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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 $\alpha$ ...
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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 ...
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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 ...
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Find a given distribution is involutive and a flat chart for the distribution

Hello I am studying distributions and foliations by myself, I got stuck on an exercise. Let $\mathcal{D}\subset T\mathbb{R}^{3}$ generated by the vector fields $X=x\frac{\partial}{\partial x}+\frac{\...
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Determine foliation given by vector field

While studying for an upcoming exam, I've found the following problem: Let $X = xz\displaystyle\frac{\partial}{\partial x} + yzxz\displaystyle\frac{\partial}{\partial y}$ and $Y = (1 + z^2)xz\...
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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 ...
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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 Definition of Leaf of Foliation in a Manifold in terms of Plaque Chains

Foliations with measure preserving holonomy, Author J. F. Plante, Annals of Mathematics, $102$, $(1975)$, $327-361$. Page $337$ it is mentioned "If $x \in M$ then $x$ is contained is some plaque $...
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Growth of a Leaf of a Foliated Bundle

Foliations I, Authors Alberto Candel and Lawrence Conlon, Chapter $12$, Page $320$, Corollary $12.2.32$. Let $(M,\mathcal{F},\pi,B,F)$ be a $C^2-$ Foliated Bundle with a Fibre $F$ compact Metric Space,...
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The Quasi-isomtery Classes of Riemannian Metrics on a Leaf of a Foliation of a Compact Manifold

Geometry of Leaves, Authors Anthony Phillips and Dennis Sullivan, Topology Volume $20$, PP. $209-218$. It is mentioned in the first section that is the introduction that "A Leaf of a foliation of ...
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Growth of Leaves of a Foliation on a Manifold

Let $M$ be a (Real or Complex) Manifold of dimension $m$ and $\mathcal{F}$ a Foliation of dimension $k$, $1 \leq k \leq m-1$, on $M$. I am actually looking for highly recommended references on the ...
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Regarding the Definition of an Orientable Foliation on a Manifold in terms of Transition Maps

Introduction to the Geometry of Foliations, Part $A$, Authors Gilbert Hector and Ulrich Hirsch, Page $15$. Let $\mathcal{F}$ be a Foliation on a Manifold $M$ defined by the atlas $\{(U_i,\phi_i)\}$. ...
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When does a singular foliation admit a resolution?

For example, Any holomorphic singular foliations on a complex manifold X of dimension n locally admits a finite resolution by finitely generated free OX-modules. On the other hand, there are more ...
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1-dimensional foliation of surfaces with prescribed graph of foliation

Definition of the graph of a foliation Let we have a $k$ dimensional foliation of an $n$ dimensional manifol $M$. One associates to this foliated manifold a (not necessarily Hausdorff) ...
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Linearisability of Holomorphic Vector Fields and Poincaré-Dulac Normal Form

Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 1, Page 36. The last two lines, Don't you think they should have written that the vector field is locally conjugate (...
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Condition for every line through a line segment $S$ to be perpendicular to $S$?

In 3 dimensional space, given a line segment $S$ which has a line $l(0)$ passing through one endpoint at right angles and another, $l(1)$, passing through the other endpoint at right angles, such that ...
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The Index of a Singular Point relative to a Singular One-Dimensional Holomorphic Foliation on a Complex Surface

Let $S$ be a Complex Surface, $C$ a Compact Non-Singular Curve and $\mathcal{F}$ a Singular One-Dimensional Holomorphic Foliation on the complex surface $S$ leaving the curve $C$ Invariant. Let $p\in ...
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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 ...
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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 ...
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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: ...
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The smooth action of $\mathbb{Z}$ on $\mathbb{R} \times F$ where $F$ is a manifold

Introduction to Foliations and Lie Groupoids, Book by Ieke Moerdijk and Janez Mrčun, Chapter 1, Page 16. The smooth action of $\mathbb{Z}$, defined on $\mathbb{R} \times F$ by $$(k,(t,x)) \mapsto (t+k,...
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3 votes
1 answer
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When is a foliation invariant under a diffeomorphism?

