All Questions
Tagged with dg.differential-geometry foliations
124 questions
11
votes
2
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303
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+500
Cohomology of foliations and closed forms along the leaves
Let $M$ be a manifold equipped with a codimension one, transversely orientable, regular foliation $F \subset M$. Let $\alpha\in \Omega^k(M)$ be a differential form on $M$ that is not closed on $M$ ...
- 2,816
4
votes
0
answers
128
views
Errata for "Foliations and Geometric Structures" by Aurel Bejancu and Hani Reda Farran
I'm reading "Foliations and Geometric Structures" (2006) by Aurel Bejancu and Hani Reda Farran and have been looking for an errata sheet. Unfortunately Prof. Bejancu has passed away. I ...
1
vote
0
answers
113
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What is the difference between the foliation of a manifold and a mere partitioning? [closed]
I'm unable to visualize a partitioning that isn't also a foliation, and I need to so I can understand The Stretched Horizon and Black Hole Complementarity
In particular, I don't understand how curves ...
- 111
0
votes
0
answers
42
views
Fiber-wise mappings composed with projection map $\pi$
Let $M^2=(0,1)^2$. Recall that a chart is a diffeomorphism $\varphi:M^2 \to M^2$. Given a chart $\varphi:(M^2,g_0)\to (M^2,g_0)$ for $g_0$ the Euclidean metric, consider the curves $\varphi^{-1}(u,t)=\...
- 383
1
vote
0
answers
53
views
The Frobenius integrability of distrbution and Hyers–Ulam–Rassias stability
Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is ...
- 356
9
votes
1
answer
415
views
Are limits of compact leaves compact?
Let $M$ be a compact smooth manifold, and $\mathcal{F}$ be a foliation on $M$. Assume that $L$ is a leaf of $\mathcal{F}$ for which there is $x\in L$ with the property that every neighborhood of $x$ ...
- 1,163
2
votes
0
answers
83
views
Simply connectedness of leaves of a foliation on an complex manifold
Now I'm searching about leaves of foliation in the following special setting.
Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...
- 328
0
votes
0
answers
77
views
Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations
Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
- 383
5
votes
0
answers
373
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A (possible) Lie algebra extension of the Lie algebra of a foliation
Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is ...
- 356
6
votes
1
answer
513
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The current situation of the Godbillon-Vey invariant conjecture
Suppose we have a compact three-manifold $M$, a codimension-one foliation $F$ of $M$, and a one-form on $\alpha$, chosen so that $F$ is tangent to $\alpha = 0$. We deduce that $\alpha \wedge d\alpha = ...
- 356
5
votes
2
answers
245
views
A smooth family of lattices on the tangent bundle?
I was recently in the cafeteria with a friend, and while having lunch I explained to him why the tangent bundle of a manifold is good at encoding geometric information of the manifold. My second ...
- 1,093
3
votes
1
answer
205
views
Looking for examples of non-singular holomorphic foliations with compact leaves
I am looking for examples (or what is known about) of the following kind of object:
X compact Kähler manifold
F a non-singular holomorphic foliation on X (so given by a holomorphic subbundle of the ...
- 280
2
votes
1
answer
241
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Leaf holonomy of Reeb foliation on Möbius strip
I am trying to understand the leaf holonomy of the Reeb foliation on the Möbius strip, the first problem being visualization. I have been unable to find a visualization of this anywhere. I am ...
5
votes
2
answers
361
views
Exterior differentiation of foliations
Let $M$ be a differentiable manifold.
Let $T^*M$ be the cotangent bundle of $M$.
Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\...
- 123
2
votes
0
answers
51
views
Transnormal foliation with non-smooth transnormal function
I am interested in results regarding transnormal foliations on a Riemannian (smooth, connected and complete) manifold $(M,g)$. More specifically, a smooth function $f:M\to{\bf R}$ is called ...
- 283
2
votes
0
answers
127
views
Foliation of $X$ by once punctured planes without any singularities
Let $n=3.$
Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$ Construct a smooth regular foliation of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-...
- 383
3
votes
0
answers
117
views
Is there a notion of representation theory of foliations?
A foliation on a manifold $M$ can be seen as a sub bundle of the tangent bundle $\mathcal{F}\subseteq TM$ that is closed under Lie bracket of vector fields.
One can think of foliation as a Lie ...
