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Let y:[0,1]-> R^n be a regular closed curve, and let h:[0,1]->[0,1] be smooth, increasing and $h^{(k)}(0)=h^{(k}(1)$ for all k=0,1,.........
Then
b=y o h is a reparametrization of y.
I want to show:
1) b is a regular closed curve so i'm guessing I needs an expression for the kth derivative
2) there is a regular homotopy between y and b. I have been guided to consider
F:[0,1] x [0,1]->R^n given by $F(u,t)=y(uh(t)+(1-u)t)$.
Then
b=y o h is a reparametrization of y.
I want to show:
1) b is a regular closed curve so i'm guessing I needs an expression for the kth derivative
2) there is a regular homotopy between y and b. I have been guided to consider
F:[0,1] x [0,1]->R^n given by $F(u,t)=y(uh(t)+(1-u)t)$.