# Questions tagged [foliations]

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

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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 ...
• 104
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• 733
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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 ...
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• 265
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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 ...
• 18.5k
1 vote
73 views

### 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 ...
• 104
1 vote
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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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### 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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### 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
1 vote
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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
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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 ...
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• 627
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### Poincaré-Dulac Normal Form for Holomorphic Foliations of Complex Manifolds of Complex Dimension $2$

Let $\mathcal{F}$ be a singular one-dimensional holomorphic foliation on a complex manifold $M$ of complex dimension $2$ and $p \in M$ a singular point for the foliation $\mathcal{F}$. Assume that the ...
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Suppose I have a smooth distribution on an open subset $M \subset R^n$; i.e. $k$ smooth functions $X^1(\cdot),\ldots,X^k(\cdot)$ on $M$ such that, at each $p \in M$, $X^i(p) \in R^n, 1 \leq i \leq k$....
It is well known that a $2-$torus foliated with lines of irrational slopes will produce dense curves on the torus. Likewise, rational slopes will lead to a periodic orbit. However, I am not seeing the ...