- 6,023
3
votes
1
answer
192
views
Holonomy of foliation with trivial normal bundle
I am wondering about the following situation. Suppose $X$ is a compact Kahler manifold and $F \subset T_X$ is a holomorphic foliation. Suppose that the ``normal bundle'' $T_X / F$ of the foliation is ...
- 3,625
0
votes
0
answers
85
views
A 1 dimensional foliation which is Riemannian foliation with respect to no Riemannian metric
What is an example of a non vanishing smooth vector field on a manifold $M$ whose corresponding foliation is a Riemannian foliation with respect to no Riemannian metric on $M$
- 356
0
votes
0
answers
161
views
A closed leaf with two different index with respect to two different Riemannian metrics
Inspired by this question about Jacobi equation. conjugate points and limit cycle theory we ask the following question:
Is there a geodesible 1 dimensional foliation $\mathcal{F}$ on a manifold $M$, ...
- 356
1
vote
0
answers
102
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Homothety vector fields generating a foliation of $S^3$
Inspired by this question on homothety vector fields we realize that non homotheticity is some how an intrinsic property of the foliation associated to the vector field. See the comment by Prof. ...
- 356
3
votes
1
answer
232
views
Vector fields $X$ and $Y$ commute on a closed set $K$. Do there exist commuting $\tilde X,\tilde Y$ with $\tilde X=X$ and $\tilde Y=Y$ on $K$?
I have a nice research idea whose proof hinges on the following question
Suppose $X_p$ and $Y_p$ are vector fields in $\mathbb{R}^3$ with $[X,Y]_p=0$ for all $p$ in some closed set $K\subset\mathbb{R}...
- 133
1
vote
0
answers
97
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about codimension two foliation
Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...
- 73
2
votes
0
answers
222
views
On "graphs" of foliations
Let $M$ be a smooth manifold and $\mathcal{F}=\{\mathcal{F}_m\}_{m\in M}$ be a (regular) smooth foliation of $M$. The leaves $\mathcal{F}_m$ are smoothly immersed and moreover weakly embedded ...
- 491
9
votes
1
answer
366
views
Which submanifolds are leaves of a foliation?
Question. Let $M^{n+1}$ be a closed manifold without boundary. Which closed submanifolds $\Sigma^n \subset M^{n+1}$ (of codimension one) are leaves of a foliation of $M$ minus some finite collection ...
- 5,038
3
votes
0
answers
85
views
Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?
Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...
- 166
2
votes
1
answer
125
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Is the orbit foliation of the Weyl chamber flow Riemannian?
$\DeclareMathOperator\SL{SL}$Fix an integer $p\geq 1$ and a cocompact lattice $\Gamma\subset \SL(p+1,\mathbb{R})$. Consider the manifold
$$
M_{\Gamma}:=\SL(p+1,\mathbb{R})/\Gamma.
$$
Let $A\subset \SL(...
- 305
4
votes
1
answer
196
views
Global transversals of a codimension one foliation
EDIT: changes to the question are in bold.
Suppose I have a smooth ($C^1$) codimension one foliation $\mathscr{F} \subset P$, the open subset of $R^n$ consisting of points with all positive ...
- 355
1
vote
0
answers
32
views
Results on compact slices in a regular foliation
Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...
- 133
4
votes
0
answers
106
views
Geodesic foliations of open manifolds foliated by hyperbolic spaces
It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700).
Suppose a complete ...
- 4,729
1
vote
1
answer
363
views
Codimension-1 foliations of Euclidean space with strictly positive normal bundle
I am interested in the following situation: Given any $n>1$ suppose I have a codimension-1 foliation of $R^n_{++}$ (i.e. the subset of strictly positive $n$-vectors) arising from an $(n-1)$-...
- 355
3
votes
0
answers
74
views
A foliation version of S.Husseini counter example in fixed point theory
In this question we are indirectly inspired by an example by S.Husseini Amer. J.Math 1977
"The Products of Manifolds with the f.p.p. Need Not have the f.p.p"
who gave an example of two ...
- 356
1
vote
0
answers
83
views
Two codimension one foliations of a Lie group whose Godbilon–Vey invariants are not the same
What is an example of a Lie group $G$ with two codimension one foliations $F_1 $ and $F_2$ such that they generate two different Godbilon–Vey invariants in $H^3(G)$?
- 356
6
votes
1
answer
289
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Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)
This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. ...
