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Questions tagged [foliations]

This tag is for questions about foliations in differential geometry and use in conjunction with the tag (differential geometry).

A foliation of a smooth manifold is a particular decomposition into connected, injectively immersed submanifolds and, these submanifolds are called the leaves of the foliation. If all of these leaves are equidimensional then the foliation is called regular or, otherwise it is called a singular foliation. According to the Frobenius theorem, a regular foliation on a smooth manifold can be equivalently expressed as an integrable distribution on the tangent bundle.

221 questions
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Complement of a foliation

I have an $n$-manifold $M$ which is foliated by leaves $F_\alpha$ of dimension $p$ and a path $\gamma:[0,1]\to M$. You can take without problems $\gamma$ to be injective. Is the following statement true? Claim: There exists a neighborhood $U$ of…
Daniel Robert-Nicoud
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12
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1 answer

The 4-sphere does not admit dimension 2 foliations

I want to know if the 4-sphere admits dimension 2 foliations. I found the following theorem in a dissertation by Jonathan Bowden (this is googleable, but I won't link it because I don't know exactly what the copyright issues might be). Anyway, here…
levitopher
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11
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Solving $\nabla \times \mathbf{b} = \mathbf{b} \times \mathbf{a}$

Suppose we are given a fixed vector field $\mathbf{a}$. I am interested in the problem of determining a vector field $\mathbf{b}$ such that $$\nabla \times \mathbf{b} = \mathbf{b} \times \mathbf{a}.$$ This has another interpretation. Suppose…
Joe
  • 143
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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 already confused. How can the orbits of $G$, which…
Katerina
  • 355
9
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2 answers

A question about the strict transform on blow-ups

I arrived at the following phrase at a material that I'm reading: Let $\pi :N'\rightarrow N$ be the blow-up of center $P$. For a given $a\in\mathcal{O}$ and $P'\in\pi^{-1}(P)$, the strict transform of $a$ in $P'$ is the ideal $str(a;P')$ of…
Marra
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8
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1 answer

Show that Hopf foliation is a foliation.

Consider $S^3 := \{(z,w) \in \mathbb{C}^2:|z|^2 + |w|^2 = 1\}$ be the unit $3$-sphere with equivalence relation $$(z,w) \sim (z',w') \iff z' = e^{i \theta }z, w' = e^{i\theta} w$$ for some $\theta \in \mathbb{R}$. My definition of…
user661541
8
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1 answer

Opens maps from topological manifolds whose fibers are not generically topological foliations

Update. I have asked this on MO, but have not yet received an answer. Proposition. The quotient map associated to a topological foliation (projecting to the leaf space) is open. However the fibers of an open map from a topological manifold need not…
Arrow
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7
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3 answers

1-dimensional foliation on a surface

Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary?
7779052
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Topology on the space of foliations

Let $(M^3,g)$ be a closed Riemannian manifold. Is there a “natural” topology on the space $\operatorname{Fol}(M)$ of smooth codimension $1$ foliations on $M$? Is there any other relevant structure on this set?
Eduardo Longa
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7
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What can be said about the leaves of a regular foliation?

I was wondering about the following. Let $M$ be a (smooth, closed, connected and oriented) manifold endowed with a regular foliation (i.e. such that all the leaves are smooth submanifolds of the same dimension and such that their tangent bundles…
Daniel Robert-Nicoud
  • 30,663
6
votes
1 answer

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 to ask for help. Hereafter is an excerpt from…
atlantropa
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6
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1 answer

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 a smooth structure on $M/\mathcal{F}$,…
koi_jp
  • 113
6
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2 answers

Codimension one foliations induced by a map to the circle.

Let $M$ be a closed connected $n$-manifold. If there exists a submersion $p:M\to S^1$, then $p$ is both proper and onto (since $M$ is compact and $S^1$ is connected). Therefore, by Ehresmann's theorem, $p$ is a fiber bundle and $M$ admits a…
Adam Chalumeau
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6
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Foliations and groupoids in algebraic geometry

I am currently studying the theory of foliations and groupoids from a differentiable viewpoint, in particular Haefliger spaces. [See Segal, Classifying spaces related to foliations, and Moerdijk, Classifying toposes and foliations.] I read that…
W.Rether
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6
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1 answer

Condition for integrability (foliation)

Let $(M^{n+1}, \langle \cdot, \cdot \rangle)$ be a parallelizable Riemannian manifold with a vector bundle isomorphism $$\varphi : TM \to M \times \mathbb{R}^{n+1}.$$ For $x \in M$, denote by $\varphi_x : T_x M \to \mathbb{R}^{n+1}$ the restriction…
Eduardo Longa
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