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211 views

### Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure?

Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure? Every paper or article I looked at, talk only about singular laminations on $\mathbb{CP}^2$. I was wondering why. If you know ...

**4**

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**0**answers

82 views

### Clarification on Étienne Ghys' “Feuilletages riemanniens sur les variétés simplement connexes” paper

I apologize for this type of question, but I'm having some trouble to understand remark 3.4(4) on page 212 of this article, that reads
The restriction of $\overline{\mathcal{G}}$ (the foliation ...

**5**

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**1**answer

123 views

### Density of non-algebraic leaves in the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has one-...

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**1**answer

63 views

### Horizontal distribution of a totally geodesic foliation

Let $\mathbb{M}$ be a $n+m$ dimensional manifold. Consider on $\mathbb{M}$ a rank $n$ sub-bundle $\mathcal{H}$ of the tangent bundle. We assume that $\mathcal{H}$ is endowed with a fiber wise inner ...

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**1**answer

247 views

### Algebraicity and non-algebraicity of leaves of the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of
$D$, the restriction of $\omega$ to $D$ has one-...

**2**

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**1**answer

170 views

### Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution

In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/...

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**1**answer

56 views

### Analytic conjugacy of vanishing holonomy groups implies analytic conjugacy of foliations

i am reading and trying to do some exercises and problems of the book Lectures on Analytic Differential Equations- Y. Ilyashenko, S. Yakovenko.
I can not solve the problem 11.6 that says
Consider ...

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**2**answers

83 views

### Nonlinear PDE for a 2D foliation

I am trying to solve a 4th order nonlinear PDE for a real function $u(x,y)$ of two variables. It is too complicated to reproduce here but it exhibits the following two very nice properties:
1) if $u(...

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**0**answers

80 views

### A specific type of first-order nonlinear ordinary differential equation

I am trying to divide $\mathbb{R}^+\times \mathbb{R}^+$ into some curves so that the integration of the function $ h(x)h(y) $(where $h(x)$ is a $C^1$ function from $\mathbb{R}^+\to \mathbb{R}^+$ that ...

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**1**answer

147 views

### Euler Class constant on Fibered Face of Unit Thurston Norm Ball?

I am reading about the Thurston norm out of Candel and Conlon's "Foliations 2" and Calegari's "Foliations and the Geometry of 3-Manifolds".
I'm trying to work through the proof of the "Fibered Faces"...

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**0**answers

88 views

### Pulled back foliation is completely integrable

There is a question that arises, while I'm trying to understand Guillemin & Sternbergs paper "On collective complete integrability according to the method of Thimm".
Assume $M$ is a symplectic ...

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**2**answers

377 views

### algebraic leaves of foliation on a product of two curves

Let $S=E\times C$ be a product of two curves, where $E$ is an elliptic curve and $C$ is a curve of genus at least two. Consider a foliation on $S$ generated by a global holomorphic 1-form $p_1^*(\...

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2k views

### What is a foliation and why should I care?

The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...

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**1**answer

108 views

### Glueing together functions defined on the leaves of a foliation

Even though my question can be asked for very general types of foliations, I am interesetd only in its answer for Poisson manifolds, which are what I am currently studying.
Consider a Poisson ...

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46 views

### Smoothness of the twistor space of a lorentzian manifold, or “convexity wrt null geodesics”

Null lines in Minkowski space form a 5-dimensional manifold, represented as a (real) quadric $\mathbf{PN}\subset\mathbb{C}\mathbf{P}^3$. This is a well-known fact, on which R. Penrose’s twistor ...

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46 views

### Integrability of the orthogonal complement of a holomorphic vector field on $\mathbb{C}^{2}$

Assume that $$\begin{cases}\dot x=P(x,y)\\\dot y=Q(x,y)\end{cases}$$ is a non vanishing holomorphic vector field on an open subset $U$ of $\mathbb{C}^{2}\simeq \mathbb{R}^{4}$. It defines a two ...

