Otdfbt

I am trying to understand geodesics and distance in hyperbolic space $H^n$ (Inverse image of $1$ under $f(x_0,x_1,dots,x_n)=-x_0^2+x_1^2+dots+x_n^2$ and with the inherited metric). More precisely, I want to show that the Riemann distance between two points $p,q$ is $d(p,q)=arccosh(-pcdot q)$.

Easy I thought, just compute the geodesics between two points $p,q$ and evaluate the length.

However, I don't really understand how to do this and would appreciate some help. How do I get a nice analytical formula which I can just "evaluate" to obtain the distance? In all the books I've read there are only very geometric arguments which don't really produce formulas and I don't really understand how to go between the two...

I will write you a sketch. First start with geometric reasoning to determine the geodesics in the hyperboloid model of hyperbolic space.

1. 
   

   Let $O(1, n)$ denote the subgroup of linear transformations of $mathbbR^n+1$ preserving the pseudo-Riemannian metric called the Lorentz metric $(cdot, cdot)$ of the . Let $O_+(1, n)$ denote the matrices $(a_ij)_i,j=0,dots, n$ in $O(1, n)$ with $a_00 > 0$. Show that
   $O_+(1, n)$ is a subgroup of $O(1, n)$. Show that elements of $O_+(1, n)$
   restrict to isometries of $H^n$.

   
   
   

   2. Determine the geodesics in the hyperboloid model of hyperbolic space:
      Let $P$ be a two-dimensional linear subspace of $mathbbR^n+1$ containing the
      point $p in H^n$. Show there is an element of $O_+(1, n)$ fixing $P$ and acting
      non-trivially on each point of $mathbbR^n+1backslash P$. Conclude that all geodesics of $H^n$ through $p$ can be obtained as intersections $P cap H^n$ for some $P$ as
      above.
      You will need to use:
      Let $(M,g)$ be a Riemannian and let $varphi:M to M$ be an isometry. Let $p in M$ such that $$varphi(p)=p$$
      and let $v in T_p M$ such that:
      $$dvarphi_p(v)=v$$
      If $gamma(t)=exp_p(tv)$ (the geodesic with velocity $v$ that start at $p$) then:
      $$varphi circ gamma(t)=gamma(t)$$
      for all $t$. You will also use the fact that geodesic in a given deirection is unique.

   
   

   3. Klein disk model of hyperbolic space:
      Let $y_1,dots, y_n$, be the standard coordinates on $D^n$. Equip $D^n$ with metric defined by
      $$g_ij =
      fracdelta_ij^2+fracy_i y_j(1-
      $$

   
   

Define a map $h : H^n_1 to D^n$ by
$$y_i=fracx_ix_0$$
Show that $h$ is an isometry.

4. Show that the geodesics of part (2) are sent by $h$ to straight lines in
   the Klein model. (The map $h$ is a stereographic projection along
   lines through the origin, to $D^n subset x_0 = 1 subset L^n+1$).




5. Now calculate the length of a straight line paramtized in arc length between the two points with the Klien metric.

This is not the fastest way to see this but it connects the geometric reasoning to a formula and you can understant the arrcosh function. Maybe now you will even be satisfied with the $arrcosh$ formula without the calculation. Because you can understand what we are measuring.