18

I have an $n$-manifold $M$ which is foliated by leaves $F_\alpha$ of dimension $p$ and a path $\gamma:[0,1]\to M$. You can take without problems $\gamma$ to be injective. Is the following statement true?

Claim: There exists a neighborhood $U$ of the image of $\gamma$ and a foliation $L_\beta$ of $U$ of dimension $n-p$ such that $F_\alpha\pitchfork L_\beta$ for all $\alpha,\beta$.

Basically what I would like to do is to have an extension of a complement of the tangent space to the leaves $F_\alpha$ in $TM$ to the tangent of local submanifolds of complementary dimension. I feel that this should be true, but I'm not sure about how to proceed. Would go to local coordinates (respecting the foliation) in charts around $\gamma$ solve the problem? How could I make the obtained complements patch together correctly?

An easy partial result: We can always find such a complement to the foliation in an appropriate chart. Indeed, by definition of foliation we know that for any point $m\in M$ we have a neighborhood $U$ of $m$ and a chart $\phi:U\to\mathbb{R}^n$ such that the leaves correspond to the $p$-planes of constant $x$, where we decompose $(x,y)\in\mathbb{R}^{n-p}\times\mathbb{R}^p=\mathbb{R}^n$. Then the preimages of the planes of constant $y$ are our complement (they are regular by the inverse function theorem and the usual arguments).

Daniel Robert-Nicoud
  • 30,663

1 Answers1

0

What might work is the following: First split the interval of your curve into three parts:

  1. The maximal open segments where the curve is tangent to the foliation
  2. The points where the curve starts/ends to be tangent to the foliation
  3. The open segments where the curve is transverse to the foliation

Then you proceed as follows:

First go through the segments of 1. and pick some trivializing sections of the normal bundle along them. Now you take the points from 2. and choose foliation charts around them. In those charts you extend the trivializing sections to along gamma such that one becomes $\dot\gamma$ and the others become a frame for $N\mathcal{F} /\dot \gamma$. Now for segments of 3. the question is can you always trivialize $N\mathcal{F} /\dot \gamma$ under the boundary conditions that this trivializing sections agree with the extension in charts chosen before. Since the oriention does not matter for our purpose the answer should be, yes, it is always possible to do that.

Now the transverse foliation is obtained by using the exponential map and the trivializing sections where $\gamma $ is tangent, in the charts you use the charts and for the segments from 3. you use $\gamma$ and the exponential map.

Florian
  • 748