Suppose that I have a well-defined smooth vector field $Y$ of the form $\sum_i f_iX_i$ on a submanifold $S$ of $M$ where each $f_i$ is a smooth function on $S$ and each $X_i$ is a smooth vector field on $S$.
Suppose also that I know $f:= \sum_i f_i$ cannot be continuously extended to all of $M$. Does this imply that $Y$ can also not be extended to all of $M$?
I don't know whether extending $Y$ would need to extend $f$ in an impossible way. Any help is appreciated.
Best Answer
No. Take $M=\mathbb R$, $S = (0,\infty)$, $X_1 = x\partial/\partial x$, and $f_1 = 1/x$. Then $f = f_1$ cannot be extended continuously to $M$, but $Y = \partial/\partial x$ can be.
Note that the key feature of this example is that $S$ is not closed in $M$. In fact, the hypotheses imply that $S$ cannot be a closed embedded submanifold of $M$, because if it were, then $f$ would necessarily have a continuous (in fact smooth) extension to all of $M$. (See Lemma 5.34 in my Introduction to Smooth Manifolds, 2nd ed.)