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### Equation of polar of chord of circle in Cartesian form [closed]

Can anybody please tell me the equation of polar of chord and its derivation. I doesn't understand homogenous joint equation of lines joining chord of two tangents to centre of circle

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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 ...

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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 ...

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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 ...

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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 ...

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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.
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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 ...

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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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### 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; ...

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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-$k$ ...