Questions tagged [foliations]
This tag is for questions about foliations in differential geometry and use in conjunction with the tag (differential geometry).
210 questions
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Hopf fiber is foliation of some distribution.
Consider $\mathbb{S}^3\subset \mathbb{H}$ where $\mathbb{H}$ is quaternions. Let
$$\pi:\mathbb{S}^3\to \mathbb{S}^2$$
s.t. $\pi(u,v)=(2u\bar{v},|u|^2-|v|^2)$ which is so-called Hopf fiber. Let $D=...
- 310
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50
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Pulling Back a Foliation through a transverse map
I am trying to show that the pullback $f^{*}F$ of a foliation $F$ is well defined for $f: N \rightarrow M$ transverse.
I see clearly that $f^{-1}(L)$ is a embedded submanifold of $N$ and so now I just ...
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Does every regular foliation admits a transversal regular foliation?
Suppose $M$ is a smooth manifold (compact, orietnted, or some other extra nice conditions), let $D\subset TM$ be an integrable distribution (i.e, $[D,D]\subset D$), then by Frobenius theorem, $D$ ...
- 612
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Existence of a foliation never tangent to a vector field?
Given a never vanishing vector field $X$ on a manifold $\mathcal M$, is there always a foliation of co-dimension 1 that is never tangent to $X$ ?
In other words, can we find a never vanishing 1-form $\...
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34
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For $\pi : M \to N$ a submersion, then $ \ker d\pi = \{ X \in TM : d\pi(X) = 0 \} $ is integrable on $M$
If $\pi : M \to N$ is a submersion, then one can show that
$ \mathcal{F}_{\pi} := \ker d\pi = \{ X \in TM : d\pi(X) = 0 \} $ is an involutive distribution on $M$.
Then one can conclude that it is ...
- 662
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Is a smooth submersion with closed fibers from a closed $3$-manifold to a non-Hausdorff $1$-manifold almost a fiber bundle over $S^1$?
Suppose $M$ is a connected and closed smooth $3$-manifold, $S$ is a $1$-dimensional locally Euclidean space with a given smooth structure, and $F : M \to S$ is a surjective smooth submersion with ...
- 1,893
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Rapidly decreasing surface? Adaptation from Schwartz functions?
A foliation $\mathcal{F}$ on a smooth manifold $ X $ is a smooth decomposition of $ X $ into disjoint connected submanifolds $ \{L_\alpha\}_{\alpha \in A} $, called leaves, where each leaf $ L_\alpha $...
- 204
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Reference request: multi-foliated systems?
Thinking about multi-foliated systems i.e. a system of foliations of the same manifold that intersect. Specificially I am curious if this type of thing is typically studied.
In which specific setting ...
- 204
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31
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Expression of [X, Y] on a local chart. Lie Bracket
Lemma. Expression of $[X,Y]$ on a local chart.
Let $x:U \subset M \rightarrow \mathbb{R}^{n}$ be a local chart. Denote by $\dfrac{\partial}{\partial x_{i}}$ the vector field on $U$ defined by $\dfrac{...
- 1
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Geometric quantisation: finding leaves of a polarisation
Let $N=\mathbb{S}^1\times\mathbb{S}^1$ be a symplectic manifold with symplectic form $\omega=\mathrm{d}\theta\wedge \mathrm{d}\phi$. The prescription of geometric quantisation is to choose a ...
- 2,803
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When is the leaf of a foliation on the quotient by a properly discontinuous free group action isomorphic to a leaf on the total space?
Let $\tilde M$ be a manifold and $G$ a discrete group acting freely and properly discontinuously on $\tilde M$. Let $M:= M/G$ and $p: \tilde M \rightarrow M$ is the projection; then $p$ is a covering ...
- 1,769
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Construct a foliation on a simply connected compact manifold with all leaves non-compact [closed]
Here's my progress so far in this problem:
There are no codimension-one foliations on even dimensional spheres (here's why). Also, by Novikov's theorem any codimension-one foliation on a simply ...
- 141
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Why is the lift of a foliate vector field to the transverse bundle a foliate vector field w.r.t. the lifted foliation?
Let $M$ be a manifold with a foliation $F$ of codimension $q$, and let $p: B \rightarrow M$ be the transverse frame bundle (which over each point of $M$ consists of frames in the quotient of the ...
- 1,769
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Two foliation atlases define the same foliation iff they induce the same partition
Let $A_1$ and $A_2$ be two different (maximal) foliation atlases on a smooth manifold $M$. I want to show, that they induce a different partition of $M$.
So far I managed to proof the following:
Let $(...
- 174
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43
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Understanding $\mathcal G$ in $\Bbb C$.
Construction:
Define $\mathcal F$ by the union of leaves:
$$\mathcal F:=\bigg \lbrace \mathcal M[\chi_t(x)]\cup \mathcal M\bigg[\frac{1}{1-\chi_r(x)}\bigg] \bigg \rbrace$$
where $\mathcal M$ denotes ...
