# Questions tagged [foliations]

The foliations tag has no usage guidance.

220
questions

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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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53 views

### Parametric general position theorem for foliations

The situation is the following: let $M$ be a manifold endowed with a smooth foliation $\mathcal{F}$ of codimension one (suppose orientable, transversely orientable) and let $F_t : S \rightarrow M$ be ...

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88 views

### 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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23 views

### Any reference on smooth foliations with only Morse-singularities but possibly admitting saddle connections?

Morse foliations are assumed to have no connection between saddles. I am looking for any reference on smooth foliations with only Morse-singularities but possibly admitting saddle connections.

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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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157 views

### 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 ...

**4**

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80 views

### Orbits space of real-analytic planar foliations

Consider a planar foliation (e.g. coming from the trajectories of a 2-dimensionnal vector field). Its orbits space (quotient of $\mathbb R^2$ by the relation "lying on the same trajectory") ...

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44 views

### 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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67 views

### 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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173 views

### 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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29 views

### 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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92 views

### 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:
...

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votes

**2**answers

153 views

### 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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106 views

### 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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156 views

### 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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128 views

### 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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119 views

### 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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492 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. ...

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130 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-...

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131 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 ...

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99 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 ...

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133 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 ...

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126 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 ...

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84 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 $...

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91 views

### 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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33 views

### 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 $\...

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140 views

### Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves

Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the ...

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195 views

### Foliation of tangent bundle arising from exponential map

We first mention our motivation:
For $M=\mathbb{R}$ with usual Riemannian metric the exp map $exp:TM\to M$ is in the form$(x,v)\mapsto x+v$
The level sets of this map define a foliation whose leaves ...

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155 views

### Volume-preserving flows with cross section

Let $M$ be an orientable closed smooth manifold of dimension n. Let $\Omega$ be a volume form for $M$, i.e., a nowhere-zero smooth n-form. A smooth $\Phi_t$ flow defined on $M$ is volume-preserving ...

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52 views

### Approximating a volume along a submersion

Here is the setup. I have a submersion $f:X \to T$, where $X$ is manifold and $T$ is a torus (you can chose a circle for the beginning if it is simpler). The manifold $X$ has a volume form $\alpha$ ...

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39 views

### Unique poine in holonomies

Let $\Lambda$ be Axiom A for $C^{1+\gamma}$ $f$. I am reading this paper. I have a problem to undestand holonomies. The holonomy mapping
$$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \...

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605 views

### Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?

More precisely, is there a criterion that decides the above question?
I am particularly interested in the smooth setting: is a smooth manifold with a smooth regular foliation by circles covered by a ...

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212 views

### On Colding-Minicozzi limit lamination theorem

Colding and Minicozzi proved the following limit lamination theorem (see Theorem 8.26 in "A course in Minimal Surfaces" by Colding and Minicozzi for the complete statement)
THEOREM: Let $\Sigma_i \...

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123 views

### A non-geodesible foliation of $S^3$ or $S^2\times S^1$

Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation?
If the answer is ...

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51 views

### Two multi-curves in a surface with the same transverse measure

Let $(\cal F,\mu)$ be the stable measured foliation of a pseudo-Anosov on an oriented surface $S$. Can there be two non-isotopic multi-loops (collections of disjoint simple loops) $L_1,L_2\subset S$, ...

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123 views

### Integrability of certain distribution associated to a connection form on the total space of a principal bundle (Principal Frobenius condition)

Let $P\to M$ be a $G$-principal bundle where $P,M$ are smooth manifolds and $G$ is a Lie group with Lie algebra $\mathfrak{g}$, whose center is denoted by $C(\mathfrak{g})$.
Let $\omega$ be the ...

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66 views

### Equivalence of two approaches to transverse measures for a foliation

Suppose that $(V,F)$ is a foliated manifold. There are three equivalent approaches to the notion of transverse measure as described in this book (see pages 65-69). I would like to understand the last ...

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64 views

### A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field

Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\...

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110 views

### An extra condition on Frobenius theorem for $1$-forms

Consinder a smooth manifold $M$ and $\omega$ a smooth $1$-form on $M$. Assume that there is an open set $U\subset M$ such that $\omega$ never vanishes on $U$. One can define a smooth distribution $\...

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621 views

### Hilbert 16th problem and dynamical Lefschetz trace formula

I would like to apply the known version of the conjectural formula (11) page !0 of the paper Number theory and dynamical Lefschetz trace formula.
Disclaimer: I do not have a complete ...

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303 views

### Leafwise de Rham cohomology (A true definition of differential forms along leaves)

For a foliated space $(M, \mathcal{F})$, one associate a leafwise de Rham cohomology. This cohomology and trace-class operators on this cohomology and trace interpretations for closed orbits of ...

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309 views

### A non-vanishing vector field on $S^3$ whose flow does not preserve any transversal foliation

Is there a non-vanishing vector field $X$ on $S^3$ which does not admit a transversal $2$-dimensional foliation? if the answer is negative, is there a non-vanishing vector field $X$ on $S^3$ which ...

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172 views

### On the definition of the Reeb foliation

To define the Reeb foliation on the sphere, one needs to fix two even functions of the from $f:(-1,1)\to\mathbb{R}$.
In the book I. Tamura, Topology of Foliations: An Introduction, the following is ...

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63 views

### Foliations with algebraic foliation chart

An algebraic foliation chart for a foliated manifold is a foliation chart for which the transition maps are polynomial maps.
What is an example of an analytic foliation of the Euclidean space $\...

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71 views

### Holomorphic Foliations of 3-manifolds with boundary

Let $M$ be a 3-manifold $M$ with boundary $\partial M$, and endow $\partial M$ with the structure of a Riemann surface. Does there exist a foliation of $M$ by Riemann surfaces such that $\partial M$ ...

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269 views

### A vector bundle associated to a codimension $1$ submanifold of a symplectic manifold

We consider the standard symplectic structure $\omega=\sum dx_i\wedge dy_i$ on $\mathbb{R}^{2n}$. To every codimension $1$ submanifold $M\subset \mathbb{R}^{2n}$ we associate a vector bundle ...

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### A Singular Foliation of $\mathbb{C}P^2$ which does not admit a global transverse submanifold

Is there a singular holomorphic foliation $F$ of $\mathbb{C}P^2$ which does not admit a global transverse holomorphic curve? More precisely there is no an immersed holomorphic submanifold ...

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234 views

### Foliation with trivial leaf holonomy

In 1960, R. Hermann showed the following:
Theorem Let $M$ be a manifold with a foliation $F$ and a bundle-like metric, if all leaves are compact and the holonomy group of each leaf is trivial, then $...

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82 views

### Circle foliations not induced by circle actions on an compact orientable manifold

It is known that if we have an orientable fiber bundle $E\to B$, with fiber a circle $\mathbb{S}^1$, then it is a principal $SO(2)$-bundle. In other words, the fibers are spanned by the orbits of a ...

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140 views

### A singular holomorphic foliation of $\mathbb{C}^2$ with a bounded leaf

Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.