As it is well-known, the geodesic sphere in the $n$-dimensional sphere $\mathbb{S}^n$ can be regarded as the intersection of $\mathbb{S}^n$ centered at the origin and a circular cone $$C=\{x\in\mathbb{R}^{n+1}:\langle x,a\rangle=const, \mbox{for some constant vector}\; a\in\mathbb{R}^{n+1}\}$$ with vertex in the origin in the Euclidean space $\mathbb{R}^{n+1}$.
But why can the geodesic sphere in the hyperbolic space $\mathbb{H}^n\subset\mathbb{L}^{n+1}$ also be regarded as the intersection of $\mathbb{H}^n$ and the similar cone $$C=\{x\in\mathbb{H}^{n+1}:\langle x,a\rangle_{\mathbb{L}^{n+1}}=const, \mbox{for some constant vector}\; a\in\mathbb{R}^{n+1}\}$$
This is the final step in the paper J.M.Barbosa and A.G Colares Stability of Hypersurfaces with Constant r-Mean Curvature, Annals of Global Analysis and Geometry 15 277-297, 1997. I can't make it through in the final step in case 2 in the proof of the main theorem, any advise and help will be really appreciated.
By a linear isometry of $\mathbb{L}^{n+1}$ (hence isometry of $\mathbb{H}^n$), we may assume the centre of the geodesic sphere is $a=(1,0,\dots,0)$. The intersection of the cone $$ C=\{x\in\mathbb{L}^{n+1}:c=\langle x,a\rangle_{\mathbb{L}^{n+1}}=x_0\} $$ for $c>1$ with $\mathbb{H}^n=\{x\in\mathbb{L}^{n+1}:x_0^2-x_1^2-\dots-x_n^2=1\}$ is the set of all $(c,x_1,\dots,x_n)$ where $x_1^2+\dots+x_n^2=c^2-1$, which is precisely the geodesic sphere in $\mathbb{H}^n$ of centre $(1,0,\dots,0)$ and radius $\cosh^{-1}(c)$.