Show Regular Homotopy thru curve and its arclength parameters

In summary, a regular homotopy Γ through closed curves with endpoints γ and an arclength parametrization can be represented as a two variable function H(x,t), with one parameter controlling the position along the interval and the other parameter adjusting the deformation of one curve into another. The function is denoted with a dash in the first slot of Γ and is used to prove properties of the homotopy.
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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!
 
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BUMP. Sorry guys but I need some help here! Anybody...?
 

What is a regular homotopy?

A regular homotopy is a continuous deformation of one curve into another, where each intermediate curve is also a regular curve. This means that the curve does not intersect itself and its tangent vector is never zero.

How is a regular homotopy represented mathematically?

A regular homotopy can be represented by a family of curves, where the parameter t ranges from 0 to 1. Each curve in this family is a regular curve, and the endpoints of the curve correspond to the starting and ending curves.

What is the significance of the arclength parameter in a regular homotopy?

The arclength parameter in a regular homotopy represents the distance traveled along the curve. This is important because it allows us to measure the change in position of the curve as it undergoes the deformation.

How is a regular homotopy useful in mathematics?

Regular homotopy is a powerful tool in topology and geometry. It allows us to study the properties of different curves and surfaces, and to determine when two curves or surfaces are equivalent. It also has applications in physics, particularly in the study of physical systems that can be represented by curves or surfaces.

Can a regular homotopy be extended to higher dimensions?

Yes, regular homotopy can be extended to higher dimensions. In three dimensions, it is known as isotopy, and in four or more dimensions, it is called h-cobordism. These concepts are used to study the properties of higher-dimensional objects, such as manifolds.

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