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A surface becomes a Riemann surface when it is endowed with an additional geometric structure.

English is not my native language, and I cant understand what does endowed mean here.

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Endowed can also be replaced with equipped there. It means the surface is given some extra structure. That is, some extra structure is defined on it.

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Mathematical structure is translated oddly into English. The words used don't invoke an object, but sound more like a cooking recipe: e.g. to define a group, you start with a set and then you mix in additional ingredients (a choice of unit, an inversion function, and a product function), and the end result is a group.

"Endowed" is just one of the many words that people use to express the English idea of starting with something and adding something more.

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I'd also suggest that this is one of those terms where mathematical terminology and popular usage are somewhat opposite.

In a populist sense it can mean 'is restricted to' (rather than having open ended possibilities). Of course mathematically, there can be no possibilities without some extra axiom or endowment.

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