[talanum1 (OP)]

See:

at timestamp: 32:05. What he's doing there is not parallel transport, it is keeping the vector pointing in the same direction on the surface in question. Real parallel transport would bring the vector back to its starting direction if transported along a closed curve in a curved manifold.

[Kryptid]

This isn't related to your model, is it?

[Eternal Student]

Hi.

at timestamp: 32:05. What he's doing there is not parallel transport

   Well, it's a useful digram to show what is happening. 

Real parallel transport would bring the vector back to its starting direction if transported along a closed curve in a curved manifold.

   I'm not sure what you think "real parallel transport" should be.   I think the problem is that you can see the vector in 3-dimensional space,  so you can see that the vector is changing as Sean moves it around the sphere.
This was done for our benefit, you were supposed to be able to see what is happening.   To do this the 2-Dimensional surface that is just the surface of the sphere was embedded in ordinary 3-D Euclidean space in a completely natural way.   To say that more simply, Sean just drew the thing as a 3-D object.
    Now we can move that vector around on the surface on the sphere and we can keep it pointing in the same direction as a vector in the 3-D space.   However, that's just something that we can do because we have the ability to see this 3-D space and to define a vector as an ordinary vector in that 3-D space.
     You have two choices when you are trying to understand what happens in parallel transport and see why Sean's diagram was useful:

OPTION 1:   You need to assume that there are entities that live in the surface of the sphere, that's all they have, that's their entire world - we'll call them the 2-D people.   You need to take a moment to think about this....
    Zoom in to a small piece of the surface of that sphere and imagine you live in it.   A 2-D person has only 2 orthogonal Directions in which they can move, that's how it is in their entire world wherever they are in that world.  They cannot move out of the surface, they cannot float up above the surface or sink down below the surface.  There is only the surface for them.
   So vectors are not written  as    (x, y, z)  with 3 components,   the 3rd component would be meaningless.   Now in a small patch of this 2-D world, it seems like everything is perfectly flat and Euclidean.  You've probably seen this before in videos, zoom in to the surface of a sphere and just look at a little patch of it.   It looks fairly flat and you could use 2-Dimensional Euclidean geometry on it and expect it all to work fairly well.   So it would be perfectly natural for the 2-D people to assume that Euclidean geometry is the most honest, truthful geometry.  I mean that's what you're probably doing right now in the 3-D world right?  You're confident you can draw two parallel lines and they will never meet aren't you?  You've never tried it but as a concept it seems perfectly natural to you.   Euclidean geometry is what we human beings assume to hold.   So the 2-D people will naturally represent a direction as some 2-component  vector  (x,y).   
    Meanwhile, us lucky 3-D people out here can see what they're describing.  In our 3-D world they are describing the vector  (x', y', z')  where z' is some value that we lucky 3-D people can determine but the sad little 2-D people have no awareness of at all.    The people in the 2-D world aren't aware that what they call the x-axis and y-axis isn't actually fixed in direction as far as we're concerned out here in the 3-D world.   When the 2-D people are up at the North pole of their sphere, their x-axis and y-axis are perfectly aligned with what we lucky 3-D people would call our x'-axis and y'-axis.   If the 2-D people move straight to the South pole of their world then their axis will also align perfectly with ours again  (well, their axis might have switched direction as far as we're concerned, but let's not worry too much about that, their  x-y plane is the same as our  x'-y' plane at least).
     If they move to the equator then the people in the 2-D world are in a bit of trouble.  They still consider everything to be described by their own x and y axis.   While in the 3-D world we can see that there is simply no way that their two permitted directions of travel - so what they consider as their x-axis and y-axis can both be aligned with what we would consider to be our x'-axis and y'-axis.  At best one of their axis might be aligned correctly but the other one is now playing the role of our z'-axis direction.
   Let's see if we can get a diagram in here, showing how the 2-D people can move in a plane with directions that are in our x'- y' axis when they are at the north pole      compared   to how they can move when they are on the equator  and their x-y plane is actually aligned with our  y' - z' plane.


tangent-planes.jpg (52.17 kB . 1258x648 - viewed 659 times)

     Now the 2-D people will parallel transport their vectors as best they can.  The best they can do is to try and keep their x and y components of the vector constant as they move around on the sphere.
     Meanwhile us lucky 3-D people can see that they really had no hope of keeping that vector pointing in the same direction.   We lucky 3-D people can see that their x and y-axis are swinging all over the place as they move around on the sphere.   The 2-D people have no way of knowing that their idea of an x and y axis isn't really a sensible Cartesian co-ordinate system, it's not a fixed and rigid reference frame as far as we lucky 3-D people are concerned.   At best, it's an intrinsic co-ordinate system and we (3-D people) can clearly see that their axis are twisting in all sorts of directions as they move around in the surface of their sphere.

