compact Riemannian manifold whose fundamental group is virtually solvable but not virtually abelian, then there are conjugate points on some geodesics in $M$ and hence geodesics between some points are not unique.

See Croke and Schroeder "The fundamental group of compact manifolds without conjugate points", Comment. Math. Helv. 61 (1986), no. 1, 161--175, MR847526, for the case when the metric is analytic, and Lebedeva "On the fundamental group of a compact space without conjugate points", here, for the general case.

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There are not unique geodesics between points in Sol geometry. The Sol metric on $\mathbb{R}^3$ may be given as $e^{-z}dx^2+e^zdy^2+dz^2$. The claim is that there is not a unique geodesic between $(0,0,0)$ and $(t,t,0)$ for $t$ large enough.

There is a rotational isometry $(x,y,z)\overset{\varphi}{\mapsto}(y,x,-z)$. This leaves invariant the line $l=\{(t,t,0)|t\in \mathbb{R}\}$, and therefore $l$ is a geodesic. For $t$ small enough, $(t,t,0)$ will lie in a normal coordinate patch about $(0,0,0)$, so $l$ will be the unique shortest geodesic between these points.

However, for $t$ large, there are much shorter paths connecting $(0,0,0)$ and $(t,t,0)$. The length of the geodesic $l$ is linear in $t$. But one may take a piecewise geodesic path, starting as a geodesic in the hyperbolic plane $x=0$ connecting $(0,0,0)$ and $(0,t,0)$, and then a geodesic in the hyperbolic plane $y=t$ connecting $(0,t,0)$ and $(t,t,0)$. Since the path $(0,u,0), 0\leq u\leq t$ is a horocycle in the hyperbolic plane $x=0$, and similarly $(u,t,0),0\leq u\leq t$ is a horocycle in the hyperbolic plane $y=t$, the length of this piecewise geodesic path is on the order of $C\log(t)$. Since the metric space is complete, there is a minimal length geodesic connecting $(0,0,0)$ and $(t,t,0)$ which is not invariant under $\varphi$, and thus there are at least two minimal length geodesic paths (and at least three geodesics) connecting the two points.

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Despite their non-uniqueness, a lot is known about geodesics of left-invariant metrics on Heisenberg groups. For example, Jang-Park [Conjugate points on 2-step nilpotent groups, Geom. Dedicata 79 (2000), no. 1, 65--80] describe conjugate points for all geodesic passing through the idenity element of a simply-connected 2-step nilpotent Lie group with 1-dimensional center. Earlier Kaplan and Eberlein obtained explicit equations for geodesics (see references in the above paper).

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