# Questions tagged [foliations]

The foliations tag has no usage guidance.

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### On certain 2 dimensional foliation of $Gl(2,\mathbb{R})$ deleted by scalar matrices

We consider the following 4 dimensional open manifold $$M=Gl(2,\mathbb{R})\setminus \{\lambda I_2 \mid \lambda \in \mathbb{R}\}$$ where $I_2$ is the identity $2\times 2$ matrix.
We consider the $2$ ...

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### Interpretation of the Schouten bracket as an integrability condition

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort ...

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174 views

### Foliation with a compact leaf

Let $M$ be a closed oriented manifold, and $F$ be a fixed foliation of $M$. We assume the dimension and codimension of $F$ are both greater than $1$.
Q Under what condition, we can say that $F$ ...

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170 views

### Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...

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### Foliation of cylinders by constant mean curvature spheres

Let the cylinder $S^{n-1}\times \mathbb R$ be equipped with the standard metric $g$. Suppose that there exists a sequence of metrics $g_{\epsilon}$ on $S^{n-1}\times \mathbb R$ such that $g_{\epsilon}$...

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161 views

### Can we foliate the space $\mathbb{R}^3$ with Frenet curves whose tangent and normal vectores span a given $2$ dimensional distribution?

Let $D$ be a $2$ dimensional distribution of $\mathbb{R}^3$.
Is there a $1$ dimensional foliation of $\mathbb{R}^3$ with Frenet curves such that for every leaf $\gamma$ of the foliation we have $\...

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169 views

### Existence of compact leaf for certain foliation of a symplectic manifold

Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^*...

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71 views

### When are two codimension-one foliations of a manifold M “diffeomorphic”?

By the question, I mean: Given two different codimension-one foliations of a manifold M, $\mathcal{F}_i$ and $\mathcal{F}_j$, when does there exist an element of Diff(M) that maps each leaf of $\...

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347 views

### A non integrable distribution which is totally geodesic

Is there a non integrable $2$ dimensional distribution $D$ of a $3$ dimensional Riemannian manifold such that the distribution is totally geodesic in the following sense:
Every geodesic whose ...

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162 views

### A Lie algebra associated to a foliation(A kind of saturation of foliations)

Inspired by the Lie algebra discussed in this answer, we consider the following Lie agebra associated to a given foliation:
Let $\mathcal{F}$ be a nontrivial foliation of a ...

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60 views

### Does there exist a leaf of this holomorphic foliation with non trivial holonomy?

Let's $\mathcal{F}$ be the holomorphic foliation of $\mathbb{C}^2$ tangent to the kernel of $\alpha=(sin x) dx -(cos x)dy$.
Are all leaves of $\mathcal{F}$ simply connected? If the answer is no, ...

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92 views

### A non integrable distribution arising from a Lie algebra of vector fields

Is there an example of a $n$ dimensional manifold $M$ and a natural number $k<n$ with a Lie subalgebra $L$ of $\chi^{\infty}(M)$ with the following property:
For every $x\in M$ the space $\{V_x ...

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345 views

### Two questions on “foliation by geodesics”

I would appreciate if you consider the following two questions on $1$ dimensional foliations whose leaves are geodesic.
1)Assume that $M$ is a Riemannian manifold which is either an open ...

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135 views

### Does a compact leaf of the smooth transversaly orientable foliation have trivial normal bundle?

In the book Geometric theory of foliations by Camacho and Neto, the following question is posed:
Let $G$ be a smooth transversaly orientable foliation. Let $F$ be a compact leaf of $G$. Prove that $F$...

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212 views

### Does every fiber bundle admits flat bundle structure?

It is well known that a bundle which admits a compatible foliation* supports a flat connection and the converse is also true (every smooth and finite dimensional). The question is:
Does every bundle ...

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255 views

### Rank of a distribution

I am reading about distributions in the context of differential geometry.
A distribution $S$ of dimension $r$ on a manifold $M$ is an assignment to each point $p \in M$ of an $r$-dimensional ...

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### Some generalizations of the graph of a foliation

The graph of a foliation $\mathcal{F}$ of a manifold $M$ is the space of all triple $(x,y,[\gamma])$ where $x,y$ lies on the same leaf $F$ and $[\gamma]$ is the equivalent class of a ...

