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Quite an impressive review!

Just 2 small correction -

1) There is another compact surface besides the 4 you have mentioned that can be constructed from a flat 2D square. Its constructed by gluing each pair of opposite edges with a 180 degree twist. It's called "Boy's Surface".

2) A 2D Torus T^2 has a only Extrinsic Curvature, but NO Intrinsic Curvature.

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