Total curveture of simple closed curve.

I curently have started reading differential geometry from these notes http://www.matematik.lu.se/matematik...igma/Gauss.pdf and I am trying to solve the exercise 2,7 which says:

Let the positively oriented parametrize a simple closed curve by arclength. Show that if the period of is then the total curvature satisfies .

Any idea?

Re: Total curveture of simple closed curve.

Hey talisman.

Can you use contour integration and the magnitude of said result by using the fact that the contour is a closed contour (so it will involve a 2*pi*i)?

Re: Total curveture of simple closed curve.

I haven't yet learn to use contour integration.

Re: Total curveture of simple closed curve.

I apologize for asking (I should have asked in the first response), but is k(s) the curvature at a parametrized s value?

Re: Total curveture of simple closed curve.

Re: Total curveture of simple closed curve.

Try looking at definition 2.11 and using the composition of taking a circle and deforming it while still keeping the curve a closed curve.

Re: Total curveture of simple closed curve.

I found it like this: becuase is parametrize by arclength then , that means that can be written as , where is the angle beetwen and , then because we have that . we get and we get the result we want.