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).

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    $\begingroup$ What does $\pitchfork$ mean? $\endgroup$ – Jim Belk Jun 4 '15 at 23:57
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    $\begingroup$ @JimBelk It means "intersects transversally", i.e. $F_\alpha\pitchfork L_\beta$ signifies: for every point $m\in F_\alpha\cap L_\beta$ we have $T_mM = T_mF_\alpha\oplus T_mL\beta$. $\endgroup$ – Daniel Robert-Nicoud Jun 5 '15 at 11:59

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