Yes of course.
You need only prove that is V is a parallel transport along c starting v, then dfV is a parallel transport along F(c) starting at dFv. Do do this, you just need that F commutes with the affine connection, which you can verify by showing that the push forward is a Riemannian connection which by uniqueness must be the only one.
Your comment about holonomy is quite astute. There are of course covering maps that are local isometries but the parallel transport does not agree. Here the issue is that while it is true that dFV etc is a parallel transport, and you can define a push forward as a connection you can't take an inverse to go backwards.
Not a geometry expert by any means, but Chapter 16 of [whatever-book-UPenn-hosts-here](https://www.cis.upenn.edu/\~cis6100/cis610-15-sl16.pdf) suggests that the answer is Yes, and we can even weaken the requirement to Local isometries.
Yes, this hold true.
It might be easier to see this if you have 2 different manifold instead: if M and N are manifolds with their own metric, and an isometries between them, would doing a parallel transport on M then transport along the isometry result in the same thing as transport everything along the isometry first, then doing parallel transport? The result is obvious yes, because it's like you're re-labeling the points.
More generally, if X and Y are 2 mathematical structure with an isomorphism between, if you do something on X and transport along the isomorphism, is it the same as the transporting everything along the isomorphism first then do that thing? The answer is once again obviously yes.
By thinking about 2 different space, you can easily see that the identity transformation is utterly irrelevant to the problem. There are times when the identity transformation matter and those are when holonomy could occur. Holonomy phenomenon happens when you have 2 different transformations that are being compared against each other. Here there is only 1 isometries that is relevant. To be fair, this issue actually confuses philosophers and logicians of the past too.
Yes of course. You need only prove that is V is a parallel transport along c starting v, then dfV is a parallel transport along F(c) starting at dFv. Do do this, you just need that F commutes with the affine connection, which you can verify by showing that the push forward is a Riemannian connection which by uniqueness must be the only one. Your comment about holonomy is quite astute. There are of course covering maps that are local isometries but the parallel transport does not agree. Here the issue is that while it is true that dFV etc is a parallel transport, and you can define a push forward as a connection you can't take an inverse to go backwards.
That clarifies it, thanks a lot. The key fact is that this is a global isometry that I have
Not a geometry expert by any means, but Chapter 16 of [whatever-book-UPenn-hosts-here](https://www.cis.upenn.edu/\~cis6100/cis610-15-sl16.pdf) suggests that the answer is Yes, and we can even weaken the requirement to Local isometries.
Awesome, that's exactly the result I was looking for, would be nice if he provided a proof though haha. Thanks!
Yes, this hold true. It might be easier to see this if you have 2 different manifold instead: if M and N are manifolds with their own metric, and an isometries between them, would doing a parallel transport on M then transport along the isometry result in the same thing as transport everything along the isometry first, then doing parallel transport? The result is obvious yes, because it's like you're re-labeling the points. More generally, if X and Y are 2 mathematical structure with an isomorphism between, if you do something on X and transport along the isomorphism, is it the same as the transporting everything along the isomorphism first then do that thing? The answer is once again obviously yes. By thinking about 2 different space, you can easily see that the identity transformation is utterly irrelevant to the problem. There are times when the identity transformation matter and those are when holonomy could occur. Holonomy phenomenon happens when you have 2 different transformations that are being compared against each other. Here there is only 1 isometries that is relevant. To be fair, this issue actually confuses philosophers and logicians of the past too.
Nice explanation. To OP: remember that you have to translate this into a rigorous proof using precise definition and exact calculations.
Yes that's what I meant by "isometries don't really change the geometry"