The foliations tag has no wiki summary.

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### A noncommutative vector bundle associated with a codimension one foliation

Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to ...

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### Product of two foliations

1.What is an example of a manifold $M$ with two foliations $F$ and $F'$ which are not topological equivalent but the product foliations $F\times F$ and $F'\times F'$, as foliations on $M\times M$, ...

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### Is this a $C^0$ foliation of $\mathbb{R}^2$?

Let $f(x)=\frac{1}{\sin(\pi x)}$ for $x\in (0, 1)$ and let
$\Gamma=\left\{(x,f(x)): x\in (0, 1)\subset \mathbb{R}^2\right\}$ be its graph.
For any set $X\subset \mathbb{R}^2$ and $\lambda>0$ and ...

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### Regularity of the taut foliation

In Eliashberg-Thurston's famous paper "Confoliations" Corollary 3.2.11, they proved that Irreducible three manifold with $b_{1}>0$ admits semi-fillable contact structure using Gabai's theorem in ...

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### A cohomology associated with a codimension one foliation(2)

What is an example of a codimension one foliation of a manifold for which this cohomology is finite dimension for all dimension $*$?
Moreover what is the description of this cohomology for ...

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### A Lie algebra associated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...

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### A cohomology associated with a codimension one foliation

Let $\alpha$ be a non vanishing one form on a manifold which which defines a codimension one foliation. With this $\alpha$ we define the following complex:
$$\phi:\Omega^{i}(M)\to ...

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### Foliation with leaves which are and are not dense

Do there exist a foliation on a closed surface (i.e. real dimension 2) which has a dense leaf and also a leaf which is not dense?

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### Lifting $SHFC$ to non singular foliations

In this question we would like to complexify the idea in the following post.
We would like to lift a "singular holomorphic foliation by curves", briefly "SHFC", of $\mathbb{C}P^{2}$ to a ...

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### Topology of the space of foliations on a 3-manifold

Denote by $\mathcal{P} (M)$ the space of smooth plane fields(oriented and transversely oriented) on a given closed and orientable 3-manifold $M$ with the $C^{\infty}$ topology, and by $\mathcal{F}(M)$ ...

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### When does a leaf space admit a (non-Hausdorff) manifold structure?

If $f:M \to N$ is a submersion with connected fibers, then the fibers of $f$ foliate $M$. This is called a simple foliation of $M$ and the leaf space can be identified with $N$. Suppose that a ...

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### Classification of complex Kronecker foliations

Let $\theta \in \mathbb{C}$ be a fixed complex number. The submersion $f:\mathbb{C}^{2}\to \mathbb{C}\; \text{with}\; f(x,y)=y-\theta x$ defines a complex foliation on $\mathbb{C}^{2}$. Consider the ...

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### A (different) foliation arising from Hopf fibration

In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...

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### A special non vanishing vector field on $S^{3}$

Is there a non vanishing vector field on $S^{3}$ with an infinite family $T_{\lambda}$ of invariant torus such that each $T_{\lambda}$ has an structure of a Kronecker foliation but for ...

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### Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...

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### On the realization of a compact surface as a leaf of an analytic foliation

Let $S$ be a compact orientable surface of genus $g \geq 2$. Is there any transversely real (or complex) analytic codimension one foliation $\mathcal{F}$ such that $\mathcal{F}$ has $S$ as a leaf with ...

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### One dimensional foliation of surfaces with prescribed graph of foliation

According to the definition of the graph of a foliation by Winkelnkemper we ask the following questions:
Let $G$ be one of the following non hausdorff 3 dim manifold
1) $G$ is a ...

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### Complementary integrable vector fields

Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric ...

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### Characterization of certain analytic vector fields on $S^{2}$

Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:
$X$ has a finite number of singularities on $S^{2}$
The equator is invariant under flow of $X$
3.$g_{*}X=X\; ...

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### Are codimension one foliations of $\mathbb{R}^{n}-\{0\}$ with compact leaves, stable at origin?

Assume that we have a codimension one foliation of $\mathbb{R}^{n}-\{0\}$ with compact leaves.
Is it true to say that the foliation is stable at origin:That is: for every neighborhood $V$ of ...

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### “The” kronecker foliation or “a” kronecker foliation?

Consider the following two foliations of torus:
1)The Kronecker foliation with slope $\sqrt{2}$
2)The Kronecker foliation with slope $\pi$
As I learn from the literature, these two foliations are ...

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### The type of a Riemann surface arising from a polynomial vector field

Consider the planar polynomial vector field $$\begin{cases} \dot x=P(x,y)\\ \dot y=Q(x,y)\end{cases}$$
It defines a singular foliation on $\mathbb{C}P^{2}$. Assume that a complex leaf contains ...

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### Holomorphic Foliations having transverse sections

In the introduction to the paper "On the Geometry of Holomorphic Flows and Foliations Having Transverse Sections" by Ito and Scardua, one reads the following "a holomorphic codimension one foliation ...

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### Hessian eigenspaces form integrable distributions on a Riemannian manifold?

Suppose $M$ is a Riemannian manifold and $f:M\to\mathbb{R}$ a differentiable function. I can form the Hessian $H$ of $f$ (with respect to the Levi-Civita connection); this is a symmetric bilinear ...

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### Foliation of the tangent bundle of $n$-sphere

Is there a smooth $n$-dimensional foliation of $TS^{n}$,( here $n\neq1,3,7$) such that the zero section be a leaf of this foliation?

