# How do level surfaces are defined?

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.

• To me, $I_c = \{ x | f(x)=c\}$ is alevel set of the function $f$. If $f$ is a submersion (i.e if $\mathrm{d}f$ has maximal rank, which is here $1$) on an open subset containing $I_c$, then $I_c$ is a submanifold of codimension $1$. But neither $I_c$ or $f$ is a level function. Jun 18, 2020 at 13:15

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).