# 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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### Dynamics on the torus

It is well known that a $2-$torus foliated with lines of irrational slopes will produce dense curves on the torus. Likewise, rational slopes will lead to a periodic orbit. However, I am not seeing the ...
1 vote
12 views

### Transverse Foliation to the Flow of a Differential Equation on a Tangent Bundle?

Let $Q^n$ be a closed manifold, $M = TQ$ its tangent bundle, $\xi$ be a differential equation on $M$ that satisfies the "canonical flip on $TTQ$" (a "second-order differential equation ...
13 views

### Foliations as equivalence relations

It is known that for an $n-$dimensional topological manifold $M$, a foliation is an equivalence relation on $M$. Is there any proof to this? I cannot find any online and would love to see one. It ...
1 vote
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### The degree of a foliation $\mathcal{F}$ on the complex projective plane $\mathbb{C}\mathbb{P}^2$

Let $\mathcal{F}$ be a Singular Holomorphic Foliation on the Complex Projective Plane $\mathbb{C}\mathbb{P}^2$. It is well-known that there are too many different equivalent ways to define the Degree ...
1 vote
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### When is the leaf space $M/\mathcal{F}$ of a foliation smooth?

Let $(M,\mathcal{F})$ be a regular smooth foliation on a manifold $M$. In general, the leaf space $M/\mathcal{F}$ is quite pathological. However, when $M$ is a Poisson manifold with compact, 1-...
1 vote
43 views

### regular analytic 1-dim. foliation of $M=(0,1)^3$

I'm investigating this question: Does there exist a regular analytic $1-$dim. foliation of $M=(0,1)^3$ s.t. all leaves include both $(0,1,1)$ and $(1,0,0)$ or approach these points in the limit? I ...
1 vote
46 views

### A particular foliation of $S^1\times S^1$

I have to construct a foliation $\mathfrak{F}$ of the 2-torus $T=S^1\times S^1$ such that some leaves of $\mathfrak{F}$ are compact while some other ones are not compact. MY IDEA. Let's fix $\alpha$ ...
1 vote
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### nonperiodic orbits of a smooth $\mathbb{R}$-action on $\mathbb{R}^2$ are embedded lines.

Let $\mathbb{R} \curvearrowright U$ be a smooth action on an open subset $U \subset \mathbb{R}^2$. Show that any non-periodic orbits under the action are embedded lines in U. My ideas so far: I have ...
36 views

### Stefan-Sussman theorem

The Stefan-Sussmann theorem is a natural generalization of the Fröbenius theorem to singular smooth distributions. So, a singular smooth distribution is integrable if, and only if, it is generated for ...
1 vote
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### What is any examples of Reeb stability theorem for germinal holonomy?

Theorem (Reeb Stability Theorem). If the compact leaf $L$ of a foliated manifold $(M, \mathcal{F})$ has trivial holonomy, then there is a neighborhood of $L$ in $M$ that is a union of leaves that are ...
21 views

### the foliations have isomorphic germs at the compact leaf iff holonomy homomorphisms are conjugate

I have a question about proof of theorem $2.3.9$ in Foliations I which is written by Candel and Conlon(publisher: AMS, GSM) I don't understand the proof and claim following the theorem. $L$ is a ...
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### Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)

First, let me state some basics mainly coming from Introduction to foliations and Lie groupoids written by I. Moerdijk and J. Mrcun. A codimension $q$ foliation $\mathcal{F}$ on a smooth n-manifold $M$...
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### Existence of a foliation on $\mathbb{R}^3$

Let $f \in C^\infty\left(\mathbb{R}^3\right)$. Let $D$ be the distribution given, at every $p \in \mathbb{R}^3$, by \begin{equation*} D_p = \text{span}\Bigg\{ \left.\frac{\partial}{\partial x}\...
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### The Harmonic Measures of Lucy Garnett

The Harmonic Measures of Lucy Garnett, Alberto Candel, Advances in Mathematics $176$ ($2003$) $187-247$. Page $208$, Proposition $5.2$ I have actually two questions: what does he mean by saying "...
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### Laminated Positive Harmonic Currents on a Compact Complex Manifold as an Integration aginst a Positive Borel Measure on a Transversal

Riemann Surface Laminations with Singularities, J. E. Fornaess and N. Sibony, J. From Anal, $2008$, $400-442$. Page $414$, Definition $7$. Let $\theta$ be a $(1,1)$ form on $M$. How can the formula, ...
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### The Eigenvalues of a Singular One-Dimensional Holomorphic Foliation at a Singular Point

Let $M$ be a Complex Manifold of complex dimension $n$. Let $\mathcal{F}$ be a Singular One-Dimensional Holomorphic Foliation on the complex manifold $M$. Let $p \in M$ be a Singular Point of the ...
1 vote
45 views

### Regarding the Definition of Holomorphic Foliation on a Complex Manifold

Geometry, Dynamics And Topology Of Foliations: A First Course, Book by Bruno Scárdua and Carlos Arnoldo Morales Rojas, Chapter 1, Page 33. Definition 1.14. The definition of Holomorphic Foliation on a ...
41 views

### Minimal sets, Perfect sets, Exceptional sets and Foliations

The first image is taken from Geometric Theory of Foliations, Book by A. Lins Neto and César Camacho, Chapter 3, Page 53. The second image is taken from Geometry, Dynamics And Topology Of Foliations: ...
1 vote
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I am studying geometric theory of foliations from Camacho and Lins and I have stumbled upon an afirmation I can't understand. There is a theorem which states: Let F be a leaf of a foliation $\mathcal{... 5 votes 1 answer 109 views ### Visualising Plaques of Foliations So I just recently started to study foliation theory and I realized I'm visualizing plaques of foliations the wrong way. Given a manifold$M$, a foliation$\mathcal{F}$of the manifold and a local ... 2 votes 0 answers 39 views ### Plane foliation with compact leaf must have a singularity I'm trying to solve the following exercise from Camacho & Neto's book$\textit{Geometric Theory of Foliations}$: Let$\mathscr{F}$be a$C^r(r\geq1)1$-dimensional foliation on$\mathbb{R}^2$... 2 votes 1 answer 31 views ### Submersions between the same manifolds having the same kernels on their derivative are locally the same I came across the following lemma in a lecture notes set that I was reading. Let$U$be an open set in$\mathbb{R}^m$,$f,g : U \to \mathbb{R}^{m-d}$be two submersions such that$Ker(Df_p) = Ker(Dg_p)...
In the proof of the Frobenius theorem , at least the one I saw, eventually we need the fact that Let $X_1,...,X_k\in \mathfrak{X}(M)$ be vector fields such that $\{X_1|_p,...,X_k|_p\}$ are linearly ...