TITLE: Do geodesics in SL2R map to geodesics in the hyperbolic plane?
QUESTION [6 upvotes]: 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 $\mathbb{R^2}\rightarrow \mathbb{R}\times \{0\}$.

REPLY [2 votes]: 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},
}