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?