Let \(\phi\colon M\to N\) be a map between two manifolds. Then \(\phi^{-1}(\{c\})\) for \(c \in N\) is called level set of \(\phi\). If \(N=\mathbb{R}\), we call \(\phi^{-1}(\{0\})\) zero set of \(\phi\).
Remark
- in general not every level set is a manifold. Indeed, every closed subset of \(M\) is a zero set of a smooth function. [@lee2013smooth_manifolds, Theorem 2.29]