Suppose I live in a 2d curved space. I can describe intrinsically the motion of a particle which moves in my 2d manifold using time "t" as a parameter.
Now suppose there is a second observer (let's call him Tom), who is also moving along a curve relative to me. I can describe also his motion intrinsically, using again the time "t" as parameter.
Now my question is... Can I (intrinsically) derive the parametric equation of motion that Tom would assign to the particle from his point of view?