straight path on a curved surface or a Riemannian manifold
In mathematics, a geodesic is the shortest path between 2 points over a curved surface. For example, an aeroplane takes the shortest path to go from one destination to the next. However the earth is not flat, it has the shape of a sphere. Displayed on a map you would see a curved line. Paths on a sphere are also called great circles, such that in this case, two points are joined by two distinct geodesics. A minimal geodesic (or segment) is a particular case where the geodesic joining two distant points has the shortest length.