Can a Submanifold Become Tangent to a Nowhere Tangent Vector Field

Question

$\newcommand{\R}{\mathbf R}$

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 $S_{t_0}$?

Answer 1

I am not sure whether I understand the question correctly, but for the diffeomorphism $\theta_{t_0}$, the derivative in $0$ will be a linear isomorphism. By the flow property this will map $V(0)$ to $V(p)$ and the tangent vector to $S$ in $0$ to the tangent vector of $\theta_{t_0}(S\cap O)$ in $p$, so these two vectors have to be linearly independent, and $ \theta_{t_0}(S\cap O)$ cannot become tangent to the flow line.

Answer 2

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 $S_{t_0}$ ?