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A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.

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Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

prove this equality without introducing any orientation, but even if you only have a proof for the oriented case the doubling arguments (mentioned in the comments) trivially extends it to non-orientable surfaces
Alex Gavrilov's user avatar