Let $\mathcal{F}$ be a Foliation on a Manifold $M$ and $g:M\longrightarrow M$ a Diffeomorphism. We say that the foliation $\mathcal{F}$ in invariant under the diffeomorphism $g$ if the diffeomorphism $...
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$C^r$ Equivalent Fibre Bundles and Discrete Structure Group

Let $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ and $(\mathbb{E}', \pi', \mathbb{B}, \mathbb{F})$ be two $C^r$ (where $r \ge 1$) equivalent Fibre Bundles, i.e., there is a $C^r$ diffeomorphism $H: \...
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When does the Fibre Bundle have Discrete Structure Group?

Let $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ be a Fibre Bundle ((where $\mathbb{E}$ is the Total Space, $\mathbb{B}$ the Base, $\mathbb{F}$ the Fibre and $\pi: \mathbb{E} \longrightarrow \mathbb{B}$...
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2 votes
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Holonomy of a Foliation that is Transverse to the Fibres of a Fibre Bundle

Let $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ be a Fibre Bundle and $\mathcal{F}$ be a $C^r$ (where $r \ge 1$) Foliation on $\mathbb{E}$ that is Transverse to the fibres of the fibre bundle $(\...
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Holonomy of a Leaf

Geometry, Dynamics And Topology of Foliations A First Course. Bruno Scardua, Carlos Arnoldo Morales Rojas. Page 59. What are $\Sigma_1$ and $\Sigma_2$? How are $\pi_1$ and $\pi_2$ defined? In general,...
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Manifolds, Vector Fields and Foliations [closed]

Let $M$ be a Manifold, $X$ be a Vector Field on the manifold $M$ and $F$ be a Foliation of the manifold $M$. When is the vector field $X$ tangent to the foliation $F$?
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The Horizontal Flow, The Time-One Map, $C^r$ conjugation and Suspension of a Diffeomorphism

Let $M$ be a $C^r$ Manifold and $f:M \rightarrow M$ be a $C^r$ Diffeomorphism, where $r\in\mathbb{N}$. Consider then the Product Manifold $$M_1:=\mathbb{R} \times M$$ and the Horizontal Flow on $M_1$ $...
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Suspension of Diffeomorphisms

There is no doubt that Suspension of Diffeomorphisms is a special case of Suspension of a Representation (that is also known as Suspension of Diffeomorphism Group). Therefore, I would be very grateful ...
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What does "the map $i$ is transverse to the vector field $X$ everywhere" mean?

Geometric Theory of Foliations, Page 28. The Lines 7 and 8 from below. What does "the map $i$ is transverse to the vector field $X$ everywhere" mean?
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Foliations induced by Vector Fields without Singularities

A well-known type of foliations is the one that is induced by vector fields without singularities. However, I have already read this type of foliations from Geometric Theory of Foliations, Page 28. ...
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2 votes
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Suspension of a Representation

Geometric Theory of Foliations, authors Cesar Camacho and Alcides Lins Neto, Chapter 5, Pages 95. Does anyone know some highly recommended references on the theorem in the image below?
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Foliations induced by Submersions

Let $T:M^m \rightarrow N^n$ be a submersion where $n \le m$. It is well-known, in this case, that a foliation (called a simple foliation) of dimension $m-n$ (or codimension $n$) is defined. Its leaves ...
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Can I find a plane in $\mathbb P^3$ to which a foliation $\mathscr F \subset T\mathbb P^3$ is tangent on a smooth curve?

In this question, we work over an unspecified algebraically closed field, not necessarily of characteristic zero. Let $\mathscr F$ be the $1$-dimensional foliation of $\mathbb P^3$ defined by the ...
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Plane Foliation and Connected Components

Let $f\in C^{2}(\mathbb{R}^2,\mathbb{R})$ such that $\nabla f(x)\neq 0$ for all $x$. Define a plane foliation $\{ \sum_{\bar{x}}^{f}\; ; x\in\mathbb{R}^2 \}$, where $\sum_{\bar{x}}^{f}=\{ x\in\mathbb{...
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5 votes
1 answer
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Why can't this submanifold be a Leaf of a Foliation?

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{...
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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 ...
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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 $\mathbb{R}^2$...
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2 votes
1 answer
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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)...
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Part of Frobenius theorem

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 ...
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