Comparing vectors at different points in general relativity In special relativity we can compare vectors at different points. For example if $e_\mu$ is a four vector at a point $p$ and $e'_\mu$ is a four vector at a point $p'$, we can parallel transport the vector $e_\mu$ to the point  $p'$ and compare it with the vector $e'_\mu$, and from this comparation extract useful information such as the relative velocity between to observers.
Mean while in this book Theoretical physics   the author claims

to compare two bases located at different points one does not jump, one follows a path and the path matters : the general relativist universe is not isotropic

But he does not give further details. My question is why in curved spacetime, can we not compare vectors at different points and to obtain useful information, such as the velocity of one observer in relation to the other?
My guess is because between two points there can be is more than one geodesic and parallel transport is path dependent.
 A: 
My question why in curved spacetime, we can not compare vectors at different points and to obtain useful information, such as the velocity of one observer in relation to the other?

The issue is that in order to compare two vectors, as you mention in the question, you must parallel transport one vector onto the other. The problem is that parallel transport is path-dependent.
Say that you want to compare two vectors on a sphere. They are both at the equator, one at longitude 0 degrees and the other at longitude 90 degrees west, both pointing due north. If we parallel transport the west one due east along the equator then we will find that the vectors are equal. But if we parallel transport it due north to the North Pole and then due south along the prime meridian to the other vector, then we will find that the vectors are 90 degrees apart.
So the only time that parallel transport is unambiguous is if you specify the path. For example you can calculate the four-momentum of a pulse of light by parallel transporting it along its worldline. But if you are just comparing two distant vectors then there is no clear path on which to parallel transport.
