# Questions tagged [foliations]

The foliations tag has no usage guidance.

287
questions

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### Smooth vs. topological: foliation into closures of orbits

Consider a (partial) map $f\colon X → X$ and the maximal closures of orbits of $f$ (i.e., closures of orbits which are not contained in larger closures of orbits). Assume that $X$ is foliated into ...

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### About the definition of singular holomorphic foliations

In [https://arxiv.org/abs/math/9809099] appears the following
Definition 1.4 : Let $M$ be a complex manifold. A singular holomorphic foliation by curves $\mathcal{F}$ on $M$ is a holomorphic ...

9
votes

1
answer

400
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### Are limits of compact leaves compact?

Let $M$ be a compact smooth manifold, and $\mathcal{F}$ be a foliation on $M$. Assume that $L$ is a leaf of $\mathcal{F}$ for which there is $x\in L$ with the property that every neighborhood of $x$ ...

5
votes

1
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226
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### Topology of windings on the two-torus

In short my question is: what can we say about the quotient topology induced by the linear flow on a two-torus?
I know that an irrational slope leads to a dense winding and hence (if I'm not mistaken) ...

2
votes

0
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78
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### Simply connectedness of leaves of a foliation on an complex manifold

Now I'm searching about leaves of foliation in the following special setting.
Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...

1
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0
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70
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### Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...

1
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1
answer

48
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### Bound on the sum of intersection number of any projectivized measured foliation with two transverse measured foliations

Let $R$ be a finite Riemann surface (having negative Euler Characteristic) without boundary (may have punctures) and $q$ be a unit area quadratic differential on $R$. We define $\mathcal{MF}_{1}=\{F \...

6
votes

1
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257
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### Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gamma_q$?

Recall that Bott's obstruction for integrability [Bott70] asserts that:
Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every ...

6
votes

1
answer

244
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### Can differential forms be exact and positive on a distribution?

Let $M$ be a manifold of dimension $d$, and let $\mathscr D$ be a distribution of rank $d - 1$ on $M$ (I would also be interested in lower rank distributions, but mainly I am interested in codimension ...

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1
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80
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### Number of ergodic transverse measures for geodesic laminations - bounded by the genus?

Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, ...

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117
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### Which holomorphic curves can be leaves of a non-singular holomorphic foliation of $\mathbb C^2$?

It is easy to see that for any entire function $f : \mathbb C \to \mathbb C$, its graph $G(f) = \{(z,f(z)) \in \mathbb C^2 \mid z \in \mathbb C\}$ can be translated by $(0,c)$ for any $c \in \mathbb C$...

5
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242
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### Aspherical space whose fundamental group is subgroup of the Euclidean isometry group

Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...

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1
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122
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### Foliation of spaces

It turns out that a very important idea to derive properties for a bigger space is to try to foliate the space, derive the same property for each leaf and patch everything up to get the desired ...

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366
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### A (possible) Lie algebra extension of the Lie algebra of a foliation

Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is ...

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### Almost Riemannian foliation

A Foliation $\mathcal{F}$ on a Riemannian manifold is called almost Riemannian foliation if $$\forall \epsilon >0 \quad \exists \delta >0 $$ such that for every leaf $L$ and every geodesic $...

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1
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136
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### Non-absolutely continuous foliation

What is a simple example of a (continuous) foliation of a manifold that is not absolutely continuous?
(A foliation is said to be absolutely continuous if holonomy maps between smooth transversals send ...

6
votes

1
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475
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### The current situation of the Godbillon-Vey invariant conjecture

Suppose we have a compact three-manifold $M$, a codimension-one foliation $F$ of $M$, and a one-form on $\alpha$, chosen so that $F$ is tangent to $\alpha = 0$. We deduce that $\alpha \wedge d\alpha = ...

