90
votes

Accepted

### What is a foliation and why should I care?

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 ...

21
votes

### What is a foliation and why should I care?

Here is another reason about why people care about foliations: If you care about dynamical systems, you should care about foliations.
For instance, if you have a nowhere singular vector field on a ...

21
votes

### What is a foliation and why should I care?

Probably there are many reasons why people care about foliations, but for someone coming from operator algebras one of the main reasons is the connection to von Neumann algebra theory. In brief, every ...

13
votes

Accepted

### Books on foliations

Geometric Theory of Foliations by César Camacho and Alcides Lins Neto in Portuguese, or in English thanks to Sue Goodman's fantastic translation.
I think it does almost everything you're asking for ...

12
votes

### Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?

For circle foliations of compact $3$-manifolds, this is essentially answered by a theorem of Epstein: every such foliation is a Seifert fibration. Most Seifert fibrations are finitely covered by a ...

11
votes

Accepted

### Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure?

It is conjectured that $\mathbb {CP}^2$ contains no embedded compact laminated set (without singularities) apart the smooth algebraic curves.
This is a strong form of the "Minimal Exceptional" ...

11
votes

### Structure of foliations of codimension 2 on three dimensional torus

[Apologies: the answer I wrote below is for $S^2 \times S^1$, not the 3-torus. Corrections added. The case of a 3-torus, or any 3-manifold, is included at the end.]
First, there are different ...

11
votes

### Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?

Dennis Sullivan constructed a non-vanishing Lispschitz vector field on a closed $5$-manifold such that all orbits are periodic but they amazingly have unbounded lengths (!). An addendum by Nicholas ...

11
votes

Accepted

### Which submanifolds are leaves of a foliation?

If the normal bundle of $\Sigma$ in $M$ is orientable, then there always exists such a foliation. The idea is that one can construct a smooth function $f$ on $M$ such that $\Sigma$ is the set of ...

10
votes

### Taut foliations and closed leaves

I'm assuming you meant homologically non-trivial. There are two very famous theorems of Gabai, which show up in quite a lot of places in 3-dimensional topology for constructing taut foliations (in ...

10
votes

### What is a foliation and why should I care?

Here's why I care about foliations. It is always interesting when a structure can be expressed in terms of simpler structures. For instance a torus is the union of circles making it into a cartesian ...

10
votes

### Exterior differentiation of foliations

I think you might be confusing $L\subset T^*M$ with $L^\perp\subset TM$, since you can't compute the Lie bracket of sections of $L$, which are $1$-forms.
The condition that $L$ define a foliation is ...

9
votes

Accepted

### Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution

Assuming there is a smooth structure on $M/D$ such that $\pi$ is a submersion. Then a function $f\colon M/D\to\mathbb R$ is smooth if and only if $f\circ\pi\colon M\to\mathbb R$ is smooth. This means, ...

9
votes

Accepted

### A non integrable distribution which is totally geodesic

Yes, the standard contact structure on the unit three-sphere in $\mathbb{R}^4 = \mathbb{C}^2$, for instance. The Legendrian great circles are the intersections of the sphere with the Lagrangian two-...

9
votes

Accepted

### Existence of a certain foliation of $\mathbb R^n$

EDIT: Originally I could prove that there is such a foliation by topological manifolds:
Clearly, if $\mathbb{Q}^n$ is the set if points with all rational coordinates, you can have a foliation by ...

8
votes

### Is every singular foliation induced by a Lie algebroid?

It is possible to define the Lie groupoid of a singular foliation and associates to it its Lie algebroid when it is smooth. This Lie algebroid satisfies the property 2.
Debord - Holonomy Groupoids of ...

8
votes

Accepted

### The concept of convex foliation

If $n>1$, and a smooth diffeomorphism $f:U\to V$ (where $U$ and $V$ are, say, convex, open subsets of $\mathbb{R}^n$) carries convex sets to convex sets, then it is easy to show that it must carry ...

8
votes

Accepted

### Rank of a distribution

(1) The boxed sentence would be better written as "A distribution $S$ of rank $r$ on a manifold $M$ is an assignment of an $r$-dimensional subspace $S_p$ of $T_pM$ to each point $p\in M$." Confusing '...

8
votes

Accepted

### Can we foliate the space $\mathbb{R}^3$ with Frenet curves whose tangent and normal vectores span a given $2$ dimensional distribution?

The answer is 'no' in general for an arbitrary Riemannian metric $g$ on a $3$-manifold $M$ and $2$-plane field $D\subset TM$.
I'll give the argument for the flat metric on $\mathbb{R}^3$ and leave ...

8
votes

Accepted

### On certain 2 dimensional foliation of $Gl(2,\mathbb{R})$ deleted by scalar matrices

Since $A^2 = \mathrm{tr}(A)\,A - \det(A)\,I_2$, and $\det(A)\not=0$ for $A\in M$, the leaves of your foliation are the same as the leaves of the foliation determined by the vector fields $X(A) = A$ ...

8
votes

### Godbillon–Vey invariant and leaf space of foliations

You should check out Thurston, William, Noncobordant foliations of $S^3$. Bull. Amer. Math. Soc. 78 (1972), 511–514.
He constructs foliations with arbitrary real-valued GV invariants. In this paper, ...

7
votes

Accepted

### Is there a $2$ dimensional foliation tangent to this particular $3$ dimensional distribution?

There is no co-dimension 2 foliation of the desired kind in a neighborhood of any point on the line $L\subset\mathbb{R}^4$ defined by $x-w=y=z=0$. Thus, if one wants to find such a foliation, it will ...

7
votes

### Two questions on "foliation by geodesics"

No to the first question. Let $M$ be a Riemannian $2$-manifold whose universal cover is the hyperbolic plane $H$, and whose fundamental group is not cyclic. For any foliation of $H$ by geodesics (...

7
votes

### Foliation with a compact leaf

There are some results concerning your question that go in the negative direction. Namely, in the following paper two results are proven:
Counterexamples to the Seifert Conjecture and Opening Closed ...

7
votes

Accepted

### A non-geodesible foliation of $S^3$ or $S^2\times S^1$

If a 1-dimensional foliation of a 3-manifold contains a Reeb component, then it is not geodesible. This follows from a theorem of Sullivan, see Section 2.5, Obstruction 1. A Reeb component is a ...

7
votes

### Kneser theorem about the Klein bottle

The answer below was given to the question as asked originally.
For a more modern, english language proof of Kneser's result, see The Poincaré-Bendixson Theorem for the Klein Bottle, by Nelson G. ...

7
votes

Accepted

### Are limits of compact leaves compact?

Here's a counterexample where the foliation has codimension two. Consider the integral curves of a flow on the 3-torus $\mathbb{R}^3 \, / \,\mathbb{Z}^3$, defined by $x' = 0, y'=\cos 2\pi x, z'=\sin 2 ...

6
votes

Accepted

### Algebraicity and non-algebraicity of leaves of the characteristic foliation

Suppose $X$ is projective manifold endowed with holomorphic symplectic form. Let $D$ be smooth divisor on $X$.
Characteristic foliation with algebraic and non-algebraic leaves.
Suppose that $X$ is ...

6
votes

### Euler Class constant on Fibered Face of Unit Thurston Norm Ball?

Have a look into Fried's Exposé: "Fibrations over $S^1$", particularly the proof of Theorem 14.6. He proves first the linearity on a neighborhood, which you understood already, then shows the ...

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