Let $D$ be the smooth distribution on $\mathbb{R}^3$ such that $$ D_{(a,b,c)}=\{\,(x,y,z)\in\mathbb{R}^3\,:\,z-bx=0\,\}. $$ How to show that for any $p,\,q\in\mathbb{R}^3$, there exists a path $\alpha$ from $p$ to $q$ tangent to $D$?
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What have you tried and what techniques do you have to understand distributions? – Ted Shifrin Nov 21 '20 at 18:54
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Basically what I have for distributions are the definitions of distributions, involutive distribution, and integral distributions in addition to Frobenius' theorem. I am not able to use differential forms techniques. The question had two parts, the first was to show that this distribution is not integral and was quite simple, I just found two generator and showed that the bracket is not in the distribution. But for this second part I had no ideia. – Edson Santos Nov 24 '20 at 14:37
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Differential forms are a powerful tool. Soldiering on without them, here's a suggestion: The vector fields $\partial/\partial y$ and $\partial/\partial x+y\,\partial/\partial z$ span $D$. Let $p=(p_1,p_2,p_3)$ and $q=(q_1,q_2,q_3)$.
(1) If $p_1=q_1$ and $p_3=q_3$, then you have an easy path contained in a single plane. Indeed, you can always move freely in the $y$ direction.
(2) If $p_1\ne q_1$, choose $b$ so that $(q_3-p_3)=b(q_1-p_1)$. Consider the points $(p_1,b,p_3)$ and $(p_3,b,q_3)$ and join them by a path tangent to $D$.
(3) If $p_1=q_1$ and $p_3\ne q_3$, first follow $D$ from $(p_1,p_2,p_3)$ to $(p_1,0,p_3)$ and then follow $D$ to $(p_1',0,p_3)$ with $p_1'\ne p_1$. Now you're back in case (2).
Ted Shifrin
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Thank you for your answer, but how can you guarantee that the final path will be differentiable at the points that you concatenate the paths? For instance, how can you guarantee that exists $\alpha'(p_1,b,p_3)$ in $(2)$? – Edson Santos Dec 04 '20 at 05:24
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Good point. If the two tangent directions are both in a plane, can you smooth the corner? (I confess that the way I always think about this question is with $1$-forms and Green’s Theorem.) – Ted Shifrin Dec 04 '20 at 05:29
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How can you solve this problem with $1$-forms and Greens' Theorem? – Edson Santos Dec 04 '20 at 05:45
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OK, so you want to project the points into the $xy$-plane and join them by a path $C$ (in the plane) with $\int_C y,dx$ equal to the change in the $z$-coordinates. Then lift the path. Do you see how to work this out? – Ted Shifrin Dec 04 '20 at 06:46
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