2
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1
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93
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### A 1 dimensional foliation of $\mathbb{R}^4$ with few compact leaves

Inspired by An algebraic Hamiltonian vector field with a finite number of periodic orbits (2) we ask if there is a 1 dimensional analytic foliation of $\mathbb{R}^4$ which has at least 1 compact ...

3
votes

1
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76
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### Can the Reeb foliation of $S^3$ be realized as stable manifold foliation of a smooth hyperbolic discrete dynamic on $S^3$?

Can the Reeb foliation of $S^3$ be realized as foliation associated to stable(or unstable) manifolds of a hyperbolic discrete dynamic on $S^3$?If yes what is a precise formulation for that ...

5
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2
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### A smooth family of lattices on the tangent bundle?

I was recently in the cafeteria with a friend, and while having lunch I explained to him why the tangent bundle of a manifold is good at encoding geometric information of the manifold. My second ...

2
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1
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259
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### Learning roadmap for holonomy theory

During my Master's thesis I encountered the theory of holonomy for the first time. Unluckily it was only tangentially related to the topic of my thesis, so I couldn't dive into it.
The book I was ...

2
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### How "big" is the space of global solutions of an eikonal Hamilton-Jacobi equation?

By an eikonal Hamilton-Jacobi equation I mean a PDE of the form
$$H(x, du(x)) = 0$$
where the Hamiltonian $H: T'M \to \mathbb R$ is given and we are solving for a function $u: M \to \mathbb R$. (...

3
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181
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### Looking for examples of non-singular holomorphic foliations with compact leaves

I am looking for examples (or what is known about) of the following kind of object:
X compact Kähler manifold
F a non-singular holomorphic foliation on X (so given by a holomorphic subbundle of the ...

3
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0
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176
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### Rational points on curves x^p + y^p = 1 in the open unit square, for p > 0

For each real p > 0, let Cp = {(x,y) ∊ (0, 1)2 | xp + yp = 1}.
It is easy to see that each Cp is topologically an open interval that is the interior of a smoothly embedded closed interval with ...

2
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1
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220
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### Leaf holonomy of Reeb foliation on Möbius strip

I am trying to understand the leaf holonomy of the Reeb foliation on the Möbius strip, the first problem being visualization. I have been unable to find a visualization of this anywhere. I am ...

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2
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351
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### Exterior differentiation of foliations

Let $M$ be a differentiable manifold.
Let $T^*M$ be the cotangent bundle of $M$.
Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\...

2
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48
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### Transnormal foliation with non-smooth transnormal function

I am interested in results regarding transnormal foliations on a Riemannian (smooth, connected and complete) manifold $(M,g)$. More specifically, a smooth function $f:M\to{\bf R}$ is called ...

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117
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### Foliation of $X$ by once punctured planes without any singularities

Let $n=3.$
Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$ Construct a smooth regular foliation of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-...

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1
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### Vector fields tangent to distributions with zero first Chern class

Let $\mathcal{F}$ be a saturated coherent subsheaf of $T\mathbb{CP}^n$. In particular, $\mathcal{F} \subset T\mathbb{CP}^n$ is a holomorphic vector subbundle outside a subset $Z \subset \mathbb{CP}^n$ ...

3
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116
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### Is there a notion of representation theory of foliations?

A foliation on a manifold $M$ can be seen as a sub bundle of the tangent bundle $\mathcal{F}\subseteq TM$ that is closed under Lie bracket of vector fields.
One can think of foliation as a Lie ...

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### Holonomy of foliation with trivial normal bundle

I am wondering about the following situation. Suppose $X$ is a compact Kahler manifold and $F \subset T_X$ is a holomorphic foliation. Suppose that the ``normal bundle'' $T_X / F$ of the foliation is ...

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83
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### A 1 dimensional foliation which is Riemannian foliation with respect to no Riemannian metric

What is an example of a non vanishing smooth vector field on a manifold $M$ whose corresponding foliation is a Riemannian foliation with respect to no Riemannian metric on $M$

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159
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### A closed leaf with two different index with respect to two different Riemannian metrics

Inspired by this question about Jacobi equation. conjugate points and limit cycle theory we ask the following question:
Is there a geodesible 1 dimensional foliation $\mathcal{F}$ on a manifold $M$, ...

