Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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
2  
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
1  
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
show 1 more comment

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.