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Questions tagged [foliations]

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65 views

What meaning of this sentence?

I study about Poisson geometry. You should be know that every Poisson structure induced singular foliation. I encountered this sentence “Two points lie in the same leaf if and only if one is ...
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44 views

Phase, Isochrons, Isochrons map and Lift

at the moment i read the following paper: https://arxiv.org/pdf/1512.04436v1.pdf I have some questions about it and i hope someone can help me. On page 4/5 they introduce isochrons and the isochron ...
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26 views

Lie group action and foliation

Let $M$ be a smooth manifold, and $G$ be a compact Lie group acting on $M$ smoothly. We assume that isotropic group of any point $x\in M$ is of dimension zero. Q How to show that $M$ admits a ...
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Integrability of distributions which are invariant under isometry group

Let $(M.g)$ be a Riemannian manifold and $D$ is a distribution on $M$. Assume that $D$ is invariant under the action of the isometry group of $M$. Under which conditions such $D$ ...
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Adapted connection on foliation

Let $(M,F)$ be a manifold with foliation, fixing a bundle-like metric $g$ we identifying $Q=TM/F\cong F^\perp$, we define the connection on $Q$, for any $s\in\Gamma(Q)$ $$\nabla_Xs=\begin{cases} ([X,...
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1answer
54 views

What “given a foliation $f:L \to M$ ” means?

I'm reading this paper. And on page $4$ the author writes the following paragraph: I am not sure about what the phrase "given a foliation $f:L\to M$" means. A foliation is an atlas, so it is awkward ...
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44 views

On morse theory and foliations

Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: ...
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32 views

Integrable system is not a level set: an example

For a manifold $M$, let $f \in C^{k}(M), k>0$, be a level set (i.e. $rank(df)=1$). I understand that this trivially implies that $\{{k.df: k \in C^{k}(M)\}}$ forms an integrable regular Pfaffian ...
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27 views

Intersection of two plaques is an open subset of both plaques.

I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from ...
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49 views

When a manifold is isometric to a product manifold?

Let $M$ be a Riemannian smooth manifold of dimension p and $\phi$ a smooth submersion from $M$ to some other smooth manifold $N$ of dimension q. Denote by $\mathcal{F}$ the foliation of leaves $\...
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Coherence of Foliated Atlases is an Equivalence Relation

I am studying foliations of manifolds and at the moment I am stuck with trying to prove that the relation of coherence of foliated atlases is an equivalence relation. I shall follow Candel and Conlon'...
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1answer
84 views

The tangent bundle of a foliation is in fact a vector bundle

If $\mathcal{F}$ is a foliation on an $n$-manifold $M$, then the tangent bundle to $\mathcal{F}$, which I'll call $\mathrm{T}\mathcal{F}$, is a $k$-plane distribution on $M$, where $k$ is the ...
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1answer
72 views

Godbillon-Vey class is independent from the choices involved

In the section 2.3 of these notes the Godbillon-Vey class is constructed. It is shown that this class does not depend from the choices involved (lemma 2.11). I have troubles understanding the ...
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36 views

Isomorphisms of complex (foliated) n-tori

From here: https://www.encyclopediaofmath.org/index.php/Complex_torus A complex torus is a complex Abelian Lie group obtained from the $n$-dimensional complex space $\mathbb{C}^n$ by factorizing ...
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42 views

Distance along leaves of a foliation is continuous

Let $\mathcal{F}$ be a codimension-1 continuous foliation of $\mathbb{R}^2$ with $C^1$ leaves. That is, a partition of $\mathbb{R}^2$ into $C^1$ curves which can locally be mapped continuously to ...
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1answer
66 views

Differentials of non-smooth functions, wedge products of currents?

In a paper of McMullen he considers foliations on a manifold determined by a closed 1-form $\rho$. He says an $L^\infty$ function $f$ is constant on the leaves of the foliation if "$df \wedge \rho = ...
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1answer
84 views

Connected components of nonempty level sets form a foliation of $M$.

