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Questions tagged [foliations]

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2
votes
1answer
74 views

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 $2$ ...
3
votes
1answer
89 views

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 ...
3
votes
1answer
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 $F$ ...
4
votes
1answer
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 ...
4
votes
0answers
45 views

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 $g_{\epsilon}$...
3
votes
1answer
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 $\...
3
votes
0answers
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^*...
0
votes
0answers
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 $\...
7
votes
2answers
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 ...
1
vote
1answer
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 ...
2
votes
0answers
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, ...
0
votes
1answer
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 ...
5
votes
1answer
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 ...
2
votes
1answer
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 $F$...
2
votes
1answer
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 ...
1
vote
1answer
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 ...
0
votes
0answers
38 views

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 ...
2
votes
1answer
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 $\tau$ (...
5
votes
0answers
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 ...
3
votes
1answer
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+(...
4
votes
1answer
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 ...
1
vote
2answers
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
3
votes
1answer
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 ...
1
vote
0answers
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 $\mathbb{C}P^1$. ...
3
votes
0answers
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 ...
1
vote
0answers
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 ...
3
votes
0answers
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 ...
4
votes
0answers
180 views

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 ...
1
vote
1answer
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,...
6
votes
1answer
150 views

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 $T_x N$...
4
votes
0answers
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 ...
7
votes
1answer
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 "...
6
votes
0answers
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. ...
4
votes
0answers
98 views

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 ...
0
votes
1answer
120 views

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 \...
1
vote
1answer
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 ...
3
votes
1answer
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 ...
2
votes
1answer
75 views

$2$ 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 ...
2
votes
0answers
94 views

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 ...
1
vote
1answer
58 views

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 ...
6
votes
0answers
136 views

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}...
1
vote
0answers
51 views

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 ...
3
votes
0answers
276 views

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 ...
6
votes
0answers
183 views

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 ...
3
votes
0answers
98 views

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 ...
8
votes
1answer
225 views

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 ...
4
votes
1answer
134 views

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?
13
votes
2answers
511 views

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 ...