From metric tensors to distances

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?

• The metric $d(p, q)$ is just the infimum of the set of lengths of paths connecting $p, q$---it's an instructive exercise to verify that this really defines a metric. Of course, it's not practical to consider all possible paths, so to compute a distance one needs to use additional observations, but what facts are available depends very much on the setting. For that reason, I suggest including a concrete example. – Travis Willse Mar 21 at 21:34
• @Travis Thanks! I edited the question with a concrete example. – user25004 Mar 21 at 23:51
• Possible duplicate of Distance in the Poincare Disk model of hyperbolic geometry – Arctic Char Mar 22 at 5:21

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.