Let $M,N$ smooth manifolds and $F:M \to N$ a smooth submersion. Show that the connected components of nonempty level sets form a foliation of $M$. My idea is to use the Global Fröbenius theorem for ...
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60 views

Arcs Contained In Continuous Injections of $[0,1)$

Suppose we have a metric space $X$ and a continuous injection from $[0,1)$ onto $X$. The case I had in mind will satisfy that $X$ is compact, but the problem I have can be stated more generally, as ...
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1answer
58 views

A foliation of the sphere

In the context of coisotropic reduction, I asked myself the following question: In $\mathbb{R}^{2n}$ with coordinates $(x_{1},y_{1},\ldots,x_{n},y_{n})$, we consider the unit sphere $S^{2n-1}\subset\...
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1answer
39 views

foliation with many tangencies

Suppose you have smooth foliation on a Euclidean ball $\mathbb{B}^{4} \subset \mathbb{C}^{2}$, whose leaves are holomorphic curves with respect to the standard complex structure. Let $(z_{1},z_{2})$ ...
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1answer
54 views

Extending foliations of submanifolds

Take a manifold $M \cong \Sigma \times \mathbb{R}$, in which there are two submanifolds (of the same dimensions), $S_1$ and $S_2$, such that $S_i \cong \sigma_i \times \mathbb R$, and both are ...
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Foliations and groupoids in algebraic geometry

I am currently studying the theory of foliations and groupoids from a differentiable viewpoint, in particular Haefliger spaces. [See Segal, Classifying spaces related to foliations, and Moerdijk, ...
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39 views

Manifolds with boundary and foliations

Is there a theory of foliations by manifolds with boundary? Particularly, Is there a generalization of the Frobenius theorem and the Stefan-Sussmann theorem in which the leaves are manifolds with ...
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113 views

Alternative definition for characteristic foliation of a surface

Given a surface $S$ in a contact $(2n+1)$-manifold $(M,\xi)$ one can define the characteristic foliation of $S$ via looking at where the tangent bundle to $S$ coincides with $\xi$ (the singular part ...
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1answer
107 views

Opens maps from topological manifolds whose fibers are not generically topological foliations

Update. I have asked this on MO, but have not yet received an answer. Proposition. The quotient map associated to a topological foliation (projecting to the leaf space) is open. However the fibers ...
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2answers
131 views

Confusion with definition of foliation

Below is the definition of foliation of a manifold appearing in the book Introduction to Foliations and Lie Groupoids by Moerdijk and Mrčun. Definition 1. Let $M$ be a smooth manifold of dimension $...
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1answer
202 views

Solving $\nabla \times \mathbf{b} = \mathbf{b} \times \mathbf{a}$

Suppose we are given a fixed vector field $\mathbf{a}$. I am interested in the problem of determining a vector field $\mathbf{b}$ such that $$\nabla \times \mathbf{b} = \mathbf{b} \times \mathbf{a}.$$...
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0answers
42 views

integrable distribution and maximal leaves

I'm trying to understand the proof of the following theorem An integrable distribution $\cal{D}$ induces a partition of the manifold into maximal leaves. The proof starts like this: Take $m\in M$,...
2
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1answer
70 views

How to determine the foliation which integrates a 1-dimensional distribution?

The question is as follows: In the 3-sphere $S^3 \subset \mathbb{R}^4 $ consider the 1-dimensional distribution defined by $$X = -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y } - \...
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2answers
164 views

How to determine the foliation of an involutive distribution?

The question is as follows: Consider the distribution $D$ in $\mathbb{R}^3$ generated by the vector fields: $$ X_1 = \frac{\partial}{\partial x} + \cos x \cos y \frac{\partial}{\partial z}, ~~~~ ~~~~ ...
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1answer
97 views

Foliation with single leaf coming from frobenius theorem

I was reading the Ambrose -- Singer theorem [2] and I came up with a question. The theorem ends with a situation like the following: Let $M$ be a connected manifold modeled on the banach space $\...
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1answer
40 views

Isomorphism of sheaves of q-forms

Consider $D \subset \mathbb{C}^{n + 1}$ be a hypersurface such that $D \cong T \times D_{0}$, where $T$ be a disc in $\mathbb{C}$ and $D_{0}$ be a hypersurface in $\mathbb{C}^{n}$ [for this isomorphim,...
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1answer
51 views

Restricting Foliations by Hopf Circles

Suppose I have $\mathbb{S}^5$, foliated by Hopf circles. I am wondering if this restricts in some way to the foliation by Hopf circles on $\mathbb{S}^3$ in the join $\mathbb{S}^5=\mathbb{S}^3*\mathbb{...
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123 views

How to show that a manifold is foliated by stable manifolds?

