Given a manifold and its metric tensor, how can I compute the distance between two points on the manifold?
What are the high level steps?
Edit: In particular, suppose the manifold is an open unit ball in $R^d$ $$B = \{ x \in \mathbb R ^d: |x| < 1 \},$$ and the metric tensor is $$\frac{2}{(1-\|x\|^2)^2} g_E$$ where $x \in B$ and $g_E$ is the Euclidean metric tensor. How should one compute the distance between two points on the the manifold?
If you have a curve on the manifold (say two-dimensional) $u_i=u_i(t)$, between $t=t_1$ and $t=t_2$, and your metric tensor is $g_{ij}$ (covariant components), then the length of the curve is given by $$ \int\limits_{t_1}^{t_2}\sqrt{g_{ij}(u(t))\partial_tu^i\partial_tu^j}\,dt. $$ If you choose your curve to be a geodesic, then you get the distance.