My question is about domains of trivializability of distributions on a smooth compact manifold. Assume that you have a sequence $\{E^k\}\_k$ of $C^r$ distributions of rank $n$ which is $C^0$ close to a fixed $C^r$ distribution $E$ of rank $n$ with respect to a given Riemannian metric. 


My question is that for each $x \in M$, can I find a uniformly sized neighbourhood $U$ on which I can trivialize all $\{E^k\}\_k$ for large $k$, owing to the fact that they are $C^0$ close to a fixed distribution (and for $U$ I take it to be the nbd on which $E$ is trivializable), or do I need higher regularity other conditions. My initial idea was the following:


For each $x \in M$, there is a closed neighbourhood $U$ on which $E$ can be locally, orthonormally, trivialized (in the sense that there is a collection of $n$ orthonormal $C^k$ vector fields $\{X\_i\}\_{i=1}^n$ spanning $E$ on $U$) . You can write $U$ as a union of finite number of $m$ close sets $\{U^k\_l\}\_{l=1}^{m}$ on which $E^k$ is orthonormally trivializable, with orthonormal basis of vector fields say; $\{X^k\_{i,l}\}\_{i=1}^n$. Then my idea was to try to inductively "transform" these vector fields so that on any intersection $U^k\_l \cap U^k\_m$, each $X^k\_{i,l}$ is close to $X^k\_{i,m}$ (up to reordering of $i$) so that one can patch these vector fields using a partition of unity subordinate to $\{U^k\_l\}\_{l=1}^{m}$ without "losing dimension". But now, I am not sure that my initial idea on how to build this transformation was correct.


So my first question is whether if this is true, or if this proof could be completed or not or any reference to such questions or issues would be also most welcome. I have (almost) never come up on such estimates about neighbourhood sizes and so on or any usage of sequence of distributions to approximate other distributions for any specific reason as it seems such approaches are not common in geometry. So any reference on such works is also welcome as an extension of my question.


Thanks 



A question about the "size" of the neighbourhoods in which bundles are trivializable
I may not have understood your question correctly, but assuming that "distribution" stands for a $n$-plane field, any vector bundle over any contractible base is trivial, so you only need to make sure that your neighborhoods are contractible. Speaking about "size", you can introduce a Riemannian metric and take all $U$ smaller than the minimal injectivity radius. 


