A metric geodesic is defined as a curve along the shortest or longest possible distance between two points. Mathematically, it is defined as a curve whose length does not change with small variations that vanish at the endpoints. This stability could be a minimum distance, a maximum distance, or a point of inflection.
Mathematically, a metric geodesic is defined by the curve
For instance on the surface of a sphere the shortest possible distance between two points is always the circumference of the sphere that runs through those two points. Those points on the sphere define exactly one "line" that runs through them. This line can't be said to be straight in the Euclidean sense of the word. However, for the curved surface of the sphere it represents the shortest possible distance and is therefore a metric geodesic of that space and represents a straight path for that space. Another possible geodesic for those two points is the other part of the circumference, which would be the longest path possible.