My intuition tells me that the shortest distance between two points on the surface corresponds to a line segment joining the two points on the map of said surface, because, the path on the surface is same as the shortest path in the map. However, this turns out to be wrong.
Take for instance, the Beltrami-Poincare half-plane model of $\mathbb{H}^2$, the shortest path between two points seems to be an arc of a semi circle centered at somewhere on the horizon. Picture:
Why is the shortest distance not a straight line in the map here?
Probably I am missing something quite basic, but I just can't seem to figure it out.
Best Answer
Here's one way to think of the issue: Suppose $$ ds^{2} = \lambda(x, y)^{2} (dx^{2} + dy^{2}) $$ for some smooth, positive function $\lambda$. Given two points $p$ and $q$, a path from $p$ to $q$ may "shorten itself" by veering off the Euclidean segment in order to travel in a region where $\lambda$ is smaller.
In the hyperbolic plane, $\lambda(x, y) = 1/y$, which grows rapidly as $y \to 0^{+}$ and which drops off as $y$ grows. As you know, the shortest paths are arcs of circles meeting the boundary $y = 0$ at right angles. Conceptually, these arcs of circles "climb just enough" into a region of larger $y$ to minimize their length. (That's not a proof, of course, just a conceptual interpretation. You're invited to sharpen the interpretation, e.g., explaining why circles meet the boundary at right angles, or why if $y_{0}$ is large the geodesic connecting two points $p = (x_{0}, y_{0})$ and $q = (x_{0} + 1, y_{0})$ is nearly a Euclidean segment.)
More vividly, we could construct a function $\lambda$ that is close to zero in some disk and very larger elsewhere. In the resulting conformal metric, a path might deviate considerably from a Euclidean segment in order to pass through the disk where $\lambda$ is small.