I am studying the basics of differential geometry and I am focusing on distrributions.
Studying from the notes of my professor, I have found that the concept of distribution and its integration is linked to the concept of level functions.
So, suppose we have a smooth function $\lambda : \mathbb{R}^{n}\rightarrow \mathbb{R}$. If we have this, we have that exist a set of points that satisfy:
$I_c = \left \{ x\in {\mathbb{R}^{n}} : \lambda (x)=c \right \} $
with $c\in Im(\lambda )$
this define level functions, of dimension $(n-1)$, but why it defines thes level functions? And why of dimension $n-1$?
And , are these foliations?
If I keep studying, I find an example, in which are considered two independent functions $\lambda _1$ and $\lambda _2$, so:
$rank\begin{pmatrix}
\frac{d\lambda _1}{dx}\\
\frac{d\lambda _2}{dx}
\end{pmatrix} =2$
and so it defines a foliation of dimension $(n-2)$ .
But why the foliation is of dimension $n-2$ and not of dimension $2$?
It ocntinues sayin that if we are considering a distribution of constant rank $k$ and the distribution $\Delta $ is involutive, then there exist $n-k$ indepedent functions such that:
$\frac{d\lambda _1}{dx}(\tau _1(x)…\tau _k(x))=0$
where $\tau _1(x)…\tau _k(x)$ are vector fields.
If it can help better understand the context, I am studying this in order to study the Frobenius theorem.
I am really confused about the topics of differential geometry, since personally I find them hard to grasp. If something is wrong or unclear please tell me and I will try to correct.
Can somebody please help me?
Best Answer
Your terminology is a little off, the set $I_c$ is not called a "level function" it is called a "level set" or, as you say in your title, a "level surface". The intuition is that it consists of all points at the same "level", where different values of $c$ define different levels.
In general there's not much you can say about the topology of $I_c$ to justify calling it a level surface. However, if $c$ is a regular value of the function then you can apply the submersion theorem, which is basically a version of the implicit function theorem, to prove that $I_c$ is a submanifold of dimension $n-1$. In this case one might say that $I_c$ is a "level manifold" of the function, in the case $n=3$ it would truly be a "level surface", but even in higher dimensions the terminology is abused and $I_c$ is called a "level surface", or you could simply call it a level hypersurface without any abuse of terminology.
If all values of $c$ are regular values, and if $\lambda$ is surjective, then yes, the set of level manifolds forms a foliation, and it is a foliation of dimension $n-1$, where the dimension refers to the dimensions of the individual leaves of the foliation, i.e. the individual level manifolds.
But if some values of $c$ are not regular values, or if $\lambda$ is not surjective, then generally speaking you cannot call the collection of level sets a "foliation" (except in highly special circumstances).
By the way, there are plenty of good books which cover these topics in detail. I like the first volume of Spivak's "Differential Geometry" for these topics in your post.