Let $M, N$ be smooth manifolds, $F:M\to N$ be a smooth map, and $A\subset$ M be a smooth submanifold of $M$. What conditions do I need to impose if I want to have that $f(A)$ is a smooth submanifold of $N$?
Some counterexample that I have in mind is $M=\mathbb{R}^3$, $N=\mathbb{R}^2$, $F$ is given by $F(x,y,z)=(x,y)$, and $A=\{(\sin(t),\sin(t)\cos(t),\cos(t))|t\in[0,2\pi]\}$. So, figure $8$ in space which is projected to a figure $8$ in the plane with a selfintersection.