Questions tagged [foliations]

This tag is for questions about foliations in differential geometry and use in conjunction with the tag (differential geometry).

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Holonomy of the Reeb foliation on the solid torus

The past few days I am trying to calculate the holonomy group of the boundary leaf(i.e. of the torus) of the Reeb foliation on the solid torus. Classic textbooks, like Moerdijk & Mrcun, cite the ...
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23 views

“Horizontal” Foliation on a fiber bundle

Before I ask my question, I would like to build up the setting. Let $\pi:E \rightarrow M$ be a smooth fiber bundle. Since $\pi$ is a smooth submersion, there is an induced foliation $\mathcal{F}(\pi)...
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75 views

Transversal and intersection of two foliations

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two foliations of a manifold. We say that $\mathcal{F}_1\pitchfork \mathcal{F}_2$ if $T_p L^{(1)}+T_pL^{(2)}=T_p M$ for any $p\in M$, where $L^{(1)}$ and $L^{...
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39 views

Can any vector be extended locally to a geodesic vector field?

Given a (pseudo-)Riemannian manifold $(M,g)$ and some vector $X_p\in T_pM$ at $p\in M$, can one always extend $X_p$, locally, to a geodesic vector field $X$, in the sense that any integral curve of $X$...
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21 views

Smooth extension of foliations which are “smooth up to the boundary”

Suppose that we have a smooth manifold with boundary $M$ and let $\mathcal{F}$ be a codimension-$1$ $C^\infty$-foliation of $M \setminus \partial M$ such that for any connected component $C$ of $\...
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23 views

Holonomy of a path

Let $(M,\mathcal{F})$ be a foliated manifold. Take a leaf $L$, two points $x,y\in L$ and a path $\gamma$ from x to y. I consider the case where $\gamma([0,1])\subset U$, $U$ is the domain of some ...
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31 views

Existence of Vector Field on Foliation

Let $M$ be a manifold with a foliation, and for some $p \in M$ denote $L$ the leaf of $M$ that contains $p$. For some tangent vector $v \in T_pL$, can we always find a global vector field $F \in \...
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52 views

Line bundle associated with a $1$-cocyle of holomorphic functions.

I'm trying to understand the following extract: A holomorphic foliation by curves $\mathcal{F}$ of degree $r$ on the projective space is a bundle map $\Omega : H_{-r+1} \rightarrow T\mathbb{P}^n$, ...
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1answer
26 views

Are singular foliations spanned by collinear vector fields equal?

Let $M$ be a compact $n$-manifold (let's say with boundary, but this isn't too important), $X$ a vector field on $M$ and $f:M \to \mathbb{R}$ a non-zero function on $M$. My question: are the singular ...
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45 views

Topology on the space of foliations

Let $(M^3,g)$ be a closed Riemannian manifold. Is there a “natural” topology on the space $\operatorname{Fol}(M)$ of smooth codimension $1$ foliations on $M$? Is there any other relevant structure on ...
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30 views

Fibration in integral manifolds

Consider a smooth manifold $M$ of dimension $n$ and an integrable tangent distribution $$ \mathcal{D} = span\{X_1,...,X_k\}$$ with $k\leq n$. Then we know that $M$ is foliated by the connected ...
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33 views

Confusion about the definition of a rectangular neighborhood

I am looking for a precise definition of a rectangular neighborhood in the definition of a foliation chart. In the book "Foliations I" by Alberto Candel and Lawrence Conlon, the definition is the ...
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22 views

Looking for reference: If a Riemanian manifold is foliated by max symmetric submanifolds, then coordinates can always be chosen such that …

In Weinberg's book on General Relativity in section 13.5 it is shown that, loosely stated, if a Riemannian manifold $(M,g)$ of dim $m$ is composed of maximally symmetric submanifolds $(N,h)$ of dim $n$...
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67 views

A foliations as a G-stucture

According to this Wikipedia entry, a foliation is a particular G-structure whose structure group reduction induced by block matrices. I tried to find more details of this approach to foliations, but ...
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30 views

Normal field to a foliation

Given a vector field $n$ on a Riemannian manifold $(M,g)$ we can define \begin{equation} \Delta_x=\{v\in T_xM : g(v,n_x)=0\} \end{equation} I have to find the condition on $n$ such that this ...
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29 views

What is the leaf space of the Kronecker foliation of the torus?

The leaves of the Kronecker foliation of the torus are diffeomorphic to the real line $\mathbb{R}$ and are dense on $T^2$. I cannot, however, describe what is the leaf space $T^2/\mathcal{F}$.
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quotient of closure foliation

Let $M$ be a smooth manifold with foliation $F$. I can understand the quotient $M/F$, i.e. $x\sim y$ if and only if $x,~y$ are in the same leaf. By the Richardson's paper, Transverse manifold and $...
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32 views

Symplectic Reduction: Why are $\mathbb{S}^1$ Orbits Tangent to $\omega$-Null Directions?

