Integrability of distributions close to a given one.

Question

In this and this papers Thurston proves that every distribution is homotopic to an integrable one (in the first one for codimension greater than one and in the other for codimension one).

Recently, I've came up with a nice paper by Burago and Ivanov which with some other hypothesis manage to start with a distribution and construct a foliation tangent to arbitrarily small cone field around it.

I am far from foliations, but I was wondering if there are examples of distributions which are not integrable and can not be perturbed in the $C^0$ topology in order to become integrable (Edit: That is, a distribution is $\epsilon$ close to other if it is contained pointwise in a cone of angle $\epsilon$ of the original one.)

References appreciated!

Answer 1

No smooth non-integrable distribution can be $C^0$ approximated by integrable ones.

For example, consider the following 2-dimensional distribution in $\mathbb R^3$: the plane at $(x,y,z)\in\mathbb R^3$ is spanned by vectors $(1,0,0)$ and $(0,1,x)$. Perturb this distribution within a small $C^0$ distance $\varepsilon\ll 1$. Consider the square in $\mathbb R^2$ with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$ and let $\gamma$ be its boundary (counter-clockwise). This $\gamma$ has a "lift", that is a curve $\tilde\gamma$ in $\mathbb R^3$ which is tangent to the distribution and whose projection to the horizontal plane is $\gamma$. The lift is found by solving an o.d.e., so it is unique if the distribution is smooth but may be non-unique if it is only $C^0$. In the non-perturbed case, the unique lift ends at $(0,0,1)$, hence in the perturbed case all lifts end near $(0,0,1)$. This implies that the distribution is not integrable - if it was integrable, there would be at least one lift (the one contained in a leaf of a foliation) that ends near the origin.

The proof in the general case is similar.