I have a specific problem, but would also like to know how to tackle the general case. I will first state the genral question. Let $M$ be an embedded submanifold of $\mathbb{R}^n$ and let $F: \mathbb{R}^n \to \mathbb{R}^n $ be a smooth map. How do I go about checking whether $F(M)$ is a smooth embedded submanifold of $\mathbb{R}^n$ or not? The specific problem I have is the following :- Let $F:\mathbb{C}^2\to \mathbb{C}^2$ the map $(z\_1,z\_2) \mapsto (z\_1+z\_2,z\_1z\_2)$ and let $M$ be the unit sphere in $\mathbb{C}^2$, i.e., $\lbrace (z\_1,z\_2) : |z\_1|^2 + |z|^2 = 1 \rbrace $. Is $F(M)$ an embedded submanifold of $\mathbb{C}^2$ (considered as $\mathbb{R}^4$)?



Checking whether the image of a smooth map is a manifold
The specific $F(M)$ is not a smooth submanifold. Here is an argument.


To simplify formulas, I renormalize the sphere: let it be the set of $(z\_1,z\_2)\in\mathbb C^2$ such that $|z\_1|^2+|z\_2^2|=2$ rather than 1. Then, as Gregory Arone pointed out, $F(M)$ is the set of $(b,c)\in\mathbb C^2$ such that the roots $z\_1,z\_2$ of the equation $z^2-bz+c$ satisfy $|z\_1|^2+|z\_2^2|=2$. I claim that it is not a smooth manifold near the point $p:=(2,1)\in F(M)$.


Let us intersect $F(M)$ with two planes: $\{b=2\}$ and $\{c=1\}$. If it is is a smooth submanifold, at least one of the intersections should be locally (near $p$) a 1-dimensional smooth submanifold of the respective plane (otherwise both planes are tangent to $F(M)$ at $p$, but this is impossible since they span the whole space). But this is not the case:


If $b=2$, the equation is $z^2-2z+c=0$, hence $z\_1+z\_2=2$, then $|z\_1|+|z\_2|\ge 2$ and therefore $|z\_1|^2+|z\_2|^2\ge 2$. Equality is attained only for $z\_1=z\_2=1$, thus the intersection is a single point $c=1$, not a 1-dimensional submanifold.


If $c=1$, the equation is $z^2-bz+1$, hence $z\_1z\_2=1$, then $|z\_1|\cdot|z\_2|=1$ and therefore $|z\_1|^2+|z\_2|^2\ge 2$. The equality is attained if and only if $|z\_1|=|z\_2|=1$, so $z\_1$ and $z\_2$ are conjugate to each other. The set of $b$'s for which this happens is the real line segment $[-2,2]$ which is not a submanifold near 2.


