Help to translate mathematical definitions

Question

I'm from a non speaking english country so many concepts I learn are translated to my native language and most of them I can easily translate or find them but these one I can´t seem to:

(I will post the direct translation followed by the definition)

1)"variety": it's the generalization of a surface. The most usual are topological "varieties" and differentiable "varieties";

2)"subvariety":the "subvariety" of a "variety" $M$ is a subset $S$ which himself has the structure of a "variety";

3)"immerse subvariety": An "immerse subvariety" of a "variety" $M$ is the image $S$ of an immersion $f:N\to M$ which need not to be injective;

4)"immersed subvariety" ou "regular subvariety": is an immersed subvariety whose immersion is a "topological dive".

5)"topological dive": is an injective immersion whose inverse function is continuous.

Also, of what area of mathematics are these terms for?

[context]In my calculus III course I thought I was going to learn about vectorial analysis, so I was expecting, well, vectors. Somehow, I got stuck with all this terminology that fell out of the sky not knowing from what branch of mathematics it is, how it relates to vectorial analysis and I have very little intuitive notions of these terms. Currently I'm on parametrizations, how does any of these terms relates to it?

Answer 1

These are all terms from differential geometry.

1)"variety": it's the generalization of a surface. The most usual are topological "varieties" and differentiable "varieties";

This is called a manifold in English. (Note that the term "variety" also exists in English, but refers instead to algebraic varieties that are defined by polynomial equations.)

2)"subvariety":the "subvariety" of a "variety" $M$ is a subset $S$ which himself has the structure of a "variety";

This is a submanifold.

3)"immerse subvariety": An "immerse subvariety" of a "variety" $M$ is the image $S$ of an immersion $f:N\to M$ which need not to be injective;

This is an immersed submanifold.

4)"immersed subvariety" ou "regular subvariety": is an immersed subvariety whose immersion is a "topological dive".

This is an embedded submanifold.

5)"topological dive": is an injective immersion whose inverse function is continuous.

This is an embedding. Note that in English there is a distinction between a "topological embedding" and a "smooth embedding", but it seems that your "topological dive" actually means a "smooth embedding", not a topological embedding. A topological embedding would be the same thing except it is merely required to be continuous, rather than an immersion. (Note, though, that a map is a smooth embedding iff it is both a topological embedding and an immersion, so in item 4) above it makes no difference since the map is already assumed to be an immersion.)