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In my language, spanish, the word for (topological) manifold and (algebraic) variety is the same: "Variedad". This happens also in some other languages, like french (variété) or portuguese (variedade).

Is there a mathematical link between these two concepts that justifies this linguistic fact?

Marco Flores
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    Sure. The common ground was surfaces in $\mathbb R^3.$ Manifold comes from the German, it is originally an adjective, Riemann's phrase amounts to "many times" or manifold extended aggregates. Compare twofold, threefold. – Will Jagy Jul 16 '14 at 04:08
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    Yes. John Nash, better known for Nash equilibrium, once proved that all (real) manifolds are homeomorphic to some algebraic variety. http://www.jstor.org/discover/10.2307/1969649?uid=3739920&uid=2&uid=4&uid=3739256&sid=21104338172697 – Adam Hughes Jul 16 '14 at 04:08
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    I think the more basic point is that every smooth variety (over $\mathbb{R}$ or $\mathbb{C}$) is naturally also a smooth manifold. – Qiaochu Yuan Jul 16 '14 at 04:23
  • Yes, but Nash's partial converse is more surprising, and has the benefit of making them come on equal footing rather than considering manifold to be the truly "more general" concept. – Adam Hughes Jul 16 '14 at 05:45
  • The name is the same also in Czech: “varieta”. – user87690 Jul 16 '14 at 07:29

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