# Questions tagged [foliations]

The foliations tag has no usage guidance.

244
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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
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### A map in cohomology for a pair of foliations

Let $M$ be a compact manifold with a pair of transverse foliations $\mathcal{F}$ and $\mathcal{G}$, i.e.
$$
TM=T\mathcal{F}\oplus T\mathcal{G}.
$$
Restriction of differential forms to the leaves of $\...

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0
answers

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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 (...

3
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0
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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}:=\...

2
votes

0
answers

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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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1
answer

148
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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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### 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 ...

2
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answers

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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-...

3
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101
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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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0
answers

73
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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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116
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### Show that the foliations $\mathcal{F}$ and $\mathcal{F}^{\prime}$ have isomorphic germs at the compact leaf iff holonomy homomorphisms are conjugate

I have a question about proof of theorem 2.3.9 in Foliations I, by Candel and Conlon (AMS, GSM)
$L$ is a compact leaf in two foliated $n$-manifolds $(M, \mathcal{F})$ and $\left(M^{\prime}, \mathcal{F}...

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0
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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 ...

4
votes

1
answer

131
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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 $\...

3
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answers

91
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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 $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...

5
votes

1
answer

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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 ...

3
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198
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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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votes

2
answers

222
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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 ...

2
votes

1
answer

186
views

### 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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votes

1
answer

257
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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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2
answers

265
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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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1
answer

130
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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 $1\leq i,j\leq 3$?

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### Existence of closed transversals for taut foliations in arbitrary codimension

There are several different definitions of "tautness" for foliations, the most widely know is probably topological tautness, which is specific to codimension one and means that the foliation ...

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184
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### Blowing up the zero section for "Chasse au Canard" (some new kind of geometric canards)

In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a ...

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1
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152
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### Orbits space of real-analytic planar foliations

Consider a foliation of $\mathbb{R}^2$, say coming from the trajectories of a vector field $X$. Its orbit space (the quotient of $\mathbb{R}^2$ by the relation "lying on the same trajectory")...

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### Clarification about the process of naturally endowing a space with a Riemann orbifold structure supported on a sphere

I am having some difficulties understanding an argument in a proof. Here is an excerpt from Lyubich–Peters - Classification of invariant Fatou components for dissipative Henon maps, first geometric ...

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### The diversity of Riemannian metrics adapted to a given foliation، A Krein Millman view point(2)

Inspired by this answer to the linked question we add a more bounded conditions to this post. This question is asked seperately because the previous one had a complete answer so we did not revise ...

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1
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### The diversity of Riemannian metrics adapted to a given (1 dimensional) foliation, A Krein Millman view point

Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. Let $K$ be the space of all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1$...

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### Complex $2$ manifold $M$ with a given real 2 dimensional submanifold which meets all complex limit cycles of foliations of $M$

Is there a $2$ dimensional holomorphic manifold $M$ with a closed $2$ dimensional real submanifold $A$ of $M$ such that for every singular holomorphic foliation of $M$, all leaf with non trivial ...

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### A quantity associated to a foliated manifold and its non-commutative interpretation

Let $M$ be a compact $n$-dimensional manifold. Assume that $F$ is a $k$-dimensional foliation of $M$.
The graph $G(M,F)$ of this foliation is a $(n+k)$-dimensional manifold. We recall its definition:
...

2
votes

2
answers

181
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### Why a Teichmuller map is not a pseudo-anosov?

Let $X$ be a riemannian surface. Suppose $f:X\to X$ is a Teihmuller map with respect to a quadratic differential $q$ on $X$. This means that, if $q=dz^2$ in local coordinates in a neighborhood of ...

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### Foliations and locally free action of $\mathbb{R}^{n-1}$

Let $M$ be a $n$-dimensional closed manifold endowed with a foliation ${\cal F}$ suppose that the leaves of ${\cal F}$ are diffeomorphic to $\mathbb{R}^{n-1}$ are the leaves of ${\cal F}$ defined by a ...

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187
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### Codimension two foliations with transverse surfaces

Suppose I have some closed $4$-manifold $X$ and a codimension-two foliation $\mathcal{F}$, as well as a closed surface $\Sigma$ of nonnegative self-intersection that is everywhere transverse to $\...

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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 ...

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### Examples of why conditions for Novikov compact leaf theorem are necessary

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^...

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### 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. ...

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### 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-...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $...

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### Existence of codimension 1 topological foliations

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 ...

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### A "singular" Tischler theorem

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 $\...