Are there unique geodesics in the NIL and SOL geometry? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:27:27Z http://mathoverflow.net/feeds/question/40453 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40453/are-there-unique-geodesics-in-the-nil-and-sol-geometry Are there unique geodesics in the NIL and SOL geometry? HenrikRüping 2010-09-29T10:53:32Z 2010-09-29T19:27:27Z <p>Is there a unique geodesics between any two points in the <a href="http://en.wikipedia.org/wiki/Geometrization_conjecture#Nil_geometry" rel="nofollow">NIL (resp. SOL) geometry</a>? 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 <a href="http://en.wikipedia.org/wiki/Hyperbolic_space" rel="nofollow">hyperboloid model</a> and the functions sinh,cosh.</p> http://mathoverflow.net/questions/40453/are-there-unique-geodesics-in-the-nil-and-sol-geometry/40495#40495 Answer by Agol for Are there unique geodesics in the NIL and SOL geometry? Agol 2010-09-29T16:54:28Z 2010-09-29T16:54:28Z <p>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. </p> <p>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.</p> <p>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. </p> http://mathoverflow.net/questions/40453/are-there-unique-geodesics-in-the-nil-and-sol-geometry/40503#40503 Answer by Sergei Ivanov for Are there unique geodesics in the NIL and SOL geometry? Sergei Ivanov 2010-09-29T17:46:22Z 2010-09-29T17:46:22Z <p>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.</p> <p>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", <a href="ftp://ftp.pdmi.ras.ru/pub/publicat/preprint/2002/05-02.ps.gz" rel="nofollow">here</a>, for the general case.</p> http://mathoverflow.net/questions/40453/are-there-unique-geodesics-in-the-nil-and-sol-geometry/40522#40522 Answer by Igor Belegradek for Are there unique geodesics in the NIL and SOL geometry? Igor Belegradek 2010-09-29T19:27:27Z 2010-09-29T19:27:27Z <p>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).</p>