Return to Answer

Without any disrespect let me say that I find it incredible that someone naturally cares about non-commutative geometry but needs convincing about actual geometry (this just goes to highlight that there is a wide variety of ways of thinking in mathematics). I would need convincing the other way around (e.g. How are C* algebras relevant in foliation theory from the geometric point of view?).

The most basic way, is when you consider the level sets of a function. If the function is a submersion you get a non-singular foliation, but this is rare. However every manifold admits a Morse function and the theory of Morse functions (which can be used for example to clasify surfaces, and to prove the high dimensional case of the generalized Poincaré conjecture) can be seen as a special (or maybe as the most important) case of the theory of singular foliations (where the singularities are pretty simple).

A notable fact generalizing the above case (the result is in papers of Sussmann and Stefan from the early 70s) is the following: Consider $n$ vector fields on a manifold. For each point $x$ consider the set of points you can reach using arbitrary finite compositions of the flows of these vector fields. The partition of the manifold into these "accessibility classes" is a singular foliation (in particular each accessibility class is a submanifold).

Hopf then asked in the 1930's if there exists a foliation of the three dimensional sphere using only surfaces. The first observation, due to Reeb and Ehresman is that if one of the surfaces is a sphere then you cannot complete the foliation without singularities. They also constructed the famous Reeb foliation and answered the question in the afirmative.

Since then there has been a whole line of research dedicated to the question of which manifolds admit non-singular foliations. In this regard the main Theorem is due to Thurston who (in the words of an expert in the theory) came around and "foliated everything that could be foliated".

But there are other lines of research. For example I know that there is a certain subset of algebraic geometry dedicated to trying to understand the foliations of complex projective space which are determined by the level sets of rational functions of a certain degree.

Also, whenever you have an action of the fundamental group of a manifold there is a natural "suspension" foliation attached (suspensions are considered the "local model" for a general foliation and are hence very important in the theory). This point of view sometimes has given results in the current area of reseach known as higher-Teichmüller theory (where basically they study linear actions of the fundamental group of a surface).

Oh, and I haven't even mentioned the special place that foliations occupy in the theory of 3-dimensional manifolds. Here there are many results which I cant say much about (but I've heard the book by Calegari is quite nice). Maybe a basic one is Novikov's theorem which basically proves that the existence of Reeb components is forced for foliations on many 3-manifolds.

Without any disrespect, let me say that I find it incredible that someone naturally cares about non-commutative geometry but needs convincing about actual geometry (this just goes to highlight that there is a wide variety of ways of thinking in mathematics). I would need convincing the other way around (e.g. How are C* algebras relevant in foliation theory from the geometric point of view?).

The most basic way is when you consider the level sets of a function. If the function is a submersion you get a non-singular foliation, but this is rare. However every manifold admits a Morse function and the theory of Morse functions (which can be used for example to classify surfaces, and to prove the high dimensional case of the generalized Poincaré conjecture) can be seen as a special (or maybe as the most important) case of the theory of singular foliations (where the singularities are pretty simple).

A notable fact generalizing the above case (the result is in papers of Sussmann and Stefan from the early 70s) is the following: Consider $n$ vector fields on a manifold. For each point $x$, consider the set of points you can reach using arbitrary finite compositions of the flows of these vector fields. The partition of the manifold into these "accessibility classes" is a singular foliation (in particular each accessibility class is a submanifold).

Hopf then asked in the 1930's if there exists a foliation of the three dimensional sphere using only surfaces. The first observation, due to Reeb and Ehresman is that if one of the surfaces is a sphere then you cannot complete the foliation without singularities. They also constructed the famous Reeb foliation and answered the question in the affirmative.

Since then there has been a whole line of research dedicated to the question of which manifolds admit non-singular foliations. In this regard, the main Theorem is due to Thurston who (in the words of an expert in the theory) came around and "foliated everything that could be foliated".

But there are other lines of research. For example, I know that there is a certain subset of algebraic geometry dedicated to trying to understand the foliations of complex projective space which are determined by the level sets of rational functions of a certain degree.

Also, whenever you have an action of the fundamental group of a manifold there is a natural "suspension" foliation attached (suspensions are considered the "local model" for a general foliation and are hence very important in the theory). This point of view sometimes has given results in the current area of research known as higher-Teichmüller theory (where basically they study linear actions of the fundamental group of a surface).

Oh, and I haven't even mentioned the special place that foliations occupy in the theory of 3-dimensional manifolds. Here there are many results which I can't say much about (but I've heard the book by Calegari is quite nice). Maybe a basic one is Novikov's theorem which basically proves that the existence of Reeb components is forced for foliations on many 3-manifolds.

Since then there has been a whole line of research dedicated to the question of which manifolds admit non-singular foliations. In this regard the main Theorem is due to Thurston who (in the words of an expert in the theory) came around and "foliated everything that could be foliated". There's also a nice theorem due to Bott pointing out an obstruction for foliations with certain regularity.

Since then there has been a whole line of research dedicated to the question of which manifolds admit non-singular foliations. In this regard the main Theorem is due to Thurston who (in the words of an expert in the theory) came around and "foliated everything that could be foliated".

TLDR: Foliations occur naturally in many contexts in geometry and dynamical systems. There may not be a very unified "Theory of foliations" but several special types have been studied in depth for different reasons and have yielded insight or participate in the proof of important results such as the Poincaré conjecture and Hörmander's bracket theorem. For this reason mathematicians have taken notice and singled out foliations as a basic object in geometry (there have even been significant efforts in producing a couple of nice treaties trying to give the grand tour, for example the books by Candel and Conlon).

And (I couldn't resist adding one last example), there are also foliations by Brouwer lines, which have recently been used (by LeCalvez and others) to prove interesting results about the dynamics of surface homeomorphisms.

TLDR: Foliations occur naturally in many contexts in geometry and dynamical systems. There may not be a very unified "Theory of foliations" but several special types have been studied in depth for different reasons and have yielded insight or participate in the proof of important results such as the Poincaré conjecture and Hörmander's bracket theorem. For this reason mathematicians have taken notice and singled out foliations as a basic object in geometry (there have even been significant efforts in producing a couple of nice treaties trying to give the grand tour, for example the books by Candel and Conlon).