Normal-bundle obstruction to being a regular level set

If an embedded codimension-$k$ submanifold $X\subset M$ is $f^{-1}(y)$ for a regular value $y$ of a smooth map $f:M\to {\mathbb{R}}^k$, then $df$ identifies its normal bundle with the trivial bundle $X\times {\mathbb{R}}^k$. A submanifold with nontrivial normal bundle therefore cannot be such a regular level set. The core circle of the Möbius band is a codimension-one example.