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 $F(M)$ and embedded submanifold of $\mathbb{C}^2$ (considered as $\mathbb{R}^4$)?