I Comparing tensors at different points

[Pencilvester]

Hello PF, I have a question about comparing tensors at different points. Carroll says, “there is no natural way to uniquely move a vector from one tangent space to another; we can always parallel-transport it, but the result depends on the path, and there is no natural choice of which path to take.” I understand that this is the general case for any two points on a curved manifold, but what if we limit ourselves with the requirement that the two points be in the same local convex neighborhood? If there’s only one unique geodesic between the two points, that seems like a pretty natural preferred path to me. So is there a reason that this method isn’t used (at least for points in the same local convex neighborhood)? Or is this method used, and I just haven’t read about it yet?

 

[Orodruin]

I understand that this is the general case for any two points on a curved manifold, but what if we limit ourselves with the requirement that the two points be in the same local convex neighborhood?

Not unless the curvature is zero in that neighbourhood.

If there’s only one unique geodesic between the two points, that seems like a pretty natural preferred path to me.

If you only had two points, that would be natural. But you do not have only two points and typically you would like comparisons to imply an equivalence relation, i.e., if $A_p \in V_p$, where $V_p$ is one of the tensor spaces at $p$, then if $A_p \sim A_q$ and $A_q \sim A_r$, then $A_p \sim A_r$. This generally does not hold if you use the geodesic between $p$ and $q$ to define the relation between $V_p$ and $V_q$, etc., simply because the geodesic from $p$ to $r$ does not need to be the composition of the geodesics from $p$ to $q$ and from $q$ to $r$.

 

[Pencilvester]

if $A_p \in V_p$, where $V_p$ is one of the tensor spaces at $p$, then if $A_p \sim A_q$ and $A_q \sim A_r$, then $A_p \sim A_r$. This generally does not hold if you use the geodesic between $p$ and $q$ to define the relation between $V_p$ and $V_q$, etc., simply because the geodesic from $p$ to $r$ does not need to be the composition of the geodesics from $p$ to $q$ and from $q$ to $r$.

Makes sense, thanks!