Somebody can correct me if I’m wrong but here is how I see it:

As far as the surface is concerned (in R2 space), it is a flat square (with u and v axis instead of x and y). If you were to remap this as a square where the u and v axis were all straight like in a cartesian grid, the shortest point from A to B is the red line.

Looking at the surface in R3 space however we can see that the u and v coordinates are not straight lines and thus the shortest path in R3 space may look quite different if it were remapped into our flat, square, R2 space.

Another example is a map of the earth. If you look at a typical, Mercator projection map of the earth (the earth in R2 space), you may think that the shortest path for a plane to take from Sydney to Buenos Aires is to travel Eastwards. If you look at it in R3 space however you can easily see that the shortest route is to go south, almost over Antartica:

So, whilst the shortest route still lies in the surface domain (we dont tunnel through the earth!), the path looks quite different in R3 space than it does in R2 space.