Suppose $N\subset M$ is an embedded submanifold, and $X\in\mathfrak{X}(M)$ a complete vector field that is tangent to $N$. Let $\phi_{X}^{1}$ denote the time $1$-flow of $X$.
What can be said about $N\cap\phi_{X}^{1}(N)$? It seems to me that in general, this should be an open subset of $N$, and therefore an embedded submanifold itself (if nonempty). But I cannot find a clean proof...