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Homework Statement
Let γ be a regular closed curve in Rn. Show that there is a regular homotopy Γ through closed curves with Γ(−, 0) = γ and Γ(−, 1) an arclength parametrization of γ
Homework Equations
The Attempt at a Solution
Hey guys,
I just posted another question about homotopy but often my questions don't seem to garner a ton of attention and so I'd like to post this here to hopefully be able to get a sense of how to represent a homotopy and how to manipulate it. So I mentioned this in my other post but long story short I was sick and missed the lectures on homotopy (if there were even any, my instructor expects us to do a lot of work on our own and doesn't post lecture slides). I understand a Homotopy is a function which, given 2 endpoints and a family of curves between those two endpoints deforms one curve into another. So I kind of understand it conceptually, doesn't seem that far out there I just have no idea how to represent that function in order to prove anything about it.
I know it has two parameters (at least two "types" of parameters) one which controls how far along the interval you are (to which there exists a mapping from the interval to the curve, i.e., how far along the curve you are) and the 2nd parameter adjusts incrementally the deformation of one curve into the next. So we could think of it kind of like time in the sense that one curve would take time to deform into another one and the 2nd parameter counts that as it ticks up.
Given that, I would imagine the Homotopy would be a two variable (or take x to be a vector if using higher dimensional curves) function like H(x,t). However I have no idea how this is represented or why exactly there's a dash in the first slot of Γ up there.
Any help would be wonderful and truly appreciated!