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.