# Questions tagged [tangent-distributions]

The tangent-distributions tag has no usage guidance.

14
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### A smooth family of lattices on the tangent bundle?

I was recently in the cafeteria with a friend, and while having lunch I explained to him why the tangent bundle of a manifold is good at encoding geometric information of the manifold. My second ...

2
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0
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### Maximal dimension guaranteed for integral manifolds of hyperplane distributions

To KSackel and anyone else has viewed this: I'm sorry my edits have been all over the place. I've tried to cut it down to my remaining curiosities, so there's less to wade through (and hopefully fewer ...

3
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### Existence of sections of a fibre bundle which are covariantly constant along certain directions

Given a vector bundle $\pi\colon E \rightarrow B$ equipped with a connection $\nabla$, it is well known that a basis of flat sections $s_i$ ($i=1,\dots,\text{rank}(E)$) (i.e. $\nabla_X s_i = 0$ for ...

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1
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381
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### Rank of a distribution

I am reading about distributions in the context of differential geometry.
A distribution $S$ of dimension $r$ on a manifold $M$ is an assignment to each point $p \in M$ of an $r$-dimensional ...

4
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2
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### Converse to Chow's theorem in sub-riemannian geometry

Chow's theorem is the statement that if $M$ is a connected smooth manifold endowed with a distribution $\mathcal{D}$ which is completely non integrable (i.e. iterated commutators of smooth sections of ...

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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 ...

2
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1
answer

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### Local (quasi-)normal form for 3-plane fields on 6-manifolds

A 3-plane field $D$ on a 6-manifold $M$ is generic if $[D, D] = TM$. I'd like to do some explicit computations with a general local plane field of this type, and so I want to find a local (quasi-)...

3
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1
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529
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### Osculating spaces and distributions on (real) Grassmannian manifold

Hello! Recenlty, doing my research, I came across a quite natural construction, and I would like to know more about it. Unfortunately, being not expert neither in Grassmannians nor in Contact Geometry,...

6
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1
answer

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### Studying non-linear PDEs with manifolds

I'm sorry if this is an inappropriate forum to ask this question on, for I fear it is pretty undergraduate-level one :) I was contemplating on the study of non-linear PDEs. Is it possible to reduce a ...

7
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2
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405
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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.
...

2
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### Differential inclusions for distributions

Given a set valued function $F$ such that for every $x\in M$ (a manifold) we have that $F(x)\subset T_xM$, a differential inclusion is the "equation", $\dot{x} \in F(x)$.
I was wondering if someone ...

2
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1
answer

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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, ...

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1
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### Does a smooth, constant-rank, integrable distribution have a basis in which the traces of the structure constants are the divergences of the corresponding basis elements?

In a previous question, I asked an utterly trivial question, which Deane Yang correctly pointed out was utterly trivial. I will now ask a similar question, which is the one I meant to ask last time; ...

0
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1
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258
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### Does every smooth integrable constant-rank distribution have a basis in which the structure constants are traceless?

My question is local and coordinate-full: I have an open neighborhood $0 \in U \subseteq \mathbb R^n$, and I'm allowed to make it smaller around $0$. On this neighborhood, I have a constant-rank-$k$ ...