Suppose we have an immersion $f$ between $S$ and $M$. Moreover, suppose that $X$ is a smooth vector field on $f(S)$. Prove that we can extend $X$ to a open neighborhood $V$ of $f(S)$ on $M$. Here $M$ and $S$ are a smooth manifold of dimension $k$ and $n+k$ respectively.
What I see is that you can extend this vector field in a 'parallel' way but I don't know how to formalize this idea.