  A geodesic curve in flat space is a straight line. Its  tangent at one point is parallel to the tangent at previous or  subsequent points. A straight line in Euclidean space is the only  curve that parallel-transports its own tangent vector. This notion of parallel transport of the tangent vector also characterizes geodesic  curves in curved space. That is, a curve \(\gamma\) in curved  space, which parallel-transports its own tangent vector along all of  its points, is called a geodesic curve. Given a manifold  with an affine structure and some arbitrary local coordinate system,  the coordinate functions (components) \(\gamma^{i}\) of a geodesic curve \(\gamma\) satisfy the second-order non-linear  differential equations 