How can I prove that "A piecewise regular simple closed curve bisects (this curve splits the unit sphere into two pieces, the area of which are equal) the area of the unit sphere if and only if it has total geodesic curvature 0."
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How your lectures is Differential Geometry may help you to begin to solve the question ? In particular, have you seen specificities of curves lying on a sphere such as http://math.stackexchange.com/q/537033 ? – Jean Marie Apr 27 '16 at 11:03
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Take a look at the Gauss-Bonnet theorem. – Christian Blatter Apr 27 '16 at 11:10
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Hi! I was just going through the same problem and I am stuck at the place where Gauss Bonnet Theorem has to be used. So I basically need to somehow prove that the integral of the geodesic curvature with the arc length element is 0. How do i go about this??? I would also like to get an intuitive idea of what we are doing. Thanks! – Nivedita Mar 01 '18 at 10:21