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I'm watching this video (Frederic Schuller) and, at timestamp 9:50 have become confused about fiber bundles: https://youtu.be/UbQS40KHkH0?t=587

He says you can imagine 'turning the tangent vectors around' such that they are all aligned, as pictured second here:

He had immediately previously referred to the vertical lines in a similar diagram as being fibers that had vertical extent / 'length' to represent being composed of multiple elements that each mapped to an element of the base space (above pictured as the blue circle).

I would have thought that, in the case of the tangent vectors to a circle, each tangent vector would be an individual element of the fiber representing its tangent space, such that there would be only one (or two, due to reversing vector direction) element(s) in each fiber, and there would be no continuous-seeming vertical extent along which to pick out distinct elements, as he did in the video. I feel like there's a small chance he's implying different positions on the same tangent vector correspond to different elements in the fiber at that point, and just want to verify that that isn't the case?

I now have two mental images of what a fiber bundle is. Which of these, or neither, is right?

I'm inclined to think the first, because it makes more sense inherently and extends well to a sphere:

If it's obvious from my diagrams/descriptions that I've misunderstood anything else, I'd be grateful to be told as much!

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  • $\begingroup$ Has just occurred to me that, though I encountered this in a physics context, maybe it should be on the mathematics StackExchange? Is it relevant enough to mathematical physics to be here? $\endgroup$ – perilousGourd May 3 '20 at 14:57
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The fibre in a tangent bundle is an entire vector space. In other words if ${\bf e}$ is a unit vector tangent to the circle at some point, then the fibre is the infinite set $\{x{\bf e}| x\in {\mathbb R}\}$ equiped with the usual vector addition and scalar multiplication operations.

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  • $\begingroup$ Ah! So, in a tangent bundle, the fiber on a manifold at a point is the set of all vectors that are tangent to the manifold at that point, where vectors of different length are different vectors? For some reason, I had a misconception that vectors of the same orientation would be the same tangent vector / that all that distinguished tangent vectors was orientation. Is what I've said here now correct, though? $\endgroup$ – perilousGourd May 3 '20 at 23:24
  • $\begingroup$ Your new understanding is correct. $\endgroup$ – mike stone May 3 '20 at 23:33

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