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78 views

### Classifying transverse curves to a surface foliation carried by a train track

Suppose that a foliation $\cal F$ on a surface $F$ is carried by a train track $\tau$. Is it possible to classify all $\cal F$-transverse multi-loops in $F$ in terms of a combinatorial data on $\tau$ (...

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52 views

### C*-algebra of a singular surface foliation

Noncommutative geometry associates a $C^*$-algebra $C^*(S,{\cal F})$ with a foliation $\cal F$ on a manifold $S.$
Did somebody study this construction for noncompact surfaces $S$?
What I am really ...

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234 views

### Is there a $2$ dimensional foliation tangent to this particular $3$ dimensional distribution?

Is there a $2$- dimensional foliation of $\mathbb{R}^4\setminus \{0\}$ whose tangent space is contained in $\ker \alpha$ where $\alpha$ is the following non integrable $1$-form?
$$\alpha=(x^2+y^2)dx+(...

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119 views

### Can every curve be made transversal to a foliation by applying a pseudo-Anosov?

Let $F$ be a compact oriented surface with a foliation $\cal F$ with $k$-prong singularities only (or, if it helps, assume that $\cal F$ admits an invariant measure). Is it true then there exists a ...

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207 views

### A strongly non-integrable distribution

What is an example of a three-dimensional smooth distribution $D$ of $\mathbb{R}^4$ with this property:
Not only $D$ is not integrable but also there is no a two-dimensional ...

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63 views

### Powers of pseudo-Anosov and the geometric intersection numbers

Let $\phi$ be a pseudo-Anosov of a compact oriented surface $F$ with boundary. Let $\beta\subset F$ be a simple closed loop and $\alpha$ either a simple closed loop or an embedded arc with endpoints ...

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78 views

### Putting a transverse measure on a surface foliation

Let $F$ be an orientable surface with a foliation $\cal F$ with $k$-prong singularities only, for $k\geq 3$.
Since I am looking for an invariant transverse measure on $\cal F$, assume that there is ...

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80 views

### Are isotopic transversal curves on a foliated surface transversally isotopic?

Let $F$ be an orientable surface (possibly with boundary) with a foliation $\cal F$ with $k$-prong saddle singularities only for $k\geq 3,$ (as in figure borrowed from Farb-Margalit book). Suppose ...

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76 views

### A singular foliation analogy of the Riemann Hilbert problem

Note:
In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$.
...

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84 views

### Causal fermion systems fromm fractal geometry

Okay, first off- I apologise if this is a stupid question. I'm mainly a very young physics guy, but this has primarily math basis. I'm trying to build a theory that is, long story short, some ...

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126 views

### A complex limit cycle not intersecting the real plane

Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow?
There is a regular leaf $L$ whose holonomy, along at least one closed curve ...

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101 views

### On the Ekedahl-Barron $F$ conjecture

Let $(X,F)$ a one-dimensional folication over a smooth variety $X$ over $\mathbb{Z}$ . Let $(X_p,F_p)$ the modulus $p$ reduction of $(X,F)$. We assume that $(X_p,F_p)$ is a foliation in positive ...

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### Singular foliations of $\mathbb{C}P^2$ that are compatible to Fubini-Study metric

Is there a complete classification of quadratic polynomial vector fields on $\mathbb{C}^2$ whose corresponding singular foliation of $\mathbb{C}P^2$ satisfies the property quoted below?
The regular ...

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75 views

### Lift of Integrable Subbundle

Let $(M,F)$ be a manifold with integrable subbundle $F$ of the tangent bundle $TM$.(foliation).
Q For a submersion $N\to M$, we can lift a subbundle $F'$ of $TN$, can we say $F'$ is still integrable,...

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### The concept of convex foliation

A $n-1$ dimensional submanifold $N\subset \mathbb{R}^n$ is called a convex submanifold if for every $x\in N$ ,ther is a neighborhood $W$ of $x$ in $N$ such that $W$ entirly lies at one side of $T_x N$...

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118 views

### Riemannian submersions with negatively curved fibers

Are there some known examples of totally geodesic Riemannian submersions:
$$
\pi : N \to M
$$
such that:
1) $N$ is a compact Riemannian manifold
2) $M$ is a compact Riemannian manifold with positive ...

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264 views

### Holomorphic vector fields tangent to a hypersuface singularity

Let $(V,0)\subset (\mathbb{C}^n,0)$, $n\geq 3$ a (germ of) hypersurface given by $V =\{f=0\}$, $f$ a germ of holomorphic function. A (germ of) holomorphic vector field $X$ on $(\mathbb{C}^n,0)$ is "...

