Can a Submanifold Become Tangent to a Nowhere Tangent Vector Field $\newcommand{\R}{\mathbf R}$
Let $M=\R^2$ and $S=\{(0, t):-1<t < -1\}$ be a submanifold of $M$. Let $V$ be a vector field on $M$ which is nowhere tangent to $S$. Let $\theta$ be the flow of this vector field defined on the flow domain $D\subseteq \R\times M$.
Let $t_0\in \R$ be such that the integral curve starting at the origin is defined on $[0, t_0]$.
It is known that there is a neighborhood $O$ of the origin such that $\theta_{t_0}:O\to M$ maps $O$ diffeomorphically onto $\theta_{t_0}(O)$, the latter being open in $M$. Write $\mathbf p=\theta_{t_0}(\mathbf 0)$ and note that $S_{t_0}:=\theta_{t_0}(O\cap S)$ is a $1$-submanifold of $M$ passing through $\mathbf p$.

Question. Is it possible that $V_{\mathbf p}$ is tangent to $S_{t_0}$?



Hint: since $\theta_{t_0}$ is a diffeomorphism, ${\rm D}\theta_{t_0} (\mathbf{0})$ has full rank. Take $V_{\mathbf{0}}$ and $(0,1)$ — these are two vectors in $T_{\mathbf{0}}\mathbb{R}^2$; morever, $(0, 1)$ is tangent to $S$ at $\mathbf{0}$. What can we say about rank of $D\theta_{t_0} (\mathbf{0})$ if $V_{\mathbf{p}}$ suddenly becomes tangent to $S_{t_0}$ ?
