All Questions
Tagged with dg.differential-geometry foliations
124 questions
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A possible sub-Riemannian structure associated to a non-symmetric matrix
Let $A=(a_{ij})$ be an invertible matrix with real entries $a_{ij}$.
We associate to $A$ the $1$-form $\alpha=\sum_i (\sum_j a_{ij}x_j)dx_i$.
The distribution $\ker \alpha$ is integrable if and ...
- 108k
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3
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References for holomorphic foliations
I'm looking for an introduction to holomorphic foliations and foliations of complex manifolds.
Any little helps, but I'm particularily interested in problems of the type where we have a hermitian ...
- 356
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Foliations, von Neumann algebras and measurability
In the excellent book Noncommutative Geometry by Alain Connes much of the first chapter is devoted to foliations. At the end of the first chapter the author discusses index theory on measured ...
- 9,330
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Foliations by circles on the 3-torus
Let $T=S^1\times S^1 \times S^1$ be the 3-torus. Let us denote by $\alpha$ the isotopy class of the loop $S^1\times pt\times pt$ and let $\mathcal F_\alpha$ be the set of all smooth oriented ...
- 3,928
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Riemannian foliations and their leaf space
Let be $(M,g)$ a riemannian manifold with a singular riemannian $\mathcal{F}$ in $M$, see [1] the definition of singular riemannian foliation.
The riemannian metric on $M$ induces a distance on $M$, ...
- 245
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Necessary and sufficient condition for the foliation to induce a fibration
Suppose that there is a locally integrable foliation on a manifold $M$ such that any of its leaves is not dense in $M$. Does that mean that we can factorize $M$ by this distribution and get a ...
- 2,305
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Non-diffeomorphic smooth structures on the quotient of a manifold by an integrable distribution
In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/...
- 6,813
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Glueing together functions defined on the leaves of a foliation
Even though my question can be asked for very general types of foliations, I am interesetd only in its answer for Poisson manifolds, which are what I am currently studying.
Consider a Poisson ...
- 3,652
9
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1
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Conformal changes of metric and geodesics
Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...
- 356
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Can we foliate the anti-de sitter space in 3 dimensions by Riemann surfaces?
Can we foliate the anti-de sitter spacetime in 3 dimensions by hyperbolic Riemann surfaces? -- I think this is possible, but got stuck at finding the particular projective mapping that does this. Can ...
- 19.6k
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Deformation of foliation
Suppose $\kappa$ is a no-where vanishing 1-form, then its kernel is integrable is equivalent to condition $d\kappa \wedge \kappa = 0$.
My question is, can such foliation smoothly deformed such that ...
- 68.9k
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A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation
Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential forms....
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1
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When does a submersion have connected fibers?
Can we characterize when a submersion $F:M \to N$ between two smooth manifolds has connected fibers? If this is too hard, what are some sufficient conditions?
CommunityBot
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3
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twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors
Given a regular (constant rank) bi-vector $\Pi \in \Gamma(\bigwedge^2TM)$ on a smooth manifold $M$ the necessary and sufficient condition for the image of $\Pi^\sharp:T^*M\to TM$ to be an integrable ...
- 108k
4
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Smooth functions tangent to the leaves of a foliation
Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space
$$T_f C^\infty(M,N) = \...
- 5,343
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Kähler Cones in $\mathbb{C}^4$ and a foliation of $\mathbb{P}^3$
Take the 3-dimensional complex projective space $\mathbb{P}^3$. Consider the action of the group $SU(2)\times SU(2)$. I have read in physics related articles that these group gives a singular ...
- 101
3
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1
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345
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extended forms from foliations [closed]
hi,
i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...
- 14.4k
4
votes
1
answer
344
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Conformally immersed Riemann surfaces and foliations
I want to show that conformally immersed Riemann surfaces in $\mathbb{R}^4$ are leaves of a 2-foliation $\mathcal{F}$. I start with the generalized Weierstrass representation of the surfaces: take 4 ...
- 1,051
0
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1
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218
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Does an abelian group acting on a riemaniann manifold define an othogonal foliation?
This question is related to my previous question. Suppose that a group $G$ acts freely and properly on a Riemaniann manifold $(M, g)$. Than the orbits form a foliation for $M$. For $p \in M$, let $V_p$...
- 4,014
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522
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Preprint of Hamilton on deformations of foliations
Does anyone have access to Hamilton's 1978 Cornell preprint 'Deformation Theory of Foliations'. It is widely quoted but I couldn't find any online copy.
- 46
3
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2
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339
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A local transitivity property of the automorphism group of a foliated manifold
Let $(M,\mathcal F)$ be a smooth foliated manifold. An automorphism of $(M,\mathcal F)$ is a diffeomorphism of $M$ that takes leaves of $\mathcal F$ onto leaves. Let now $L$ be a leaf of $\mathcal F$. ...
CommunityBot
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7
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2
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408
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Hypersurfaces orthogonal to a cone
This question is somewhat related to Differential inclusions for distributions but I am asking for something rather more specific, so I hope it is alright to leave this as a separate, new question.
...
CommunityBot
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3
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1
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500
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Integrability of distributions close to a given one.
In this and this papers Thurston proves that every distribution is homotopic to an integrable one (in the first one for codimension greater than one and in the other for codimension one).
Recently, ...
- 32.4k
14
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2
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Frobenius Theorem for subbundle of low regularity?
Frobenius Theorem says that a subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ is tangent to a foliation if and only if for any two vector fields $X, Y \subset E$ the bracket $[X,Y]\subset E$...
CommunityBot
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