imagine all the wind on the earth's surface right now. that's a vector field or gauge (as physicists call it). there are many units that it could denominated in

imagine all the wind on the earth's surface right now. that's a vector field or gauge (as physicists call it). there are many units that it could denominated in (meters per sec, feet per second, etc). changing from one to the other is a gauge transformation (unit change, rescaling). okay now suppose you want to compare the wind at two points. you can't just take the vectors at those two points and take a dot product because they're not in the same vector space (each of them is in a vector space tangent to the point they're anchored at). to figure out the issue with doing this watch this video https://youtu.be/p1tfZD2Bm0w?t=170 https://youtu.be/p1tfZD2Bm0w?t=170 you have to perform parallel transport. one way to do this is to track how much moving the vector changes it (messes with its orientation). that's the connection and the covariant derivative. you can then take a look at how that connection itself changes over the surface of the earth. if it's curl free (recall curl free vector field from multivariable calculus) then you don't have eddies in your wind charts i.e. you can't get faster by going around in a loop. absolutely none of this applies to currency or equities because the surface of the earth is smooth and curved. neither of this is true about equities or securities or currency.