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Suppose $M$ is a smooth manifold with a smooth vector field $X$ on it. If $X$ is not a complete vector field (a complete vector field is one for which all integral curves exist for all time) is it possible to embed it in another smooth manifold and extend $X$ to a complete vector field on the new manifold? More precisely, can one find a smooth manifold $N$, a complete vector field $Y$ on $N$, and a smooth embedding $F : M \to N$ such that $F_*X = Y$, i.e. for all $p \in M$, we have $dF_p (X_p) = Y_{F (p)}$?

For example, if $M = (0, 1) \subset \mathbb R$ and $X = \partial_x$ then clearly $X$ is not complete, but $M$ embeds inside $N = \mathbb R$ with the extension $Y = \partial_x$, which is complete.

Chaitanya Tappu
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  • This seems a little bit tricky. Does this help at all: https://math.stackexchange.com/questions/378783/completing-a-vector-field-on-a-non-compact-manifold-m? (btw: Hi Chaitanya!) – Alekos Robotis Mar 28 '20 at 21:35
  • Hi @AlekosRobotis! Yeah I saw that one while trying to see if this had already been asked. But no, I don't want to change the vector field in any way where it's already defined. I want to extend it. – Chaitanya Tappu Mar 28 '20 at 21:45
  • What if you try the following: take $\mathbb{R}^2\setminus {(0,0)}$ and the unit vector field pointing radially inward. – Alekos Robotis Mar 28 '20 at 21:48
  • I think Lee has some extension Lemmas for vector fields from a submanifold into an ambient manifold which will help you in a lot of cases. This might go wrong in situations like the above where no matter how you embed you might be forced to make the integral curves collide?

    I haven't thought about these things in a long time, though.

     – Alekos Robotis Mar 28 '20 at 21:54
  • In the above one perhaps you could embed the punctured plane as a half cylinder in a cylinder. Then the integral curves go up parallel to the axis of the cylinder, and perhaps can be continued indefinitely – Chaitanya Tappu Mar 28 '20 at 21:57
  • Yes, that seems to work. Maybe this always works? Perhaps you can always foliate the domain by the integral curves and come up with a similar way of pasting them together into a bigger manifold where there is "enough space." – Alekos Robotis Mar 28 '20 at 22:00

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