The foliations tag has no wiki summary.
2
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1answer
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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 $n-2$ ...
4
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309 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
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0answers
47 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 ...
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0answers
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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
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1answer
93 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
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1answer
105 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 ...
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99 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 ...
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98 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 ...
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64 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 ...
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0answers
105 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 ...
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0answers
81 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
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1answer
100 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), ...
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2answers
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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 ...
7
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1answer
138 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 ...
6
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3answers
358 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 ...
0
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1answer
93 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 ...
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2answers
211 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
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1answer
92 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 ...
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3answers
428 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
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0answers
84 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
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1answer
350 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 ...
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3answers
348 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 ...
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0answers
121 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
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1answer
242 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
$$T_f C^\infty(M,N) = ...
1
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1answer
292 views
When are $k$-sectors of a Lie groupoid a manifold?
Let ${\mathcal{G} = \lbrace s,t:G_1 \to G_0 \rbrace}$ be a Lie groupoid. Define
$$(\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
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1answer
370 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 ...
4
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3answers
635 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 ...
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0answers
133 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
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0answers
145 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
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1answer
771 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
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1answer
480 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
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1answer
283 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
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1answer
172 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
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1answer
237 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 ...
9
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1answer
816 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
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1answer
195 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 ...
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2answers
620 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
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2answers
454 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
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1answer
308 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
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371 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
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2answers
309 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
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0answers
114 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
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1answer
291 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, ...
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1answer
983 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
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1answer
131 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 ...
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3answers
263 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 ...
5
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1answer
412 views
When does a submersion have connected fibers?
Can we characterize when a submersion $F:M \to N$ between two smooth manifolds has connected fibers? If this is too hard, what are some sufficient conditions?
5
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1answer
173 views
Local product structure horseshoes
I am reading the paper:
The Hausdorff dimension of horseshoes of Diffeomorphism of surfaces, Bulletin Brazilian Mathematical Society, Ricardo Mañe,(1990). Roughly speaking the author state that, the ...
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2answers
239 views
A local transitivity property of the automorphism group of a foliated manifold
Let $(M,\mathcal F)$ be a smooth foliated manifold. An automorphism of $(M,\mathcal F)$ is a diffeomorphism of $M$ that takes leaves of $\mathcal F$ onto leaves. Let now $L$ be a leaf of $\mathcal F$. ...
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3answers
747 views
Cohomology of a sheaf of functions locally constant along a foliation
Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology ...
