Extending foliations of submanifolds

Question

Take a manifold $M \cong \Sigma \times \mathbb{R}$, in which there are two submanifolds (of the same dimensions), $S_1$ and $S_2$, such that $S_i \cong \sigma_i \times \mathbb R$, and both are disjoint. Each of these submanifolds has a foliation $\sigma_{i}(t)$. Does there exist a foliation of $M$, $\Sigma(t)$, such that it is compatible with both? That is, for $t \in \mathbb R$, we have $\sigma_i(t) \subset \Sigma(t)$.

If this is not true, what are possible conditions for this to be true? It's obviously true in some cases at least (when the submanifolds inherit the foliation from the manifold). I suspect that for instance taking two copies of $(0,1) \times \mathbb R$ in $\mathbb R^2$ along the axis $y$ would not admit any such foliation if the first copy was foliated by lines according to their coordinates $y$ while the other was foliated by the reverse $-y$.

Answer 1

Goggle my paper "on extending local foliations" in the quarterly journal in 1977. I addressed the following general question: given a compact smooth manifold X and a finite collection of codimension q embedded submanifolds each with trivial normal bundles when could the local product foliations be extended to a global foliation. While in general the answer is no, there are a number of positive results.

Answer 2

Check this article:

https://doi.org/10.1016/j.geomphys.2018.01.021

I think that, by using the characteristic distribution, you may prove the following:

Let $M$ be a manifold with $N_{1}$ and $N_{2}$ as embedded submanifolds which may be separated by two open subsets of $M$ (for example closed embedded submanifolds). Assume that both manifolds have associated a foliation $\mathcal{F}_{N_{1}}$ and $\mathcal{F}_{N_{2}}$, respectively. Then, there exists a foliation of $M$ extending both which, furthermore, satisfies some condition of maximality.