# Questions tagged [foliations]

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173 views

### 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 ...
167 views

### 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 ...
85 views

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

### 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 ...
110 views

### 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 $M$
153 views

### 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 ...
1 vote
97 views

### 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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### 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 ...
1 vote
64 views

### 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  \...
181 views

### 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 ...
194 views

### 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 ...
67 views

### 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 ...
1 vote
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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 $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
201 views

### 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 ...
447 views

### 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 ...
270 views

### 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 ...
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-...