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196 views

### The connection between Lie algebroids and foliations

I need a bit of clarification about some of the geometry underlying the connection between Lie algebroids and foliations. In case of any confusion I'm using the definition of Lie algebroid from here.
...

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### Road to holomorphic foliations?

I want to know a "knowledge road" to holomorphic foliations. I assume that differential geometry and complex analysis is needed, but, what else? For example, I want to be able to read Lins Neto's book ...

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### A complete classification of linear foliations of $\mathbb{R}^n \setminus \{0\}$

A linear $1$-form on $\mathbb{R}^n$ is a $1$-form $\alpha=\sum_i P_i(x_1,x_2,\ldots,x_n)dx_i$ such that each $P_i$ is in the linear form $P_i=\sum_j a_{ij}x_j$. A linear foliation of $\mathbb{R}^n \...

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142 views

### A possible sub-Riemannian structure associated to a non-symmetric matrix

Let $A=(a_{ij})$ be an invertible matrix with real entries $a_{ij}$.
We associate to $A$ the $1$-form $\alpha=\sum_i (\sum_j a_{ij}x_j)dx_i$.
The distribution $\ker \alpha$ is integrable if and ...

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128 views

### Obstructions for a foliation to be transformed to a Frenet foliation

Assume that we have a $1$ dimensional foliation of $\mathbb{R}^2$. Is there a global diffeomorphism of the plane which maps all leaves of the foliation to curves with non zero curvature?
One can ...

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### $2$ dimensional foliations of space whose leaves contain the trajectories of a given vector field

Assume that $X$ is a non-vanishing vector field on $\mathbb{R}^3$.
Is there a $2$-dimensional foliation of space such that every trajectory of $X$ is contained in a leaf of the $2$-dimensional ...

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### Surgery and Curvature on Foliation

Let $X$ be an oriented closed smooth $4$-manifold. Suppose that $TM$ admits a foliation $\mathcal F$ of dimension two, and admits a positvescalar curvature.
Q: If we do the surgery on $X$ to reduce ...

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### The entire parametrization of leaves of singular holomorphic foliation of $\mathbb{C}P^2$

What is an example of an entire non constant holomorphic function $\gamma: \mathbb{C} \to \mathbb{C}P^2$ such that the image of $\gamma$ is a leaf of a singular holomorphic foliation of ...

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### Existence of complementary pairs of foliations on spheres

Let $M$ be an $n$-manifold, $0\leq k\leq n$. We define a $(k,n-k)$-bifoliation on $M$ to be a pair $(\mathscr{E},\mathscr{F})$ consisting of ($C^\infty$ nonsingular) foliations $\mathscr{E},\mathscr{F}...

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### Which planar smooth foliations are not smooth equivalent to a foliation arising from level sets of a harmonic function?

Is there an smooth foliation of the plane which is not smoothly equivalent to a foliation $dH=0$ where H is a harmonic function without critical values?
If the answer is negative then we conclude ...

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### A certain generalization of the Poincare Bendixon theorem

Assume that we have a $n-1$ dimensional integrable distribution $D $ on $\mathbb{R}^n \setminus \{0\}$ which generates a foliation $\mathcal{F}$. We fix an orientation for $D$.(For $n=2$ we ...

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### Reference request: Complex geodesic flow

Can someone suggest a book on complex geodesic flow? I am interested in it mainly because I was told these form a very useful class of Riemann surface laminations. Of special interest to me is the ...

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### Foliations, von Neumann algebras and measurability

In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured ...

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### 1-Bridge Braids in Solid Tori (Berge-Gabai knots), dual knots, & knot exteriors

Let $K$ be a knot in a solid torus. Combining results of Berge and Gabai, we know that if $K$ admits a solid torus surgery, then $K$ is a 1-bridge braid. Using Gabai's result, we can figure out what ...

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### A non vanishing vector field on $S^3$ with a periodic attractor

Is there a non vanishing real analytic vector field $X$ on $S^3$ such that $X$ has an attractor periodic orbit(An asymptotically stable periodic orbit) ? What about the smooth case?

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### Is every singular foliation induced by a Lie algebroid?

Let $M$ be a smooth manifold.
A smooth distribution $D$ on $M$ is the union of a family $\{D_p \leq T_p M : p\in M\}$ of vector spaces such that there is a family $\mathcal C $ of smooth vector ...