I know that in a general curved manifold it is possible to describe parametric curves. Then, the length of the curve will be a indipendent from the coordinate system used --> the metric tensor transforms properly to guarantee this.
So far so good. Now, let's take the time "t" as parameter for a generic curve on a curved manifold. If I consider two observers which are in relative motion on the manifold, their coordinate system relation will be a function of t. So the coordinate transformation depends on the parameter chosen to parametrize the curves on the manifold.
I know how to deal with this problems in 3d euclidean space --> relative kinematics, with concepts like relative velocity etc. But what if the manifold is curved? Is there a relative kinematics notion also in this scenario? In this case, where could I study deeper this concept? (I could not find this on the web)