- 204
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55
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Foliation by complex hypersurfaces
I am learning about the Levi flat hypersurfaces. Let $M \subseteq \mathbb{C}^n$ be a real smooth hypersurface, i.e. for every point $p$ in $M$ there is an open set $U_p$ in $\mathbb{C}^n$ containing $...
- 1,025
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A question on Lee's Proof of the Global Frobenius Theorem (Lemma 19.22)
I'm afraid this is a stupid question — I'm not a mathematician, so please correct me when I'll be saying something wrong — but I've been stuck at this point for so long that I thought it would be wise ...
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Existence of special transversal on foliation
This is a somewhat technical question about a line in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Program in the proof of the structure theorem, Theorem 8.3 in ...
- 4,873
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143
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Euler-Poincaré formula for foliations
Does someone have a nice proof for Proposition 11.14 in Farb&Margalits "Primer to Mapping Class Groups", which states the following:
Let $S$ be a closed surface with a singular foliation ...
- 1,432
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23
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foliations subdivided into maximally distinct classes that preserve some global metric?
It's simple to construct an analytic codimension one foliation of $\Bbb R^2_{\gt 0}$ with one class of functions. I wonder if there are pros and or cons of using more distinct classes of functions for ...
- 204
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A "length function" on measured lamination
I am a little confused about Corollary 4.13 in paper of Leininger and Aramayona. Let's keep everything simple that we denote by $S,\tau(S),\mathcal{S}(S),ML(S)$ closed orientable surface of genus $g\...
- 733
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63
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Evolution of second fundamental form
For hypersurfaces, one classically computes the evolution (in the normal direction) of the area form as mean curvature, and the evolution of the mean curvature as the stability operator. I'm lucky ...
- 1,817
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82
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Existence of Complementary Integrable Distribution
Let $M^n$ be a smooth manifold. Let $D$ be a (regular) integrable distribution on $M$.
The question is whether there always (i.e. for any $M$ and $D$) exists an integrable distribution $D'$ on $M$ ...
- 111
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2
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195
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Locally free circle actions on Euclidean space
I've been wandering whether or not there are locally free smooth circle actions on $\mathbb{R}^n$, or, more specifically, if an Euclidean space would admit a smooth nonvanishing vector field $X$ whose ...
- 587
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57
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How do I prove that left-invariant foliations are the “same” as Lie subalgebras $\mathfrak h \subset \mathfrak g$?
I found the following statement in my book:
Let $G$ be a Lie group with Lie algebra $\mathfrak g$. Left-invariant foliations are the “same” as Lie subalgebras $\mathfrak h \subset \mathfrak g$ ?
I ...
- 3,508
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87
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Measured foliations on surfaces
Is the measure of a simple closed curve consisting of two arcs transversal to a foliation equal to the sum of the measures of these arcs? Let A be an arc that is the union of arcs B and C, where arc B ...
- 1
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63
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Classification of good foliations of a pair of pants
The following is a proposition from FLP (Thurston's work on surfaces).
Proposition 6.7 (Classification of good foliations of a pair of pants) The function $\mathcal{MF}_0(P^2)\to\Bbb R^3_+$, which to ...
- 5,157
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107
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Foliation of $S^3$ with Leaves $S^1$
I'm attempting to construct a foliation of $S^3$ with leaves diffeomorphic to $S^1$.
My initial thoughts were to construct an involutive distribution generated by $X_1,\dots X_n$ such that $\langle ...
- 265
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88
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Reference for "every foliation satisfies a system of PDEs"?
I know foliations as a particular topic in differential geometry. I understand the definition and basic properties of a foliation from the DG point of view, including the Frobenius theorem. While ...
- 18.7k
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76
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foliational reciprocity
While thinking about foliations I thought about a property that a pair of manifolds could have, which I will call "foliational reciprocity."
Question:
Does there exist a pair of smooth ...
- 204
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102
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Intuition for flat charts of a distribution
In Lee's Introduction to Smooth Manifolds he gives the following definition:
Given a rank-$k$ distribution $D \subset TM$, let us say that a smooth coordinate chart $(U,\varphi)$ on $M$ is flat for $...
- 6,641
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Singular foliation of a particular singular distribution
Let's consider the manifold $\mathbb{R}^2$ and the singular distribution $D$ given by
$$ D_{(x,y)}=\left\langle\frac{\partial}{\partial x},\varphi(y)\frac{\partial}{\partial y}\right\rangle, \ \ \ \...
- 414
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65
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What are the distance between the leaves and the dimension of a foliation?
I'm reading an article on singular Riemannian foliations and I have a few very basic questions:
(1) Apparently, the transnormallity condition is equivalent to say that the leaves are locally ...
- 317
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When are the coordinate functions of a distribution independent of the leaf in the foliation?
