What type of object is a map from 2D to 4D?

Question

A vector function $\mathbf V : \Bbb R^2 \to \Bbb R^4$ I think is a surface because it as a parameterization has two parameters. But if I think for example if it is projected into $\Bbb R^3$ it may appear as a volume?

Answer 1

If the function is suitably nice, it will indeed be a surface, and projecting into $\mathbb R^3$ will at most again give a surface, which however may intersect itself. However in special cases, the projection might instead be a curve (if the surface happens to be orthogonal to the $\mathbb R^3$ it is projected to) or even a curve near some points and a surface near others. It cannot, however, be a volume (but see below).

As analogy, consider curves in $\mathbb R^3$, which are continuous functions from $\mathbb R$ to $\mathbb R^3$. If they are suitably nice, then projecting to $\mathbb R^2$ will still give a curve, not an area.

However note that there are non-nice functions. For example, you can have a surjective function from $\mathbb R^2$ to $\mathbb R^4$. Projecting that to $\mathbb R^3$ of course gives all of $\mathbb R^3$. Such a surjective function cannot be continuous, though.

But there can be also continuous functions that are "pathological". Consider space-filling curves which are continuous, but densely fill a two-dimensional subset of $\mathbb R^2$. The Cartesian product of two such functions give an analogous map from $\mathbb R^2$ to $\mathbb R^4$, and the projection to $\mathbb R^3$ then of course also will be space-filling.

Answer 2

A surface in math is a "two dimensional manifold", meaning each point "Looks a lot like $\mathbb R^2$". By that definition, if you have a map $f:\mathbb R^2\to\mathbb R^4$ that's "nice" in a suitable way (continuous with continuous inverse), it will be a "two dimensional manifold".

Now, there can be two dimensional manifolds with very weird "projections into $\mathbb R^3$". The prototypical example of this is the Klein Bottle - this is a two dimensional manifold (a surface) that can't be put in $\mathbb R^3$ without intersecting itself. It can be put into $\mathbb R^4$ with no issue though.

So, it depends on what your map is - if your map isn't assumed continuous at all, we no longer can say any of the above.