A cubic surface, fibered by elliptic curves.pic.twitter.com/fbKhRYO9ef
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so the fiber essentially tells you how the cross section changes increase some parameter?
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This is where connections come in! All the fibers in our bundle look the same, and so we might want a way to rigorously encode this, to connect different fibers. Let’s think about vector bundles, so you have a vector space tacked on to your base manifold at every point.
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And a vector field is a rule picking out one vector from each of these vector spaces, to be the vector based at that point of the manifold (fancy math word here is a “section of the vector bundle”)
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I am still not sure I get what this is.
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