I'd like to get some information and references, starting with a name, for the following quite common situation, for a smooth (i.e. $C^\infty$) $n$-manifold $M$. A partition $\mathcal{L}$ of $M$ is given such that for any $p\in M$ there is an open nbd $U$ of $p$ and a homeomorphism $\phi:\mathbb{R}^k\times \mathbb{R}^{n-k}\to U$ such that
- $\big\{L\cap U: L\in\mathcal{L}\big\} = \big\{\phi(\mathbb{R}^k\times\{y\}):y\in \mathbb{R}^{n-k}\big\}$
- For all $\alpha=(\alpha_1,\dots,\alpha_k)\in\mathbb{N}^k$, the map $\phi$ has partial derivatives $\partial^\alpha\phi$, which are continuous on $\mathbb{R}^k\times \mathbb{R}^{n-k}$.
For all $(x,y)\in\mathbb{R}^k\times \mathbb{R}^{n-k}$ the partial differential $D_1\phi(x,y)$ of $\phi$ (w.r.to its first variable $x\in\mathbb{R}^k$) is injective.
In other words, $M$ is partitioned into smooth embedded $k$-submanifolds varying continuously (in the sense of the $C^\infty$ topology in the space of smooth embeddings). If we asked $\phi$ to be a smooth diffeomorphism, that would be a smooth foliation of $M$, but here there is no differentiability in directions transverse to the leafs, so that, for instance, the local distance between two close leafs may be $\epsilon$ somewhere and $\sqrt\epsilon$ somewhere else.
