# When does variété mean manifold?

Following advice from this post, I am in the process of translating Ehresmann's 1934 paper "Sur la Topologie de Certains Espaces Homogènes" from French to English.

French-English dictionaries online and Google translate are helping me out quite a bit. However, I'm confused about the standard usage of the word variété in French mathematics writing. It can be translated as either variety or manifold.

Does the phrase variété algébrique translate to algebraic variety, or algebraic manifold?

The difference is subtle, and maybe not all that important for my purposes in studying the Grassmann manifolds which are simultaneously algebraic varieties and topological manifolds, whence algebraic manifolds. Of course, I could think of Grassmann manifolds all the time as algebraic manifolds, but in translating the paper I want to have the same frame of mind as Ehresmann did when he wrote it.

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It translates to algebraic variety. Unqualified I believe variété translates to many things in English. In the topology literature it tends to translate to manifold. Outside of strict topology usage is more flexible. – Ryan Budney May 19 '11 at 20:03
I'm a native speaker, and I totally agree with Ryan on this! – Henri May 19 '11 at 20:35
I just want to confirm Ryan, click at "English" in the left column of these French Wikipedia articles: fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_alg%C3%A9brique and fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_(g%C3%A9om%C3%A9trie) - add the end-parenthesis by hand to the URL - Variete without adjectives is mostly manifold, but with the algebraic adjective it is mostly variety. In Czech, the difference is too subtle for all but 5-10 people haha but "manifold" is translated as "varieta", too. – Luboš Motl May 19 '11 at 20:38
Great! This is what I thought to be true. Thanks. – wckronholm May 19 '11 at 20:48

Nowadays (note the caveat!) the translation into English of "variété algébrique" is definitely "algebraic variety". To give an explicit example, the equation $y^2=x^2+x^3$ describes "une variété algébrique affine" and in English "an affine algebraic variety" , not "an algebraic manifold". Traditionally, if you wanted to translate "algebraic manifold" into French you would say "une variété algébrique non-singulière": the adjectives "régulière" is more recent and "lisse" even more recent (due to Grothendieck I would guess). I even remember reading the clumsy sentence "une variété algébrique non nécessairement non-singulière" just meaning "an algebraic variety" !

A standard pun/joke is "une variété algébrique n'est pas une variété" which translated becomes "an algebraic variety is not a manifold", a true but not sidesplitting statement.

Finally, in (differential ) topology I have never heard anything but variété suitably qualified: topologique, différentiable (différentielle), $\mathcal C^k$, à bord. The translations would be : topological, differentiable (differential), $\mathcal C^k$, manifold and manifold with boundary .

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Just for completeness: in topology, for example, one rarely uses the complete term all the time; you might see a proper definition the first time, and the rest of the text, just the word 'variété' is used (par abus). – Gerben May 19 '11 at 22:09
It seems like Ehresmann is using the term "variété plane" to mean "linear space." Is this common? – wckronholm May 30 '11 at 17:02
The English translations of the course in differentiable calculus on manifolds given decades ago by Henri Cartan causes similar confusion-I wish the translators weren't so literal in translating terminology when it's so obvious from the context what the proper terminology in English should be! – Mathemagician1234 Jan 2 at 9:13
Well said, @Mathematician1234 – Georges Elencwajg Jan 2 at 9:20