# Questions tagged [foliations]

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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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### 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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### Vanishing of leafwise cohomology with coefficients

Let $M$ be compact manifold with a foliation $\mathcal{F}$. The Bott connection gives a representation of $T\mathcal{F}$ on the normal bundle $N\mathcal{F}$, defined by  \nabla_{X}\overline{Y}:=\...
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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 ...
1 vote
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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 $(n-1)$-...
1 vote
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### Are $f_1,f_2$ necessarily pseudo-Anosov?

Suppose $F: S_1\times S_2\to S_1\times S_2$ is an Anosov diffeomorphism (assume there is one), and $S_1,S_2$ are closed surfaces. If $F$ is homotopic to $f_1\times f_2$, a product of two surface ...
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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 $\mathbb{R}^2$. ...
1 vote
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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 ...
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 $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
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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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### Coisotropic foliation of regular Poisson manifolds

A Lagrangian foliation of a Poisson manifold (M, P) is a foliation F of M for which TF = P♯(AnnTF). It implies that P is regular, of rank twice the dimension of F. Now, Let (M, P) endowed with ...
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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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### 3-manifold foliated by circles is Seifert fibered

Let $M$ be a compact 3-manifold with boundary equipped with a 1-dimensional foliation all of whose leaves are circles. An old theorem of Epstein says that $M$ is a Seifert fibered space. The proof of ...
1 vote
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Let $M^3$ be a smooth, closed 3-manifold. Given a smooth codimension-one foliation $\mathcal{F}$ of $M$, the Novikov compact leaf theorem asserts that, when the universal cover $\widetilde{M}^3$ of $M^... 6 votes 1 answer 722 views ### Books on foliations I am looking for resources (books, notes, lecture video, etc. anything will do although printed material in English is preferable) on foliations which satisfy some or all of the following constraints. ... 1 vote 2 answers 166 views ### Transverse invariant measures to vector fields Given a smooth, non-vanishing vector field on a compact manifold, when does the 1-dimensional foliation given by its integral curves admit a transverse invariant measure? I've seen examples of higher-... -1 votes 1 answer 158 views ### Can you give me an example of a totally disconnected subgroup of a topological group that is not a topological group? [closed] In the book The topology of fiber bundles, Steenrod characterize bundle over a base$X$and totally disconnected structural group$G$as follows. Theorem: Let$X$be arcwise connected, arcwise ... 5 votes 0 answers 103 views ### Is there an orbit map without path lifting property? I am looking for an example of a topological group$G$acting by homeomorphisms on a metrizable space$X$such that the orbit map$X\to X/G$doesn't have the path lifting property, that is, there is a ... 4 votes 1 answer 153 views ### Leaves of stable foliation of holomorphic Anosov diffeomorphism I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a ... 2 votes 0 answers 128 views ### Elliptic foliations of the plane A$1$dimensional foliation of the plane$\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field$X$with the following properties: The differential operator ... 2 votes 0 answers 94 views ### Principal circle bundles that are smooth foliated Let$\xi=(\pi,E,B)$be an orientable circle bundle, i.e., a bundle with fiber$\mathbb{S}^1$and structural group$G=\textit{Diff}^+(\mathbb{S}^1)$. Claim 1: The bundle$\xi$admits a structure of$...
One of the many famous theorems proven by William Thurston is that a closed connected smooth manifold $M$ admits a codimension 1 smooth foliation if and only if $\chi(M)=0$: W.P. Thurston, Existence ...
The Tischler theorem says that a compact manifold $M$ admitting a closed nowhere vanishing $1$-form $\alpha$ fibers over the circle. I was wondering if anything could be said about the case where \$\...