So i am given a normal curve that is simple and closed and that is on the 2-sphere. I need to prove that this curve bisects the sphere into 2 parts of equal area 2pi. I know I need to use the Gauss Bonnet Theorem. And so, i need to somehow prove that the integral of the geodesic curvature with respect to the arc length element is 0. But how do i go about doing this??
Could someone also give me an intuitive picture of what we are trying to do, in order to prove this! Thanks!