# Questions tagged [foliations]

The foliations tag has no usage guidance.

267
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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 ...

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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 $u: M \to \mathbb R$. (...

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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 ...

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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 ...

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### Leaf holonomy of Reeb foliation on mobius strip

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

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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(\Lambda^...

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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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### 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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### 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 $Z \subset \mathbb{CP}^n$ ...

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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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### 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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### 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. ...

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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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### 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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### 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(...

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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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### 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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### 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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### 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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### 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 (...

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### Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence

My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...

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### Geodesic foliations of open manifolds foliated by hyperbolic spaces

It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700).
Suppose a complete ...

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### Codimension-1 foliations of Euclidean space with strictly positive normal bundle

I am interested in the following situation: Given any $n>1$ suppose I have a codimension-1 foliation of $R^n_{++}$ (i.e. the subset of strictly positive $n$-vectors) arising from an $(n-1)$-...

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### A foliation version of S.Husseini counter example in fixed point theory

In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977
"The Products of Manifolds with the f.p.p. Need Not have the f.p.p"
who gave an example of two ...

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### Name of a foliation on an open subset of $\mathrm{PSL}(2,\mathbb R)$

To a representative $\left(\begin{array}{cc} a&b\\c&d\end{array}\right)$ of an element in $\mathrm{PSL}(2,\mathbb R)$ we associate
the point $\tau=\frac{b+id}{a+ic}$ of the Poincaré upper half-...

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### An algebraic foliation of $\mathbb{C}^2$ which admits a non-algebraic complex limit cycle $L$ with $L\cap \mathbb{R}^2=\emptyset$

Is there a polynomial vector field on $\mathbb{R}^2$ whose corresponding singular complex foliation of $\mathbb{C}^2 $ admits a complex limit cycle $L$ such that $L$ does not intersect the real ...

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### An algebraic foliation of $\mathbb{C}^2$ with real coefficients whose all complex limit cycles intersect the real plane $\mathbb{R}^2$

Is there a polynomial vector field $X$ on $\mathbb{R}^2$ such that every complex limit cycle
of the corresponding foliation of $\mathbb{C}^2$ must necessarily intersect the real plane $\mathbb{R}^2$.
...

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### Two codimension one foliations of a Lie group whose Godbilon–Vey invariants are not the same

What is an example of a Lie group $G$ with two codimension one foliations $F_1 $ and $F_2$ such that they generate two different Godbilon–Vey invariants in $H^3(G)$?

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### Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)

This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. ...

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### A foliation with prescribed graph of foliation

**I have already asked this question on MSE https://math.stackexchange.com/questions/4272279/1-dimensional-foliation-of-surfaces-with-prescribed-graph-of-foliation **
Definition of the graph of a ...

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### Minimal sets of foliations in the plane (generalisation of Poincaré-Bendixson)

Let $F$ be a $1$-dimensional foliation of an open subset of the plane defined by a
locally Lipschitz line field. Suppose $C$ is a compact minimal set of $F$ (i.e. $C$ is non-empty, compact, a union of ...

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### Is a linear vector field a geodesible vector field?

I have already asked this question in MSE; I repeat it here at MO.
Assume that $A\in M_n(\mathbb{R})$ is a non-singular matrix.
Is the flow of linear vector field $X'=AX$ a geodesible flow on $\...

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### Liouville property in the Bost theorem on foliations

Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...

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### Which elements of the fundamental group can be realized as transversals of a taut foliation?

Specifically in a closed, orientable 3-manifold. I'm not necessarily looking for a complete answer, as I don't expect one. Is there prior literature on this question? Also interested in this question ...

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### What is a holomorphic foliation?

For a smooth foliation $F$, there are three equivalent definitions:
the leaves of $F$ are tangent to a smooth vector field;
the foliation chart $\phi:U\to \mathbb R^k\times \mathbb R^{n-k}$ is ...

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### Smooth rank one foliations with closed leaves

Let $F$ be a smooth rank one foliation on a manifold $M$. Suppose that all leaves of $F$ are compact (that is, circles). Then its leaf space (edit: when additional assumptions are taken) is an ...

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### Complex fibration over complex torus

Let $M$ be a 3-dimensional complex manifold, and $\Lambda$ a discrete lattice in $\mathbb C^2$. Suppose there is a holomorphic submersion $f:M\to\mathbb{C}^2/\Lambda$ with fibers given by 1-...

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

Notation:
We say two $C^1$ manifolds are $C^1$-homeomorphic if they are homeomorphic via a $C^1$ homeomorphism with $C^1$ inverse.
Question:
Let $n \geq 2$. Given a countable dense set of points $P \...

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

I recently got to know about the existence of the so-called Godbillon–Vey invariant, and I am interested in its relationship with foliation theory in 3-manifolds. I briefly recall here the definition:
...

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### Orthonormal frame on $\mathbb{S}^3$ orthogonal to foliations [duplicate]

Does there exist a smooth orthonormal frame $X_1,X_2,X_3$ on $\mathbb{S}^3$ such that the distribution spanned by $X_i$ and $X_j$ is integrable for all $1\leq i,j\leq 3$?