Reference for the geometry of horospheres I am looking for a reference to a proof of the following well-know fact (cited for example by
B.Farb in ``Relatively hyperbolic groups'',  Geom. Funct. Anal.  8  (1998),  no. 5, 810--840); MR1650094, 
DOI:10.1007/s000390050075.
Suppose $X$ is the universal covering of a negatively curved Riemannian manifold, let $O$ be an open horoball in $X$ and let $H=\partial O$ be the horospherical boundary of $O$. 
Also suppose that $\gamma\colon [0,1]\to X\setminus O$ is a rectifiable path such that $d(\gamma (t), H)\geq k>0$ for every $t\in [0,1]$, and let $\pi\colon X\setminus O\to H$
the (well-defined) nearest-point projection. Then, there exists $\alpha>0$ (only depending on the curvature of $X$) such that the length $L(\pi\circ\gamma)$ of $\pi\circ\gamma$ is bounded above by $e^{-\alpha k} L(\gamma)$.
Of course, this fact can be reduced to the computation of the Lipschitz constant of the projection of a horosphere onto another horosphere having the same basepoint.
When $X$ is the real hyperbolic $n$-space, such a computation is very easy, and it is likely that the variable curvature case can be reduced to the hyperbolic case via some comparison theorem. However, I was wondering if there is some standard reference I could rely on.
 A: Try Geometry of horospheres by
Heintze and Im Hof.
A: I think you can find this in Chapter II.8 of Bridson and Haefliger.
A: I am now also reading Farb´s article your cited, and I would like to propose an answer of mine for your question and help that we can discuss it.
Let $\gamma$ be a geodesic such that $d(γ(t),H)≥k>0$ for every $t∈[0,1] $ where H is a horosphere. Let $p = \gamma(0), q = \gamma(1)$, then length of $\gamma$,that is $L(\gamma)$, equals to $d(p,q)$. Let $p^\prime, q^\prime$ be the projection points of p,q respectively on horosphere. Suppose that $d(p,p^\prime)=k, d(q,q^\prime) \ge k$. Then choose a point $m$ in geodesic connecting $q,q^\prime$ so that $d(q^\prime,m)=k$ and connect $m$ to p by geodesic by curve $\delta$ which constructed by push geodesic $\gamma$ along geodesic with distance k. 
By Proposition 4.1 in Farb´s``Relatively hyperbolic groups'', Geom. Funct. Anal. 8 (1998), no. 5, 810--840, $d(p^\prime,q^\prime) \le L(\delta ) \times e^{-ak}$. On the other hand, $L(\delta) \le L(\gamma) $ which can prove use the definition of the length and the fact that $\gamma$ is a geodesic.(I don´t know whether $\delta$ in this case is a geodesic, but it not affect the result).
