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I am looking for a reference/proof/disproof of the following statement.

Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle A,B\rangle_e :=tr(AB^*)$. Let $pr:SL_2(\mathbb{R})\rightarrow SL_2(\mathbb{R})/SO_2(\mathbb{R})=\mathbb{H}^2$ be the canonical projection.

Is is true that $pr\circ \gamma$ is a (constant speed) geodesic in $\mathbb{H}^2$ for any geodesic $\gamma$ in $SL_2(\mathbb{R})$?

The speed might even be $0$. Another similar example, where geodesics map to constant speed geodesics would just be the orthogonal projection $\mathbb{R^2}\rightarrow \mathbb{R}\times \{0\}$.

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  • $\begingroup$ Geodesics that point in the direction of the SO(2) subgroup cannot map to geodesics. I believe that these map to the constant path. If so, do you allow the constant path as a geodesic? $\endgroup$ – Sam Nead Dec 21 '14 at 10:10
  • $\begingroup$ Tanks, Yes I would allow those. I will clarify my question now. $\endgroup$ – HenrikRüping Dec 21 '14 at 10:15
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Exactly horizontal geodesics (perpendicular to the isotropy subgroup of the point in hyperbolic space) project to geodesics. Others do not, they give rise to more general curves (called ballistic curves here).

Edit (twice)

To be specific, my remark applies to the situation that $pr:SL_2(\mathbb R)\to \mathbb H^2$ (given by acting on $i\in \mathbb H^2$) is a Riemannian submersion. Since we have a left action on $\mathbb H^2$, a suitable right invariant metric on $SL_2$ would furnish this. In such a situation one has: a geodesic in $SL_2$ which is perpendicular to the fibers of $pr$ at one point, is perpendicular everywhere and it projects to a geodesic on $\mathbb H^2$.

You look at a left invariant metric, but one which is given by an $SO_2$-invariant inner product on the Lie algebra. Inversion on $SL_2$ is an isometry from right to left invariant metric fixing the isotropy $SO_2$ of $i\in \mathbb H^2$. The right action of $SO_2$ on $SL_2$ is thus by isometries for the left invariant metric you use.

In this case, let me describe the geodesic equation on $SL_2$ (which differs just by a sign from the better known right invariant case) A smooth curve $t \mapsto g(t)\in SL_2$ is a geodesic iff $$a(t) = g(t)^{-1}.g'(t) \in \mathfrak{sl}_2$$ satisfies $$\partial_t a(t) = ad^\bot_{a(t)}a(t)$$ where the adjoint $ad_X^\bot:\mathfrak{sl}_2\to \mathfrak{sl}_2$ is with respect to the inner product $\langle\quad,\quad\rangle_e$ so that $\langle ad_X^\bot Y,Z\rangle_e = \langle Y,[X,Z]\rangle_e$.

Moreover, for $X\in \mathfrak{so}_2$, the left invariant vector field $L_X$ on $SL_2$ is the infinitesimal generator for the right $SO_2$ action; the momentum mapping (for the lifted action) implies that $\langle X, a(t)=g(t)^{-1}.g'(t)\rangle_e$ is constant in $t$ for each geodesic $g(t)$ on $SL_2$. If the constant is 0 you get horizontal geodesics, and going down to $\mathbb H^2$ is at the same time the Riemannian orbit space and symplectic reduction. If the constant is not 0, you get a conservation law for ballistic curves.

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  • $\begingroup$ Can ballistic curves be reparameterized to become geodesics? (Perhaps allowing variable speed?) $\endgroup$ – Sam Nead Dec 21 '14 at 12:37
  • $\begingroup$ No they cannot, in general. $\endgroup$ – Peter Michor Dec 21 '14 at 12:56
  • $\begingroup$ @Sam : look at the example of $S^1\times S^1=\mathbb{C}^*$ acting on $\mathbb{R^4}=\mathbb{C}^2$. $\endgroup$ – HenrikRüping Dec 21 '14 at 13:16
  • $\begingroup$ I am a bit confused. In my example, $\mathbb{H}^2$ is already the quotient and there is no isotropy. I could not find that specific example in the paper and for certain other examples, like $\mathbb{R}$ acting on $\mathbb{R}^2$ by translation, every geodesic maps to a (constant speed) geodesic and not only the ones that are perpendicular to the tangent space of the orbit $T_xGx\subset T_xM$. $\endgroup$ – HenrikRüping Dec 21 '14 at 13:27
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This is studied in the more general context of the unit tangent bundle in P. T. Nagy's 1977 paper. He shows that the geodesics have constant geodesic curvature in this setting. The most general version is discussed in a later paper

@article {MR2027641,
    AUTHOR = {Berndt, J. and Boeckx, E. and Nagy, P. T. and Vanhecke, L.},
     TITLE = {Geodesics on the unit tangent bundle},
   JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A},
  FJOURNAL = {Proceedings of the Royal Society of Edinburgh. Section A.
              Mathematics},
    VOLUME = {133},
      YEAR = {2003},
    NUMBER = {6},
     PAGES = {1209--1229},
      ISSN = {0308-2105},
     CODEN = {PEAMDU},
   MRCLASS = {53C22 (53C25 53C35)},
 MRNUMBER = {2027641 (2004k:53054)},
 MRREVIEWER = {Andrea Spiro},
       DOI = {10.1017/S0308210500002882},
       URL = {http://dx.doi.org/10.1017/S0308210500002882},
}
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