Manifolds which are not realized with the regular value theorem

Question

Are there smooth/holomorphic manifolds which cannot be defined using the regular value theorem? That is, they are not the preimage of a regular value?

Answer 1

Let $M$ be a smooth manifold and consider the projection $p : M \times\mathbb{R} \to \mathbb{R}$. Note that $p$ is a submersion, so every value is a regular value. For any $r \in \mathbb{R}$, the preimage of $r$ is $M\times\{r\}$ which is diffeomorphic to $M$.

For a complex manifold $X$, we can similarly consider the projection $p : X\times\mathbb{C} \to \mathbb{C}$.