# Invariant submanifolds

Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical assumptions) that the following is true:

If $V(p)$ is tangent to $N$ for all $p\in N$, then $N$ is an invariant submanifold of $\Phi_t$.

Is this true? What sorts of technical assumptions would I need to worry about to make the statement rigorous? I imagine, for example, that there could be global topological issues so that perhaps the statement only holds locally.

Is there a good (basic) reference on invariant submanifolds?

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It will work if the submanifold is compact. To prove it, you can remark that $V$ induces a vector field on $N$, and the integral curves on $N$ will still be integral curves on the ambient manifold $M$. –  Olivier Bégassat May 18 '13 at 8:33
You can get by with closed -- slightly weaker than compact. –  Ryan Budney May 18 '13 at 8:35
Hmm ok. Would y'all mind commenting on what could go wrong if the submanifold were not closed? –  joshphysics May 18 '13 at 15:40
Let $M = \mathbb R^2$ and $V = \partial/\partial x^1$. Then $V$ is tangent to the submanifold $N = (-\infty,0)\times\{0\}$, but $N$ is not invariant under the flow. –  Jack Lee May 18 '13 at 15:44
@JackLee Ah ok thanks. –  joshphysics May 18 '13 at 15:59
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