I am trying to read an article by Gromov and Thurston in which they construct compact negatively curved manifolds with pinched curvature arbitrary closed to - 1 (http://www.ihes.fr/~gromov/PDF/7[55].pdf)
For this purpose, they write the hyperbolic metric on the hyperbolic n-space $\mathbb{H}^n$ as $dr^2+\sinh(r)^2d\theta+\cosh(r)dx^2$ where $r\geq 0$ and $\theta\in[0,2\pi]$ and $dx^2$ stands for the hyperbolic metric of a totally geodesic $\mathbb{H}^{n-2}\subset \mathbb{H}^n$.
I don't see where this come from. In fact, in a survey (http://archive.numdam.org/ARCHIVE/TSG/TSG_1985-1986__4_/TSG_1985-1986__4__101_0/TSG_1985-1986__4__101_0.pdf), Pansu name these coordinates $(r,\theta,x)$ the exponential coordinates normal at $\mathbb{H}^{n-2}$ so I thought it was link to the parametrization which maps $(r,\theta,x)$ to $\gamma(r)$ where $\gamma$ is the geodesique with $\gamma(0)=x$ and unit tangent vectors $\cos(\theta)e_1+\sin(\theta)e_2$ where $(e_1,e_2)$ is an orthonormal basis of $(T_x\mathbb{H}^{n-2})^{\perp}$. But, when I write it down, I find a tensor metric with somme $dx_idr$ terms.
If someone can help me, it would be great ! Thank you,
Karas.