I'm trying to understand how to show explicitly that a manifold may be foliated by the stable manifolds, using the following statements of stable manifold theorem and foliations. I feel like it should ...
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68 views

Extending a hypersurface to a foliation

Given a smooth embedded submanifold of codimension 1, i.e. it is a hypersurface, say $H \hookrightarrow M$, is it possible to show that there exists an open nbd. $U\supset H$ in $M$ s.t it is ...
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85 views

Diffeomorphisms of generic foliations

Given a "generic" foliation with 1-dimensional leaves on a closed manifold M, I would like to claim that there are no diffeomorphisms of the manifold fixing the foliation, other than flowing along the ...
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2answers
116 views

Foliations of spheres

Reading a paper, I came across the statement that there are no transversely orientable codimension-one foliations on even dimensional spheres $\mathbb{S}^{2n}$. Could anyone explain why?
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How can one define singular foliations in terms of charts?

Let $\mathcal F$ be a (regular) codimension $q$ foliation. A foliated chart is an open set $U$ and a local diffeomorphism $\varphi:U\to R^{q}\times R^{n-q}$, where $\varphi^1=c^1,\dots,\varphi^q=c^q$ ...
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48 views

Local condition on differential form satisfying Frobenius so that the quotient is Hausdorff.

Let $M$ be a smooth manifold and let $\mathcal D$ be an integrable distribution of codimension one. Then by Frobenius theorem there exists a foliation $\mathcal F^{\mathcal D}$ such that the leaves ...
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1answer
76 views

Union of compact leaves of foliation is open and closed

Let $M$ be a manifold that has a transversely orientable foliation with leaves of codimension one. I am trying to show that the union of all compact leaves is closed and open. I got the following ...
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1answer
89 views

Existence of a local transverse embedded submanifold for a flow

Let $M$ be a smooth manifold and $\Phi$ the flow of a non-vanishing vector field. There always exists around every point $x$ at least locally an embedded submanifold $S_x$ that is transversal to $\Phi$...
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239 views

Reeb Foliation for 3-Sphere

I was trying to prove the following statement: Let $\mathcal{F}$ be the Reeb foliation of $S^3$ and let $\phi$ : $S^3$ $\rightarrow$ $N$ be a continuous map whose restriction to each leaf of $\...
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1answer
156 views

Integrability of almost product structure

An almost produsct structure on a smooth manifold $M$ is a pair $(D,D')$ of complementary distributions. One says that the structure is integrable if there is a local map $(U,\phi^i,\psi^\alpha)$ for ...
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Condition for integrability (foliation)

Let $(M^{n+1}, \langle \cdot, \cdot \rangle)$ be a parallelizable Riemannian manifold with a vector bundle isomorphism $$\varphi : TM \to M \times \mathbb{R}^{n+1}.$$ For $x \in M$, denote by $\...
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1answer
351 views

Submersion with compact level sets is a fiber bundle

In the book "Foliations I" Candel & Conlon, the exercise 1.1.3 is as follow: If $\partial M = \emptyset = \partial B$ and $B$ is connected, prove that the submersion $f:M\rightarrow B$ with ...
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1answer
68 views

Foliation dense if $G = \textbf{R}$, where $G$ is a subgroup of a Lie group $G'$.

I have the following statement: Let $G$ be a subgroup of a lie group $G'$, and the action is left multiplication. The leaves are then the left cosets of $G$ in $G'$. If for example, we let $G = \...
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1answer
134 views

Foliated vector fields span tangent bundle

Suppose we have a foliation on a manifold $M$ which will be call $F$. Foliated vector fields are those $X$ for which: $$[X,T] \in TF$$ for all $T \in TF$. It is easy to see that locally if we have a ...
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1answer
525 views

Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
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40 views

Proper surjective holomorphic function is a topological bundle

If I have $u:M\rightarrow D\setminus 0$, $M\subseteq (\mathbb C^2,0)$ compact, $\mathbb C^2 = \mathbb C \times \mathbb C$ and $D$ the Poincare disk, $u$ is holomorphic, surjective, proper and every ...
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230 views

Quotient map for space of leaves of a foliation

I have been told the following fact: Given a foliation $F$ of a smooth manifold $M$, then the projection $\pi : M \to M /F$ onto the space of leaves (points on the same leaf are identified) is open. ...