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### Leaves of bounded genus

Let $\mathcal{F}$ be a codimension one foliation in a closed $3$-manifold $M$. Does there exist an upper bound for the genus of the compact orientable leaves? That is, does there exist $G >0$ such ...

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### Homothety vector fields generating a foliation of $S^3$

Inspired by this question on homothety vector fields we realize that non homotheticity is some how an intrinsic property of the foliation associated to the vector field. See the comment by Prof. ...

3
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### Vector fields $X$ and $Y$ commute on a closed set $K$. Do there exist commuting $\tilde X,\tilde Y$ with $\tilde X=X$ and $\tilde Y=Y$ on $K$?

I have a nice research idea whose proof hinges on the following question
Suppose $X_p$ and $Y_p$ are vector fields in $\mathbb{R}^3$ with $[X,Y]_p=0$ for all $p$ in some closed set $K\subset\mathbb{R}...

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84
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### about codimension two foliation

Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...

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220
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### On "graphs" of foliations

Let $M$ be a smooth manifold and $\mathcal{F}=\{\mathcal{F}_m\}_{m\in M}$ be a (regular) smooth foliation of $M$. The leaves $\mathcal{F}_m$ are smoothly immersed and moreover weakly embedded ...

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### Which submanifolds are leaves of a foliation?

Question. Let $M^{n+1}$ be a closed manifold without boundary. Which closed submanifolds $\Sigma^n \subset M^{n+1}$ (of codimension one) are leaves of a foliation of $M$ minus some finite collection ...

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### Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?

Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...

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### Is the orbit foliation of the Weyl chamber flow Riemannian?

$\DeclareMathOperator\SL{SL}$Fix an integer $p\geq 1$ and a cocompact lattice $\Gamma\subset \SL(p+1,\mathbb{R})$. Consider the manifold
$$
M_{\Gamma}:=\SL(p+1,\mathbb{R})/\Gamma.
$$
Let $A\subset \SL(...

3
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1
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### Do taut foliations leafwise branch covering S^2 yield foliations by circles?

In this paper, Danny Calegari shows that taut foliations in (let's say closed for simplicity) 3-manifolds are precisely those which admit a map $f: M \to S^2$ which restricts to a branched cover on ...

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### Deformation cohomology of a foliation

Let $\mathcal{F}$ be a foliation on a manifold $M$, and denote by $N\mathcal{F}$ its normal bundle. There is a flat $T\mathcal{F}$-connection on $N\mathcal{F}$, called Bott connection, given by
$$
\...

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1
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### Global transversals of a codimension one foliation

EDIT: changes to the question are in bold.
Suppose I have a smooth ($C^1$) codimension one foliation $\mathscr{F} \subset P$, the open subset of $R^n$ consisting of points with all positive ...

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### Contact structures associated to taut foliations

Eliashberg and Thurston showed that a taut foliation may be deformed to tight (positive and negative) contact structures. Vogel proved that for a taut foliation without torus leaves, the associated ...

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### Reference for foliation on orbifolds

Can anyone recommend a good reference for foliation on orbifolds ? Thanks!

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396
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### Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...

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### Calabi–Yau theorem and complex Monge–Ampère equation for transversally Kähler manifolds

Let $M$ be a compact smooth
manifold, and $F\subset TM$ a smooth
foliation. It is called transversally Kähler
if the normal bundle $TM/F$ is equipped with
a Hermitian structure (that is, a complex ...

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0
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### Results on compact slices in a regular foliation

Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...

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147
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### Survey or good reference of taut foliations

I am interested in the topology of foliations.
In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows.
I guess that
A. Candel and L. Conlon, Foliations I (...