The foliations tag has no wiki summary.

**3**

votes

**1**answer

133 views

### 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 ...

**3**

votes

**0**answers

53 views

### 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 ...

**10**

votes

**2**answers

1k views

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?
Please see : Most interesting mathematics mistake?
Added 1: According to their method, ...

**1**

vote

**0**answers

122 views

### 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?

**2**

votes

**1**answer

138 views

### 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 $n-2$ ...

**4**

votes

**2**answers

428 views

### Limit cycles as closed geodesics(geodesiable 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 ...

**2**

votes

**0**answers

51 views

### 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 ...

**3**

votes

**0**answers

153 views

### What is a pure algebraic interpretation for this dynamical property?

According to comments of Yves Cornulier to the previous version of this question I revise the question as follows:
To what extent the following types of Lie algebras $A$ are classified? And what is ...

**0**

votes

**1**answer

94 views

### 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 ...

**0**

votes

**1**answer

109 views

### 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 ...

**2**

votes

**0**answers

106 views

### 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 ...

**0**

votes

**0**answers

106 views

### 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 ...

**0**

votes

**0**answers

65 views

### 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 ...

**2**

votes

**0**answers

109 views

### 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 ...

**1**

vote

**0**answers

82 views

### 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, ...

**2**

votes

**1**answer

109 views

### 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), ...

**8**

votes

**2**answers

453 views

### 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 ...

**11**

votes

**2**answers

174 views

### 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 ...

**5**

votes

**3**answers

388 views

### 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 ...

**1**

vote

**1**answer

107 views

### 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 ...

**3**

votes

**2**answers

230 views

### 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 ...

**0**

votes

**1**answer

98 views

### 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 ...

**10**

votes

**3**answers

448 views

### 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 ...

**2**

votes

**0**answers

87 views

### 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 ...

**2**

votes

**1**answer

379 views

### 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 ...

**8**

votes

**3**answers

353 views

### 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 ...

**0**

votes

**0**answers

129 views

### 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 ...

**3**

votes

**1**answer

250 views

### 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) = ...

**1**

vote

**1**answer

294 views

### 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 ...

**8**

votes

**1**answer

383 views

### 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 ...

**5**

votes

**3**answers

675 views

### Examples and non-examples of Riemannian foliations

Recall a tranverse metric on a (regular) foliated manifold $(M,F)$ is a positive symmetric $C^\infty (M)$-bilinear form $g$ such that
1) $Ker(g_x)=T_x F$
2) It is invariant with respect to lie ...

**1**

vote

**0**answers

137 views

### Linearization of singular foliation in the plane

Hello,
I would like to obtain a smooth model around a singular point of a foliation (in my case the model should be the linearization of the foliation). It seems that the answer could be hidden in ...

**2**

votes

**0**answers

150 views

### Kähler Cones in $\mathbb{C}^4$ and a foliation of $\mathbb{P}^3$

Take the 3-dimensional complex projective space $\mathbb{P}^3$. Consider the action of the group $SU(2)\times SU(2)$. I have read in physics related articles that these group gives a singular ...

**10**

votes

**1**answer

822 views

### Question about Thurston's paper “A norm for the homology of 3-manifolds”

I have a question about the proof of Theorem 5 in Thurston's paper "A norm for the homology of 3-manifolds". It's the theorem that asserts that the fibered faces are, well, fibered. More precisely, ...

**2**

votes

**1**answer

496 views

### Orbits of Lie Algebra Actions

It is well known that the image of a free Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$ on a manifold $M$ is an integrable distribution of constant rank $(=\rm{dim}\;\mathfrak{g})$. Thus ...

**3**

votes

**1**answer

285 views

### extended forms from foliations [closed]

hi,
i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...

**1**

vote

**2**answers

240 views

### Extension of integrable distribution over a subset

Let $M$ be a smooth manifold and $G_k(M)$ be the $k$-dimensional Grassmian bundle of $M$. Let $K\subset M$ be a compact subset and $E:K\to G_k(M)$ be a continuous distribution on $K$.
We say $E$ is ...

**2**

votes

**1**answer

244 views

### Conformally immersed Riemann surfaces and foliations

I want to show that conformally immersed Riemann surfaces in $\mathbb{R}^4$ are leaves of a 2-foliation $\mathcal{F}$. I start with the generalized Weierstrass representation of the surfaces: take 4 ...

**10**

votes

**1**answer

865 views

### Reebless and taut foliations

Suppose we are given a closed oriented 3-manifold.
It is well known that taut foliations are Reebless, and if a Reebless foliation isn't taut then the leaves which don't admit a closed transversal are ...

**0**

votes

**1**answer

197 views

### Does an abelian group acting on a riemaniann manifold define an othogonal foliation?

This question is related to my previous question. Suppose that a group $G$ acts freely and properly on a Riemaniann manifold $(M, g)$. Than the orbits form a foliation for $M$. For $p \in M$, let ...

**6**

votes

**2**answers

641 views

### Orthogonal foliations

Consider the manifold $\mathbb{R^2}\setminus \{0\}$, on which the group of rotation acts. The orbits of the group are the circles centered in the origin, and form a foliation of $\mathbb{R^2}\setminus ...

**4**

votes

**2**answers

462 views

### Is there a “geometric” language that describes the equivalence groupoid of a foliated manifold?

Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions.
Here, ...

**3**

votes

**1**answer

318 views

### Preprint of Hamilton on deformations of foliations

Does anyone have access to Hamilton's 1978 Cornell preprint 'Deformation Theory of Foliations'. It is widely quoted but I couldn't find any online copy.

**10**

votes

**0**answers

376 views

### Codimension 2 foliations on simply connected 4-manifolds

Are there examples of codimension 2 foliations on simply connected compact 4-manifolds such that
Every leaf is diffeomorphic to $\mathbb R^2$
Every leaf is dense?
Same question for 5-manifolds ...

**7**

votes

**2**answers

313 views

### Hypersurfaces orthogonal to a cone

This question is somewhat related to Differential inclusions for distributions. but I am asking for something rather more specific, so I hope it is alright to leave this as a separate, new question.
...

**0**

votes

**0**answers

115 views

### Differential inclusions for distributions.

Given a set valued function $F$ such that for every $x\in M$ (a manifold) we have that $F(x)\subset T_xM$, a differential inclusion is the "equation", $\dot{x} \in F(x)$.
I was wondering if someone ...

**2**

votes

**1**answer

301 views

### Integrability of distributions close to a given one.

In this and this papers Thurston proves that every distribution is homotopic to an integrable one (in the first one for codimension greater than one and in the other for codimension one).
Recently, ...

**5**

votes

**1**answer

1k views

### Differential forms, PDE's and Élie Cartan

Hello everybody, I would like to know about the work of Élie Cartan of PDE's that relate to the theory of foliations and differential forms.
I am interested in the subject and will be happy to ...

**2**

votes

**1**answer

135 views

### Some more questions about regularity of homeomorphisms of foliations

This is a continuation of A question about regularity of foliations>this question answered by Dmitri.
Let $F$ and $F'$ be smooth ($C^\infty$) foliations of a manifold $M$. Assume that there is a ...

**5**

votes

**3**answers

267 views

### A question about regularity of foliations

Let $F$ be a smooth foliation of a torus. Assume that $F$ can be mapped by a homeomorphism to an irrational-straight-line foliation $L$. Does it follow that $F$ can be mapped to $L$ by a ...