TITLE: Is integral curve a embedded 1 dimensional submanifold of the given manifold?
QUESTION [5 upvotes]: I can easily see a proof that shows its going to be an immersed submanifold . (I am removing the case if the vector field at that point is 0). I am not able to see if it's a embedded submanifold or not? Thank you.

REPLY [3 votes]: In general, no, absolutely not! Consider $M = \Bbb R^2/\Bbb Z^2$ and $X= \partial/\partial x + a\partial/\partial y$, where $a$ is irrational. Then an integral curve of this is a line with irrational slope, which is dense in $M$. About as far from embedded you can get! You should expect that the "generic" vector field has non-embedded integral curves.