- 295
1
vote
0
answers
70
views
Minimal sets of foliations in the plane (generalisation of Poincaré-Bendixson)
Let $F$ be a $1$-dimensional foliation of an open subset of the plane defined by a
locally Lipschitz line field. Suppose $C$ is a compact minimal set of $F$ (i.e. $C$ is non-empty, compact, a union of ...
- 11
4
votes
1
answer
149
views
Is a linear vector field a geodesible vector field?
I have already asked this question in MSE; I repeat it here at MO.
Assume that $A\in M_n(\mathbb{R})$ is a non-singular matrix.
Is the flow of linear vector field $X'=AX$ a geodesible flow on $\...
- 356
2
votes
1
answer
276
views
Complex fibration over complex torus
Let $M$ be a 3-dimensional complex manifold, and $\Lambda$ a discrete lattice in $\mathbb C^2$. Suppose there is a holomorphic submersion $f:M\to\mathbb{C}^2/\Lambda$ with fibers given by 1-...
- 661
8
votes
2
answers
775
views
Godbillon–Vey invariant and leaf space of foliations
I recently got to know about the existence of the so-called Godbillon–Vey invariant, and I am interested in its relationship with foliation theory in 3-manifolds. I briefly recall here the definition:
...
- 521
-1
votes
1
answer
230
views
Orthonormal frame on $\mathbb{S}^3$ orthogonal to foliations [duplicate]
Does there exist a smooth orthonormal frame $X_1,X_2,X_3$ on $\mathbb{S}^3$ such that the distribution spanned by $X_i$ and $X_j$ is integrable for all $1\leq i,j\leq 3$?
- 89
2
votes
0
answers
191
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Blowing up the zero section for "Chasse au Canard" (some new kind of geometric canards)
In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a ...
- 356
3
votes
0
answers
73
views
The diversity of Riemannian metrics adapted to a given foliation، A Krein Millman view point(2)
Inspired by this answer to the linked question we add a more bounded conditions to this post. This question is asked seperately because the previous one had a complete answer so we did not revise ...
- 356
5
votes
1
answer
204
views
The diversity of Riemannian metrics adapted to a given (1 dimensional) foliation, A Krein Millman view point
Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. Let $K$ be the space of all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1$...
- 356
2
votes
0
answers
110
views
A quantity associated to a foliated manifold and its non-commutative interpretation
Let $M$ be a compact $n$-dimensional manifold. Assume that $F$ is a $k$-dimensional foliation of $M$.
The graph $G(M,F)$ of this foliation is a $(n+k)$-dimensional manifold. We recall its definition:
...
- 356
6
votes
0
answers
219
views
Foliations and locally free action of $\mathbb{R}^{n-1}$
Let $M$ be a $n$-dimensional closed manifold endowed with a foliation ${\cal F}$ suppose that the leaves of ${\cal F}$ are diffeomorphic to $\mathbb{R}^{n-1}$ are the leaves of ${\cal F}$ defined by a ...
- 3,652
1
vote
0
answers
186
views
Examples of why conditions for Novikov compact leaf theorem are necessary
Let $M^3$ be a smooth, closed 3-manifold. Given a smooth codimension-one foliation $\mathcal{F}$ of $M$, the Novikov compact leaf theorem asserts that, when the universal cover $\widetilde{M}^3$ of $M^...
6
votes
0
answers
267
views
Elliptic foliations of the plane
A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties:
The differential operator ...
- 356
5
votes
1
answer
206
views
Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves
Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the ...
- 135
2
votes
1
answer
342
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Foliation of tangent bundle arising from exponential map
We first mention our motivation:
For $M=\mathbb{R}$ with usual Riemannian metric the exp map $exp:TM\to M$ is in the form$(x,v)\mapsto x+v$
The level sets of this map define a foliation whose leaves ...
- 356
2
votes
0
answers
61
views
Approximating a volume along a submersion
Here is the setup. I have a submersion $f:X \to T$, where $X$ is manifold and $T$ is a torus (you can chose a circle for the beginning if it is simpler). The manifold $X$ has a volume form $\alpha$ ...
- 21
13
votes
2
answers
700
views
Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?
More precisely, is there a criterion that decides the above question?
I am particularly interested in the smooth setting: is a smooth manifold with a smooth regular foliation by circles covered by a ...
- 183