Suppose we have a smooth function $f: \mathbb{R}^{d+e} \to \mathbb{R}^{e}$, and let $\mathrm{x}_0 \in \mathbb{R}^{d+e}$ is a point at which we have $f(\mathrm{x}_0) = 0$ and the Jacobian matrix of $f$ has a constant rank which is not full.
In the view of the Implicit Function Theorem, does this mean anything to $f^{-1}(\mathrm{0})$? Are there extra conditions we can use to show that $f^{-1}(\mathrm{0})$ is/is not a smooth submanifold of $\mathbb{R}^{d+e}$?