If I have a smooth function $f\in C^\infty(M)$ where $M$ is compact, and a smooth integrable distribution $E$ of $M$ (so a smooth assignment of vector spaces to $TM$). For simplicity, assume $E$ is one dimensional, is $|df|_E|$ smooth (or at least Holder) on M with respects to the Riemannian metric?
It will be very helpful if someone can give some me some counter-examples to work on.
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It is pointed out in the comment that it is not clear what a smooth distribution means. There are two definitions of the regularity of distribution that will be helpful to me.
(1) The regularity of the plane fields corresponding to the distribution, defined similarly to vector fields. Take local coordinates and define regularity of plane fields in charts. For vector $v\in\mathbb{R}^n$ and $k$-dimensional subspaces $E, A, B \subset \mathbb{R}^n$, define $$d(v,E) = \min_{w\in E} d(v,w)$$ which is the length to the projection. Then define $$d(A,B)=\max \left\{ \max_{v\in A, \|v\|=1} d(v,B), \max_{w\in B, \|w\|=1} d(w,A) \right\}. $$ I am requiring this regularity measured with respects to the distance in plane fields and the Riemannian metric of $M$ to be Holder or smooth.
(1.5) I think the following definition is equivalent to (1): for $r\in \mathbb{R}_+$, $C^r$ regularity of plane fields means $C^r$ sections of the Grassmanian bundle $G^k M$.
(2) Mostly helpful to my contexts are the $C^r$vector fields tangent to $C^r$ foliations of $M$. A foliation is a division of $M$ into leaves $\mathscr{F}_p$, injectively immersed $k$-dimensional submanifolds, for points $p\in M$. They are given by foliation boxes $\phi: D^k\times D^{n-k}\to U$ such that $\phi(D^k\times y)\subset \mathscr{F}_q$, where $U\subset M$ is a neighborhood of $q=\phi(0,y)$ and $y\in D^{n-k}$. Require $C^r$ foliations to have $C^r$ foliation boxes.