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

Came across this reading The Browser (a site that curates the 6 best links every day). Worth checking out. https://thebrowser.com/