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Dears
I have a question abourt some calculus in an article, I found it in the internet but I didn't understood some steps. his is the link of the article;

http://arxiv.org/abs/math/0311352

In the first paragraph of page 19 the author says :

Let Σ⊂Hn+1 be an orientable (n−1)-dimensional compact submanifold in a geodesic sphere S of Hn+1 of center a in Hn+1 and geodesic radius r, and let ψ:Mn→Hn+1 be an orientable compact hypersurface with boundary Σ =ψ(∂M).Consider the vector field in Hn+1 represented in this model as Y(p) =−a−g(a,p) wher g is the metric define on H n+1.for every p∈Hn+1. Observe that Y is a conformal vector field in Hn+1 which is orthogonal to the geodesic spheres centered at the point a, with φ(p) =g(a,p)=cosh(distance(a,p)))
and|Y(p)|= sinh(distance(a,p)),for every p∈ of Hn+1.Therefore, along the sphere S we have Y= sinh̺ξ.
I don't understand why (a,p)=cosh(distance(a,p)))   and  |Y(p)|= sinh(distance(a,p)).
Please help me because I will presnet this work in my class next week and I have no idea about the detail of calculus