Let $\mathcal{M} := \{ (x, y, z) \in \mathbb{R}^3 : x^2 + y^2 = 1 \}$ and $p_0 = (1, 0, 0)$. Show that there is an open neighborhood $W \subset \mathbb{R}^3$ of $p_0$ and a smooth submersion $h : W \rightarrow \mathbb{R}$, such that $h^{-1} (0) = W \cap \mathcal{M}$.
Obviously, $h : (x, y, z) \mapsto x^2 + y^2 - 1$ has $h^{-1} (0) = \mathcal{M}$ (kudos to @Matematleta for pointing this out.) It seems to me that a large enough neighborhood would thus trivially solve the problem. For example, $W := {(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 < 2}$ is open and $W \cap \mathcal{M} = \mathcal{M}$.
Is there also a more local solution—i.e., with a smaller set $W$? This question follows up on a question about parametrizing $\mathcal{M}$: Embedding of cylinder submanifold in $\mathbb{R}^3$), for which I used a smaller $W$.