This is nice. If this topic is of interest, I highly recommend spending some time with Do Carmo's differential geometry. Particularly Gauss-Bonnet (pg 264) as i

This is nice. If this topic is of interest, I highly recommend spending some time with Do Carmo's differential geometry. Particularly Gauss-Bonnet (pg 264) as it is possibly the most profound result for surfaces. http://www.maths.ed.ac.uk/~aar/papers/docarmo.pdf http://www.maths.ed.ac.uk/~aar/papers/docarmo.pdf