A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator).

In general relativity, a geodesic generalizes the notion of a “straight line” to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

2 Answers. The short answer is no. In practice, for a generic metric space one must make specific requiriments (like: goedesics exist, geodesics are unique and so on…) You can build many counterexamples to the local uniqueness of geodesic by simple means.

…the shortest natural paths, or geodesics—much as the shortest path between any two points on Earth is not a straight line, which cannot be constructed on that curved surface, but the arc of a great circle route.

Of course geodesic does not always exist (as explained by Igor) and may be is not unique (as explained by Narasimham) or/and the second procedure may converge to a geodesic which is not a minimal one.

1. The procedure for solving the geodesic equations is best illustrated with a fairly. simple example: finding the geodesics on a plane, using polar coordinates to.
2. First, the metric for the plane in polar coordinates is. ds2 = dr2 + r2dφ2.
3. Then the distance along a curve between A and B is given by. S =

A triangle formed by the arcs of three geodesics on a smooth surface. SEE ALSO: Integral Curvature, Spherical Triangle.