Suppose we have two smooth manifolds $M_1$ and $M_2$ and a smooth map $i:M_1 \rightarrow M_2$ that is an embedding of $M_1$ into $M_2$. Moreover we have another submanifold $N \subset M_2$ that has a non empty intersection with the embedding $i(M_1)$. Then,in what situation is the preimage set $i^{-1}(N)$ a submanifold of $M_1$?
Or in other words, what do we have to assume so that the preimage set is a submanifold?
One hypothesis that works is that $N$ is transverse to the image of $i$, but this is not a necessary condition. For example, the preimage under the inclusion of the $x$ axis into $\mathbb R^2$ of the parabola $\{(t,t^2):t\in\mathbb R\}$ is a submanifold...