Is the vector field smooth?

Question

Let $X$ be a smooth vector field over $M$ and $S$ an embedded submanifold. If $X_p$ is in $T_p S$ for all $p$ in $S$ then there is a function $Y$ such that $di(Y_p)=X_p$.

Is $Y$ a smooth vector field?

Answer 1

Yes, this is pretty much immediate from looking in local coordinates. Near any point of $S$, we can pick local coordinates $x_1,\dots,x_n$ on $X$ such that $S$ is just the subset where $x_{m+1},\dots,x_n$ vanish, so $x_1,\dots,x_m$ are local coordinates for $S$. So your question is just: given a smooth map $\mathbb{R}^n\to\mathbb{R}^n$ which takes values in $\mathbb{R}^m$ when restricted to $\mathbb{R}^m$, is the restriction a smooth map $\mathbb{R}^m\to\mathbb{R}^m$? The answer is obviously yes.