What is the distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ given a metric $g_{xx},g_{xy},g_{yy}$ which depends on $x,y$?
Edit
In general, when the metric tensor. $\mathbf{g}$ is general, the problem consists of finding a geodesic passing through the two points and to work out the length of this piece of geodesic through $A$ and $B$. However, if $\mathbf{g}$ is the flat Euclidean metric, this is easy to find and we have $$ \operatorname{d}(A,B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Also when the metric is the Minkowski one, there exists a formula for the distance.
My question ha become therefore: under which hypothesis on the metric on a surface $S$ can I find a formula for the distance between two points?