My book defines, the Riemannian Metric (below) for hyperbolic geometry.
Given a set of points say in R^2 defined by the Riemannian Metric, is there always a geodesic between them? Furthermore, is it unique. (My book never mentions that and I am curios to know).
C is some smooth curve joining two points in the set.
There's no need to restrict this conversation to H^2.
If a manifold with a metric (a.k.a Riemannian manifold) is connected, then any two of its points can be joined by a curve - this won't necessarily be a geodesic.
Geodesics are defined as "curves of least length", say like a great circle on a sphere. This is not so precise, as there are great circles that do not minimize length. What actually goes, is that on a geodesic, the tangent vector can be transported parrallel to itself - and this condition is given by a system of differential equations.
This explains the local nature of geodesy - Using standard ODE arguments. For given initial conditions, such as a point on the manifold and a direction on the tangent space, the system can be solved locally (i.e. there is always a geodesic between that point and the ones sufficiently close to it) and the solution is unique.
Hope this clarifies things a bit. :confused:
Ah! But what about the sphere? There is more than one geodesic between the poles.
Yes, but only one along a specified direction on the tangent plane ;)
ps. Anyhow, the solution to a first order system of ODEs is not necessarily unique, if you just specify two points the solution must cross.
Okay, whatever you say, I do not know differencial geometry.
Originally Posted by Rebesques
Hm, maybe some more explanations are in order :eek:
The equations for a geodesic lead to a system of the form ,with f differentiable - thankfuly. For a unique solution, we must specify (a point on the manifold) and (a tangent vector at this point).
ODE theory grants us a unique solution for t sufficiently small. This is our geodesic!