Term for a Non-Curving Path that is Not a Geodesic

Question

Take that square and turn it into a torus. The red line segments link up to form a straight path connecting the points. However, the blue line segment is the geodesic between those points. Is there a term for paths like the red one?

Answer 1

The blue line is a geodesic connecting the two points, and is also the shortest path, but it is not the geodesic, i.e. there is not just one geodesic.

A geodesic is locally the shortest path between two of its points. I.e. at any point on the geodesic, there is some open neighborhood of that point such that the shortest path between any two points within that neighborhood is the geodesic. Thus the red line is also a geodesic.

Answer 2

This is a piecewise-geodesic curve.