   So that's it....  that's why parallel transport breaks down and why that diagram that Sean was drawing was supposed to be useful.      This is already a long post and might take a moment to understand, so I'll end here and part a "part 2" of the discussion as another post.

Best Wishes.

[Eternal Student]

Hi again.

    The next question, or idea that most people ask is something like this:       So, the problem is caused by not defining the vector in enough dimensions.  The 2-D people were unaware that 3-Dimensions exist and space is only Euclidean when viewed in those 3 dimensions.    Well, that's progress but still not quite correct.   We can play the same game again but this time assume vectors are really supposed to be considered and defined in a 4-Dimensional space.   Only in that 4-D space is geometry Euclidean.   We lucky 3-D people were confident that our space behaves properly, it never crossed our mind that it might not but maybe we are just like the sad 2-D people, over confident.   Maybe our space ("our world") is just a curved 3-D surface embedded in a 4-D space.  There are only 3 mutually ortogonal directions we can move in   etc.etc.   but   we are  unaware that our  x, y and z -axis are actually twisting all over the place if you could only view it from 4-Dimensions.    In other words we can be the sad 3-D people while the 4-D people can really see what is happening.   There is no end to this....  maybe the 4-D people were over confident and there are 5 dimensions   etc. etc.      Eventually you should reach an appreciation that assuming Euclidean geometry really is the most correct, honest, truthful geometry  is just un-necessary.  We do not need to assume that space is Euclidean if you go to enough dimensions.... all you need to appreciate is that our 3-Dimensional space does not have to be Euclidean.     There is nothing special about Euclidean geometry and no reason why it is more turthful then any other gemoetry.  It doesn't matter if there are more dimensions or if space is Euclidean in high enough dimensions,  it doesn't matter a jot,  all that matters is just that we don't assume our 3-dimensional space is Euclidean.   
    On a side-line, it just so happens that you can always embedd a Riemannian manifold (a curved space) of n dimensions into a Euclidean manifold (a perfectly sensible Euclidean space) of  M dimensions,  where  M > n and often much larger than n.   So if you're determined then you can always view a curved space from a space of higher dimension where everything would be Euclidean.   Anyway, that's just of interest to mathematicians (see Nash embedding theorems -  https://en.wikipedia.org/wiki/Nash_embedding_theorems ).   It's not necessary to work in Euclidean space and the fact that it can always be done is a separate issue probably worthy of another thread.  Maybe there is some significance to the other dimensions required and maybe there is some good reason for Euclidean geometry to hold - but let's just push all of that to one side for the moment.  That's not what Sean was trying to demonstrate or develop.  We just need to accept that Euclidean geometry isn't guranteed to hold in our real space and there is no need to go to higher dimensions, we can just continue to work with the intrinsic geometry that exists.
   So, all that exists for us is an ability to describe something (like some velocity an object has) by a vector that you might describe with 3 numerical components  (x, y, z).   We can do our best to pick up that vector and drop it somewhere else in our world but if all we've done is just keep the same numerical values in the x,y,z components then we could have a problem.   The x, y and z axis all the way over there in our world may not be in the same direction as the x,y, z axis over here in our world.   There isn't one way to move a vector from one place to another, there are lots of ways you could do it.  Parallel transport just describes one approach and what is probably the most sensible way to try and do it.  In parallel transport you make sure that the vcetor exists and is sensibly defined in every part of the space you move through as you travel from where the vector was located to where you want to move it to.  The sad thing is that the path you take from here to there matters, in very curved space it matters a lot.

OPTION 2
    Study the mathematics, then the diagram Sean was drawing makes sense.  It's probably easier this way round.   Sean has that same diagram at the start of section 3.2 in his book (Spacetime and Geometry)  but it was no help when I first read the book.  After you've read the chapter you can go back and look at the diagram and appreciate how it was supposed to help.

Best Wishes.

[talanum1 (OP)]

This isn't related to your model, is it?

No.