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**2**answers

131 views

### When are simple foliations strictly simple?

Any submersion $f: M → N$ defines a foliation of M whose
leaves are the connected components of the fibres of $f$. Foliations
associated to the submersions are called simple foliations. The foliations
...

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87 views

### On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid:
Let $\mathcal{F}\subset M$ be a ...

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**1**answer

229 views

### Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...

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95 views

### A finiteness question for integrable polynomial distributions on $\mathbb{R}^3$

This question is motivated by the finitness of limit cycles for polynomial vector fields on $\mathbb{R}^2$
Assume that $X,Y$ are two independent polynomial vector fields on $\mathbb{R}^{3}$ such ...

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**1**answer

145 views

### Closed leaves of a foliation

Let $M$ be a differentiable manifold of dimension $n + k$, let $\Delta$ be an $n$-dimensional integrable distribution (à la Frobenius), let $N$ be an $n$-dimensional connected integral manifold of $\...

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115 views

### The complex leaves containing real limit cycles of Lienard equation

According to the answer and comments to this question we realize that a useful approach to such type of questions is to consider algebraic vector field which possess algebraic solutions. On the ...

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62 views

### Can we foliate the anti-de sitter space in 3 dimensions by Riemann surfaces?

Can we foliate the anti-de sitter spacetime in 3 dimensions by hyperbolic Riemann surfaces? -- I think this is possible, but got stuck at finding the particular projective mapping that does this. Can ...

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**1**answer

226 views

### Three-manifolds having a Reebless foliation but not a taut one

A straightforward argument reveals that a taut foliation is Reebless, and of course there are many examples of Reebless foliations that are not taut. I guess that there are many examples of three-...

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133 views

### Taylor expansion in Riemannian foliations

Take:
$M$ a Riemannian manifold, ${X_0}\in M$,
$N_{X_0}$ a submanifold of $M$ going through ${X_0}$,
and $Z \in N_{X_0}$ in a neighborhood of ${X_0}$.
At ${X_0} \in N_{X_0}$, we consider the ...

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**1**answer

181 views

### Smooth conditional measures for strong stable foliations of Anosov flows

I am trying to prove an analytic result for gesodesic flows on negatively curved manifolds and I encountered the following dynamical-system porblem.
Let $B^n$ be $n$-dimensional balls and $h:B^{n-k}\...

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**1**answer

126 views

### Stability of singularity in singular holomorphic foliation

For an open subset $U$ of $\mathbb{C}^{2}$ containing $0$ and a holomorphic map $f:U\to \mathbb{C}^{2}$ which has a unique zero at the origin we associate a natural singular holomorphic ...

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**1**answer

57 views

### Comparision of two $C^{*}$ algebras associated to a non vanishing vector field on a compact manifold

Let $X$ be a non vanishing vector field on a compact manifold $M$ so we have a one dimensional foliation $F$ of $M$ with orbits of $X$.
This foliation defines a $C^{*}$ algebra $C^{*}(F)$. On the ...

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96 views

### A noncommutative vector bundle associated with a codimension one foliation

Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to \...

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117 views

### Product of two foliations

1.What is an example of a manifold $M$ with two foliations $F$ and $F'$ which are not topological equivalent but the product foliations $F\times F$ and $F'\times F'$, as foliations on $M\times M$, ...

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342 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 $\...

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95 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 ...

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55 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 ...

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202 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 $C(...

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**1**answer

102 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 \Omega^{i+2}(...

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207 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?

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258 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 $\mathcal{F}(M)$ ...

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292 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 ...

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258 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 ...

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137 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 ...

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242 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 ...

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256 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 ...

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**1**answer

124 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 ...

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215 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 ...

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169 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.$g_{*}X=\pm X$ where $g$ ...

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**1**answer

224 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 $0$...

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255 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 ...

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**1**answer

200 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 ...

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**1**answer

334 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 ...

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71 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 ...