Is any path connected component of an $n$-surface in $\mathbb R^{n+k}$ again an $n$ -surface?

Question

Let $S$ be an $n$-surface in $\mathbb R^{n+k}$ i.e, there is an open set $U \subseteq \mathbb R^{n+k}$ and a smooth map $f : U \to \mathbb R^k$ such that $S=f^{-1}(\{0\})$ and $rank Df(a)=k , \forall a \in S$ , then is it true that any path connected componenet of $S$ is again an $n$ -surface ?

Answer 1

If you read the answer by Manuel Araújo to this MSE question, you learn that a submanifold $M^n\subset {\mathbb R}^{n+k}$ is the regular level set of a smooth map $f: {\mathbb R}^{n+k}\to {\mathbb R}^{k}$ is and only if the normal bundle $\nu_M$ of $M$ is trivial. If $U\subset M$ is an open subset then the restriction of the normal bundle $\nu_M$ to $U$ is the normal bundle $\nu_U$ of $U$. If $\nu_M$ is trivial, so is $\nu_U$. Therefore, if $M$ is the regular level set of a smooth map $f: {\mathbb R}^{n+k}\to {\mathbb R}^{k}$, so is every connected component $U$ of $M$.