Integrability of distributions close to a given one. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:11:27Z http://mathoverflow.net/feeds/question/37130 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37130/integrability-of-distributions-close-to-a-given-one Integrability of distributions close to a given one. rpotrie 2010-08-30T08:34:58Z 2010-08-30T10:09:07Z <p>In <a href="http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=213742" rel="nofollow">this</a> and <a href="http://www.jstor.org/pss/1971047" rel="nofollow">this</a> 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). </p> <p>Recently, I've came up with a <a href="http://www.pdmi.ras.ru/~svivanov/papers/coherence.pdf" rel="nofollow">nice paper</a> 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. </p> <p>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.) </p> <p>References appreciated! </p> http://mathoverflow.net/questions/37130/integrability-of-distributions-close-to-a-given-one/37139#37139 Answer by Sergei Ivanov for Integrability of distributions close to a given one. Sergei Ivanov 2010-08-30T10:09:07Z 2010-08-30T10:09:07Z <p>No smooth non-integrable distribution can be $C^0$ approximated by integrable ones.</p> <p>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.</p> <p>The proof in the general case is similar.</p>