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Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known about the following

Question. When can $N$ be moved by an isotopy to be nowhere-tangent to $X$?

A necessary condition for existence of such an isotopy is that $X$ be homotopic through nowhere-zero vector fields to one nowhere-tangent to $N$. Are there invariants that can be used to imply that the desired isotopy does not exist even when this necessary condition is satisfied?

I am also interested in the particular case that there is a diffeomorphism from $M$ to $N\times \mathbb{R}$ sending $X$ to the vector field $(0,1)$. In this case, the question becomes: given a smooth embedding $h:N\to N\times \mathbb{R}$, when does there exist an isotopy $h_t:N\to N \times \mathbb{R}$ such that $h_0=h$ and $h_1$ is a section of the trivial bundle $N\times \mathbb{R}\to N$?

Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known about the following

Question. When can $N$ be moved by an isotopy to be nowhere-tangent to $X$?

A necessary condition for existence of such an isotopy is that $X$ be homotopic through nowhere-zero vector fields to one nowhere-tangent to $N$. Are there invariants that can be used to imply that the desired isotopy does not exist even when this necessary condition is satisfied?

Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known about the following

Question. When can $N$ be moved by an isotopy to be nowhere-tangent to $X$?

A necessary condition for an affirmative answer is that $X$ is homotopic through nowhere-zero vector fields to one nowhere-tangent to $N$. Are there invariants that can be used to imply a negative answer to the question even when this necessary condition is satisfied?

I am also interested in the particular case that there is a diffeomorphism from $M$ to $N\times \mathbb{R}$ sending $X$ to the vector field $(0,1)$. In this case, the question becomes: given a smooth embedding $h:N\to N\times \mathbb{R}$, when does there exist an isotopy $h_t:N\to N \times \mathbb{R}$ such that $h_0=h$ and $h_1$ is a section of the trivial bundle $N\times \mathbb{R}\to N$?

Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known about the following

Question. When can $N$ be moved by an isotopy to be nowhere-tangent to $X$?

A necessary condition for existence of such an isotopy is that $X$ be homotopic through nowhere-zero vector fields to one nowhere-tangent to $N$. Are there invariants that can be used to imply that the desired isotopy does not exist even when this necessary condition is satisfied?

I am also interested in the particular case that there is a diffeomorphism from $M$ to $N\times \mathbb{R}$ sending $X$ to the vector field $(0,1)$. In this case, the question becomes: given a smooth embedding $h:N\to N\times \mathbb{R}$, when does there exist an isotopy $h_t:N\to N \times \mathbb{R}$ such that $h_0=h$ and $h_1$ is a section of the trivial bundle $N\times \mathbb{R}\to N$?