The foliations tag has no wiki summary.
3
votes
2answers
272 views
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 ...
1
vote
0answers
61 views
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 ...
1
vote
0answers
50 views
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 ...
1
vote
0answers
183 views
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 ...
1
vote
1answer
81 views
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 ...
7
votes
1answer
140 views
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?
0
votes
0answers
43 views
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 ...
4
votes
0answers
189 views
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)$ ...
3
votes
0answers
114 views
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 ...
2
votes
0answers
169 views
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 ...
2
votes
0answers
108 views
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 ...
3
votes
0answers
137 views
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 ...
0
votes
0answers
191 views
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 ...
1
vote
1answer
112 views
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 ...
0
votes
0answers
44 views
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 ...
2
votes
2answers
103 views
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 ...
0
votes
0answers
64 views
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\; ...
2
votes
0answers
109 views
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 ...
5
votes
1answer
132 views
“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 ...
1
vote
0answers
112 views
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 ...
4
votes
1answer
194 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
0answers
61 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 ...
1
vote
0answers
139 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
1answer
147 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
2answers
514 views
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 ...
2
votes
0answers
59 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 ...
0
votes
1answer
100 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
1answer
124 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
0answers
120 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
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0answers
119 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
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0answers
66 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
0answers
112 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 ...
2
votes
1answer
206 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
1answer
119 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
2answers
464 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 ...
12
votes
2answers
192 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
votes
3answers
453 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
1answer
145 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
2answers
257 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
1answer
108 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
3answers
504 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
0answers
89 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 ...
3
votes
1answer
462 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 ...
9
votes
3answers
365 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
0answers
133 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
1answer
269 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
1answer
302 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
1answer
409 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
3answers
815 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
0answers
154 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 ...