Given a symplectic manifold $(M, \omega)$, a smooth function $H:M \to \mathbb{R}$, we can show that for a regular level set $H^{-1}(\lambda)$, if there is a free action $\mathbb{S}^1 \times H^{-1}(\...
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38 views

Reference needed: examples of where smoothness fails

Building up on a previous question of mine, I'm looking for examples of foliations where the following equivalence relation is not smooth (as a manifold): $$ \Gamma(\mathcal{F}) = \{(x,y) \rvert x,y\...
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36 views

Non-smoothness of equivalence relation on Möbius band

Consider the open Möbius band $M$. This has a well-known foliation by circles, letting the central circle 'go around once' and all other circles go around twice. Suppose we wish to consider the ...
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213 views

Show that Hopf foliation is a foliation.

Consider $S^3 := \{(z,w) \in \mathbb{C}^2:|z|^2 + |w|^2 = 1\}$ be the unit $3$-sphere with equivalence relation $$(z,w) \sim (z',w') \iff z' = e^{i \theta }z, w' = e^{i\theta} w$$ for some $\...
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207 views

Prove this partition of the plane is not a foliation

Let us define a partition of the plane as follows: for the points $(x_0,y_0)$ with $y_0\leq0$ we have leaves that are straight lines ($y=y_0$) and for $y_0>0$ we have leaves $e^{x+\operatorname{ln}...
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59 views

Is the partition of a manifold with embedded submanifolds a foliation?

Let $\{S_a\}_{a \in A}$ be a partition of embedded connected submanifolds of a manifold of $M$ where each $S_a$ has the same dimension, say $k$. Is it true that this a foliation of $M$? I guess so: ...
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45 views

Sufficient condition for a quotient to inherit a foliation

Let $M$ be a manifold, equipped with a (regular) foliation $\mathcal{F}$. Let $R \subset M \times M$ be an equivalence relation. Godement's theorem states under some conditions, $M/R$ is a smooth ...
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78 views

Reeb foliation of the plane and the Palais-Smale condition

Definition. Let $X$ be a manifold. A smooth map $f:X\to \mathbb R$ is said to satisfy the Palais-Smale condition over $y\in \mathbb R$ if whenever $y_n\to y$, any sequence $x_n\in f^{-1}(y_n)$ such ...
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36 views

The intersection of a transversal section and a plaque of a foliation

I have the following question about the transversal sections of a foliation. Let $M$ be a manifold and $F$ a foliation of codimension $q$. We know that if $\phi:U \to U_1 \times U_2 \in F$ then the ...
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48 views

Are levelsets of a scalar field without critical points a foliation?

Let $f$ be a differentiable scalar field on an $n$-dimensional Riemannian manifold $X$ without critical points, i.e. $\nabla f \neq 0$ everywhere on $X$. (Assuming $X$ has properties as required to ...
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34 views

When a map is transverse to a foliation

I am reading Geometry Theory of Foliations by Camacho and Neto and came upon this definition: "Let $N$ be a manifold. We say that $g:N \to M$ is transverse to a foliation $F$ when $g$ is transverse ...
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118 views

Equivalent definitions of foliation

I am having trouble proving the equivalence of two different definitions of foliation. The first one is the one where you define a k-foliation on a smooth manifold $M$ with k-dimensional leaves such ...
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93 views

Codimension one foliations induced by a map to the circle.

Let $M$ be a closed connected $n$-manifold. If there exists a submersion $p:M\to S^1$, then $p$ is both proper and onto (since $M$ is compact and $S^1$ is connected). Therefore, by Ehresmann's theorem,...
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41 views

2-Plane Tangent Field

I want to show that the $\mathbb{S}^{5}$ does not support two-dimensional foliation. According to the authors in Geometrical Theory of Foliations, it suffices to show that the sphere $\mathbb{S}^{5}$ ...
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32 views

Complete Stability Theorem

I am studying Geometrical Theory of Foliations, more specifically: the Complete Stability Theorem, which says: If $\text{Cod}(\mathscr{F}) = 1$, $M$ is an compact and connected manifold, and there is ...
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64 views

Show that the isotropic foliation of a spheroid in $\mathbb R^4$ is regular if and only if the spheroid is actually a sphere

This is part (i) of exercise 5.4.4 from McDuff and Salamon's book. Consider $\mathbb R^4$ with its standard symplectic structure $\omega = dx_1 \wedge dy_1 + dx_2 \wedge dy_2$, and the coisotropic ...
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49 views

Equivalence between measured foliations and laminations on surfaces

A well known theorem of Thurston asserts the existence of a homeomorphism between the spaces of measured foliations and of measured laminations of any hyperbolic surface. Where can I find a ...
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71 views

Foliation of $\mathbb{R}^3$.

A foliation of $\mathbb{R}^3$ can be given by subspaces of the form $\{p\} \times \mathbb{R}^2$ where $p$ spans over $\mathbb{R}$. I cant see how any open neighbourhood in $\mathbb{R}^3$ intersects ...
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96 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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349 views

Example of an equation that generates solutions?

Q: Is this an example of an equation that generates solutions? And is everything below correct? Part $1:$ Suppose that $x$ and $z$ are independent of each other. $$\phi(x,z)= e^{\frac{z^2}{\ln(x)}}...
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46 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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76 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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70 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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65 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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36 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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63 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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34 views

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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149 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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82 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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144 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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48 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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87 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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195 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 ...