2
$\begingroup$

Let $M$ be a smooth manifold and $N$ a submanifold of $M$. Consider the set of vector fields on $M$ that are tangent to $N$. I denote it by $\mathcal{F}$.

I am trying to prove that $\mathcal{F}$ is singular foliation on $M$, i.e. a $C^{\infty}(M)$-submodulde of $\mathfrak{X}(M)$ that is closed under the Lie bracket and locally finitely generated.

It is easy to see that the first condition is satisfied and after working with the local formula of the Lie bracket of vector fields I managed to show that the Lie bracket of two vector fields tangent to $N$ is also tangent to $N$.

I am struggling however to deal with the local generators. If $p$ is a point of $N$ then there exist local coordinates $(U,x^1,...,x^n)$ around $p$ such that $N\cap U =\{x^{k+1}=...=x^n=0\}$, where k is the dimension of $N$.BUT vector fields $\partial{x^1},...,\partial{x^k}$ do not necessarily generate $\mathcal{F}|U$. In addition, I cannot think of way to obtain generators of $\mathcal{F}$ at a neighborhood of a point that does not lie on $N$. I have also tried to work the case where $N$ is a closed submanifold, but I end up to linear combinations of vector fields on $M$ with coefficients smooth functions on a neighborhood of N. Maybe I can extend these to a neighborhood of M but I do not see how.

A submodule $\mathcal{F}\subset \mathfrak{X}(M)$ is locally finitely generated if for each point $x \in M$ there exists a neighborhood $U$ of $x$ and vector fields $X_1,...,X_k \in \mathfrak{X}(M)$ such that $\mathcal{F}|U = C^{\infty}(U)X_1|U +...+ C^{\infty}(U)X_k|U$

$\endgroup$

1 Answer 1

2
$\begingroup$

Throughout, Greek indices will run from $1$ to $n$, roman indices $i,j,k,\cdots$ will run from $1$ to $k$ and $a,b,c,\cdots$ will run from $k+1$ to $n$, with summation convention in effect. I'll also assume that by "submanifold" you mean "embedded submanifold", since you're using adapted coordinates.

Working in adapted local coordinates $x^1\cdots,x^n$, $\mathcal{F}|_U$ consists of vector fields $V^\alpha(x^1,\cdots,x^n)$ such that $V^i(x^1,\cdots,x^k,0,\cdots,0)=0$. I claim that $\mathcal{F}|_U$ is generated by the vector fields $\partial_a$ and $x^i\partial_j$.

To see this, we may write any vector field $V\in\mathcal{F}$ as $V=V^a\partial_a+V^j_ix^i\partial_j$, and define $V^j_i$ by $$ V_i^j(x^1\cdots,x^n)=\int_0^1\partial_iV^j(tx^1,\cdots,tx^k,x^{k+1},\cdots,x^n)dt $$ It's straightforward to show that $V_i^j$ are well defined and smooth (assuming without loss of generality that the coordinate chart is convex) and that the decomposition above holds.

It takes a bit more work to fit this into your definition of locally finitely generated, but it can be done by finding a smooth bump function $\psi$ around the point of interest and a a restriction of neighborhoods $U'\subset U$ such that $\psi|_{U'}=1$.

$\endgroup$
1
  • $\begingroup$ Thanks for your reply and sorry for the late answer. I didnt have much time to check your answer thourouhly, but it seems that your arguement holds. $\endgroup$
    – mits314
    Feb 4, 2021 at 18:07

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .