# Questions tagged [foliations]

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

113 questions
Filter by
Sorted by
Tagged with
12 views

### Holonomy of the Reeb foliation on the solid torus

The past few days I am trying to calculate the holonomy group of the boundary leaf(i.e. of the torus) of the Reeb foliation on the solid torus. Classic textbooks, like Moerdijk & Mrcun, cite the ...
23 views

39 views

### Can any vector be extended locally to a geodesic vector field?

Given a (pseudo-)Riemannian manifold $(M,g)$ and some vector $X_p\in T_pM$ at $p\in M$, can one always extend $X_p$, locally, to a geodesic vector field $X$, in the sense that any integral curve of $X$...
21 views

52 views

### Line bundle associated with a $1$-cocyle of holomorphic functions.

I'm trying to understand the following extract: A holomorphic foliation by curves $\mathcal{F}$ of degree $r$ on the projective space is a bundle map $\Omega : H_{-r+1} \rightarrow T\mathbb{P}^n$, ...
26 views

### Are singular foliations spanned by collinear vector fields equal?

Let $M$ be a compact $n$-manifold (let's say with boundary, but this isn't too important), $X$ a vector field on $M$ and $f:M \to \mathbb{R}$ a non-zero function on $M$. My question: are the singular ...
45 views

### 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 ...
30 views

### Fibration in integral manifolds

Consider a smooth manifold $M$ of dimension $n$ and an integrable tangent distribution $$\mathcal{D} = span\{X_1,...,X_k\}$$ with $k\leq n$. Then we know that $M$ is foliated by the connected ...
33 views

### Confusion about the definition of a rectangular neighborhood

I am looking for a precise definition of a rectangular neighborhood in the definition of a foliation chart. In the book "Foliations I" by Alberto Candel and Lawrence Conlon, the definition is the ...
22 views

### Looking for reference: If a Riemanian manifold is foliated by max symmetric submanifolds, then coordinates can always be chosen such that …

In Weinberg's book on General Relativity in section 13.5 it is shown that, loosely stated, if a Riemannian manifold $(M,g)$ of dim $m$ is composed of maximally symmetric submanifolds $(N,h)$ of dim $n$...
67 views

### A foliations as a G-stucture

According to this Wikipedia entry, a foliation is a particular G-structure whose structure group reduction induced by block matrices. I tried to find more details of this approach to foliations, but ...
30 views

### Normal field to a foliation

Given a vector field $n$ on a Riemannian manifold $(M,g)$ we can define \begin{equation} \Delta_x=\{v\in T_xM : g(v,n_x)=0\} \end{equation} I have to find the condition on $n$ such that this ...
29 views

### What is the leaf space of the Kronecker foliation of the torus?

The leaves of the Kronecker foliation of the torus are diffeomorphic to the real line $\mathbb{R}$ and are dense on $T^2$. I cannot, however, describe what is the leaf space $T^2/\mathcal{F}$.
18 views

38 views

46 views

### Phase, Isochrons, Isochrons map and Lift

at the moment i read the following paper: https://arxiv.org/pdf/1512.04436v1.pdf I have some questions about it and i hope someone can help me. On page 4/5 they introduce isochrons and the isochron ...
76 views

### Lie group action and foliation

Let $M$ be a smooth manifold, and $G$ be a compact Lie group acting on $M$ smoothly. We assume that isotropic group of any point $x\in M$ is of dimension zero. Q How to show that $M$ admits a ...
70 views

### What “given a foliation $f:L \to M$ ” means?

I'm reading this paper. And on page $4$ the author writes the following paragraph: I am not sure about what the phrase "given a foliation $f:L\to M$" means. A foliation is an atlas, so it is awkward ...
65 views

### On morse theory and foliations

Consider a manifold $M$ with a morse function $t$. For the regular points, $t^{-1}(c)$ are embeddings, and the homotopy changes as one jumps from one regular point to another. But, my question is: ...
36 views

### Integrable system is not a level set: an example

For a manifold $M$, let $f \in C^{k}(M), k>0$, be a level set (i.e. $rank(df)=1$). I understand that this trivially implies that $\{{k.df: k \in C^{k}(M)\}}$ forms an integrable regular Pfaffian ...
### Connected components of nonempty level sets form a foliation of $M$.
Let $M,N$ smooth manifolds and $F:M \to N$ a smooth submersion. Show that the connected components of nonempty level sets form a foliation of $M$. My idea is to use the Global Fröbenius theorem for ...