# Integral curves on immersed submanifold

An exercise of the book "Introduction to smooth manifolds - John M. Lee" asks to prove that if $S$ is a closed embedded submanifold of a manifold $M$, and $X$ is a vector field on $M$ tangent to $S$, then every integral curve of $X$ that intersect $S$ is contained in $S$.

Can someone show me a counteresample in the "closed-immersed" case?

I believe the issue here is more of a semantic nature: what exactly is the definition of an "immersed submanifold"? Ordinarily, they are defined as the image of an immersion map $f : S' \to M$, and the image is not required by definition to be a "submanifold" of $M$ (see, for example, https://en.wikipedia.org/wiki/Submanifold#Immersed_submanifolds). As an example, consider the usual self-intersecting Klein bottle model in $\mathbb{R}^3$. As per the usual definition, that will be an immersed submanifold, which will fail what you want for the problem.
• For me an immersed submanifold is the image of an injective immersion, so for me the Klein bottle is the image $M$ of the "figure 8 immersion" en.wikipedia.org/wiki/Klein_bottle#The_figure_8_immersion , but with domain $(0,2\pi) \times (0,2\pi)$ (so that map is injective). Why is so obvious that it fails what I want? Intuitively, a vector field $X$ in $\mathbb R ^3$ that is tangent to $M$ has to vanish on the "self-intersecting" circle $C$ (i.e. images of points with $v=0$ ), so $M$ contains every integral curve of $X$. – Ervin Jul 1 '15 at 6:26