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### Anti_symplectic 2-forms

A two form $\alpha$ on a n- manifold $M$ is called anti symplectic if for every $x\in M$, $\{ v\in T_{x} M \mid i_{v} \alpha=0 \}$ is a $n-2$ dimensional subspace of $T_{x}M$. So we obtain a $n-2$ ...

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### Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$:
\begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation}
This equation defines a foliation on ...

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### Extendability of Contact Structures; Foliations of $S^2$

I am currently reading Eliashberg's paper on the classification of overtwisted contact structures (http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf). In it, there is the following ...

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### Can an analytic set admit such a foliation?

I confess to be not an expert of analytic geometry, but I have come across the following problem, for which I need an help from experts in this specific field.
I was wondering myself if it is ...

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### An absolutely continuous foliation, which is not transversely absolutely continuous

Currently I'm studying "Introduction to dyamical systems" by Stuck and Brin. In chapter 6 they define absolutely continuous and transversely absolutely continuous foliations. By proposition 6.2.2 if ...

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### Foliation of surface all of whose leaves are circles

I'm having trouble locating a reference for the following basic fact. Let $S$ be a compact orientable surface with boundary. Assume that $\mathcal{F}$ is a foliation of $S$ all of whose leaves are ...

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### A question on lie groups( Lie algebras)

What is an example of a compact Lie group $G$ which is not isomorphic to a product of two nontrivial lie groups and satisfies the following property:
There are two non zero vector fields $X, Y \in ...

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### Foliation values of Exotic spheres

In the following question, we defined the foliation values of an smooth manifold;
Foliation values of a manifold
Let $S_{i}$'s, $i\in \{0,1,\ldots,27\}$, be the smooth structures of topological ...

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### Foliation values of a manifold

Let $M$ be a smooth n dimensional manifold. The foliation values of $M$, denoted by $F(M)$, is defined as
\begin{equation} F(M)=\{ 1\leq k\leq n\mid \text{there exist an smooth $k$ dimensional ...

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### Frobenius rank of a manifold

The rank of an smooth manifold M is defined by Milnor, as follows:
"The maximum number of independent commuting vector fields on M"
For example it is well known that the rank of $S^{3}$ is 1 (Lima, ...

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### the union of local stable manifolds along local unstable manifolds

Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s_\delta(x), ...

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### Can we foliate the punctured space by tori?

Is it possible to have a 2 dimensional foliation of $\mathbb{R}^{3}-\{0\}$ such that each leaf is homeomorphic to the torus? what algebraic topological obstruction exist?
Another question: is there ...

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### Is $\mathbb{H}^n$ quasi-isometric to a leaf of a codimension 1 foliation of a compact manifold?

If we extend the action of $\pi_1(\Sigma_g), g\geq 2,$ from $\mathbb{H}^2$ to its boundary $\partial_{\infty}\mathbb{H}^2=S^1$, the surface bundle corresponding to this action of $\pi_1(\Sigma_g)$ on ...

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### Deformation of foliation

Suppose $\kappa$ is a no-where vanishing 1-form, then its kernel is integrable is equivalent to condition $d\kappa \wedge \kappa = 0$.
My question is, can such foliation smoothly deformed such that ...

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### Existence of bundle-like metrics to a given foliation

Let $(M, \cal F)$ be a (compact) foliated smooth manifold. I would like to know if there always exists a bundle-like Riemannian metric $g$ for $\cal F$ (i.e. $({\cal L}_U g)(X,Y) = 0$ for all $U \in ...

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### Unstable Foliations

Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$.
Definition. $\mathcal{F}$ is said to be a unstable ...

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### taut foliations and the existence of total transversals

A codimension one foliation $\cal F$ on a smooth manifold $M$ is taut if every leaf of $\cal F$ meets a closed transversal (i.e., a simple closed curve that is everywhere transversal to the leaves of ...

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### Foliations by holomorphic curves on complex surfaces

On a complex surface, does there exist a non-singular foliation by holomorphic curves that is NOT a holomorphic foliation, i.e. transversally holomorphic foliation?
The surface should be compact and ...

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### A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation

Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential ...

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### C* Algebras, Foliations and Dynamical Systems

I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles so one ...

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### twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors

Given a regular (constant rank) bi-vector $\Pi \in \Gamma(\bigwedge^2TM)$ on a smooth manifold $M$ the necessary and sufficient condition for the image of $\Pi^\sharp:T^*M\to TM$ to be an integrable ...

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### Foliation over characteristic positive

Ekedahl wrote about foliation in characteristic positive, over the field $/frac{Z}{pZ}$ as a subsheaft of the tangent sheaft, that is closed over involution and $p$-power, My question is if there ...

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### Smooth functions tangent to the leaves of a foliation

Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space
$$T_f C^\infty(M,N) = ...

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### When are $k$-sectors of a Lie groupoid a manifold?

Let ${\mathcal{G} = \lbrace s,t:G_1 \to G_0 \rbrace}$ be a Lie groupoid. Define
$$(\mathcal{G}^k)_0:=\lbrace (a_1,\dots,a_k) \in G_1^k\mid s(a_1)=t(a_1)=\dots=s(a_k)=t(a_k) \rbrace$$
(This is the ...

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### A concept of dynamical coherence

I'm trying to make an overview of the study of partial hyperbolicity and there is an interesting concept of dynamical coherence which appears there. Some call it mild (see the Thesis of Pablo ...