Another Differential Geometry-Curve Theory This is another problem that keeps arising year after year that none is able to solve. Any help is very appreciated. 
Let $r(s)$ be a regular closed curve which lays in sphere $S^2$. Prove that: $$\int_y \tau(s)ds=0 $$ where $\tau $ is the torsion function $y$ the map of the this curve. (integral over that map...)

Sketch of the proof: Assume $\tau$ doesn't vanish; we let $\tau$ be positive. Show first from $\langle r,r\rangle=1$ that 
$$r=-\frac{1}{\kappa}\mathbf{n}-\left(\frac{1}{\kappa}\right)'
\cdot\frac{1}{\tau}\mathbf{b},$$
hence
$$\left(\frac{1}{\kappa}\right)^2+\left(\left(\frac{1}{\kappa}\right)'\right)^2\cdot\left(\frac{1}{\tau}\right)^2=1.
$$
If additionally $\kappa^2\neq1$ we conclude that
$$\tau=\pm\frac{\left(\frac{1}{\kappa}\right)'}{\sqrt{1-\left(\frac{1}{\kappa}\right)^2}}.
$$
From here 
$$\int\tau=\pm\sqrt{1-\left(\frac{1}{\kappa}\right)^2}.$$
Does this help?