By the Frobenius theorem, given an involutive distribution $\{X_i\}_{i = 1}^m$ on a manifold $M^n$ we can find a coordinate patch $(x^i)$ s.t. for some $q$ the set $x^{q+1} = \cdots = x^n = \text{...
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131
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Transition functions of foliation charts
Let $M$ be a smooth $n$-manifold (without boundary), and let $E \subset TM$ be an integrable (smooth) distribution on $M$ of codimension $q$. From the Frobenius theorem, we know that this distribution ...
- 23
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Developing map for foliations
I am reading Ghys' paper Holomorphic Anosov System. In the beginning of section 5, there is the following argument:
Let $\phi$ be a holomorphic Anosov diffeomorphism of a compact manifold $M$. Assume ...
- 177
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Constructing a pseudo-Anosov homeomorphism
Let S be a genus 2 closed surface. Consider the two non-separating red curves (say $\alpha$ & $\beta$) in the picture. I want to construct a pseudo-Anosov homeomorphism $\phi: S \rightarrow S$ ...
- 560
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About naturality of Godbillon-vey class [closed]
This is a problem from Lawrence Conlon's differential manifolds a first course. I do not know how to prove in the following problem
If $f: N \rightarrow M$ is transverse to $\mathcal{F}$, prove that
$$...
- 21
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In which paper did J. F. Plante introduce the notion of Holonomy Invariant Transverse Measure?
In which paper did J. F. Plante introduce (for the first time) the notion of Holonomy Invariant Transverse Measure?
I do appreciate any help can be provided.
Thanks in Advance.
- 2,566
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Can a family of linear connections defined on each leaf of a foliation be (uniquely) extended?
Say we have an involutive vector subbundle $\pi:E\subset TM\to M$, where $\pi:TM\to M$ is the tangent bundle to a paracompact smooth manifold $M$. Then $E_p=\pi^{-1}(\{p\})$ is the tangent space to ...
- 73
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156
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Detail in Lee's Proof of the Global Frobenius Theorem
I am taking a second pass at Lee's book, dotting the i's in the proofs and setting myself the objective of doing all the exercises in the body of the text. My question has to do with a step in the ...
- 29.6k
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73
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Fundamental groupoid of a foliation is not Hausdorff
Let $F$ be a foliation of a smooth manifold $M$. The fundamental groupoid $\Pi(F)$ of $F$ is the set
$\Pi(F)=\frac{\{\alpha:[0,1]\to M\text{ path cointained in a leaf}\}}{\text{homotopy with fixed end ...
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Using Godement's criterion to prove that leaf space of a foliation carries a smooth structure compatible with the quotient topology.
I am trying to prove the following from Differential Geometry by
Rui Loja Fernandes:
Let $\mathcal{F}$ be a foliation of a smooth manifold $M$. The following
statements are equivalent:
There exists ...
- 113
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Is a spherically symmetric space-time isometric to a warped product?
A spherically symmetric spacetime is a Lorentian 4-dimensional manifold $(M, g)$ whose isometry group contains a subgroup $G$ isomorphic to $\text{SO}(3)$ and whose orbits are 2-spheres. Here I am ...
- 345
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133
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Ensuring that a curve $\gamma : [0,1] \to M$ lies on an integral manifold of a distribution $E \subset TM$
Let $M$ be a smooth manifold, $E \subset TM$ a smooth distribution of codimension $k$, and $\gamma : [0,1] \to M$ a smooth curve whose tangent vector is in $\gamma^\star E$, and whose individual ...
- 1,775
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270
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When is the leaf of a foliation the level set of a function?
Suppose I have a smooth (say $C^1$) codimension one foliation of $P^n$ (open subset of $R^n$ consisting of vectors with all positive components) arising from a smooth $(n-1)$-plane field satisfying ...
- 627
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79
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Finding whether a rank-$k$ subbundle of $TM$, specified as a line subbundle of $\bigwedge^k TM$, is integrable
Let $M$ be a complex manifold or smooth algebraic variety, and let $X$ be a global vector field which is nonzero over “most” of $M$. Then I know that $X$ determines a foliation of $M$ by curves, ...
- 1,775
4
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263
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Order on the leaves of a foliation when each leaf separates an open subset of Euclidean space.
The setting is a codimension one foliation $\mathscr{F}$ of what I call $P$, the open subset of $R^n$ consisting of vectors with all positive components. I know that each leaf $L$ $\in \mathscr{F}$ is ...
- 627
5
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479
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The pullback of a foliation
I would like to know how to prove that the pullback of a foliation is actually a well-defined concept.
To be precise, if $M$ is an $n$-dimensional smooth manifold, a $k$-dimensional foliation $\...
- 2,793
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45
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Monotonic mapping from one-dimensional manifold M to R - What is the order on M?
First a bit of background: I have a particular $C^1$ codimension one foliation $\mathscr{F}$ of $P$ (n-dimensional Euclidean space with all positive coordinates) which I have equipped with a total ...
- 627