For $n>1$ let $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$. We define the following distribution $D$ on $\mathbb{R}^{2n}\setminus\{0\}$:

For $Z\in \mathbb{R}^{2n}\setminus\{0\}$ we define $D_Z=\{V\in \mathbb{R}^{2n}\mid \omega(V,Z)=0\}$

This is a nonintegrable distribution of codimension $1$. We define a meteic on $\mathbb{R}^{2n}\setminus\{0\}$ as follows: The distance $d(x,y)$ is the infimum of the Euclidean length of all $D$- horizontal curves joining(connecting) $x$ to $y$.

Is this metric well defined(i.e. is this distribution totally non integrable)?Does this metric arise from a Riemannian metric on $\mathbb{R}^{2n}\setminus\{0\} \}$?

What about if we consider the same question but we restrict all necessary structures to $S^{2n-1}$?(Intersection of above D$ with tangent space of spher and and comoutation of length of curves on the standard geometry of sphere)


Suppose we have a non-integrable distribution and a riemannian metric on a manifold. Then, one can define the metric using your construction, the resulting object is called sub-riemannian metric. It does not come from any riemannian or finslerian metric. Sub-riemannian metrics are important in the theory of nilpotent groups (and vice versa), as always with such geometric subjects there is a book of M. Gromov on it; see https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/carnot_caratheodory.pdf

It also arises in some applied area, namely, geometry of vision, on this I can not find exact reference now.

  • $\begingroup$ Thank you very much for your very interesting answer. $\endgroup$ May 13 '20 at 21:16
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    $\begingroup$ Considering this applied thematic the thing to google is "sub-riemannian model of visual cortex". $\endgroup$ May 14 '20 at 10:08
  • $\begingroup$ thank you again for your answer. i confess that i did not read the book of Gromov yet. I just brows it but i will read it. But just a few more questions: Is the distribution I considered totally non integrable that is the liealgebra of vector field tangent to D is thewhole algebra of vector field? Moreover is there a 1 dimensional folition of S^3 tangent to D(The intersection of D with S^3) such that the foliation has no closed leaf? On the opposite extrem, is there a 1 dimensional foliation of 3 sphere tangent to D whose all leaves are closed? $\endgroup$ May 18 '20 at 18:03
  • $\begingroup$ Note that non of the the three standard vector fields of S^3, which produce foliation by circles, ,are tangent to D. $\endgroup$ May 18 '20 at 18:04
  • $\begingroup$ I think if you restrict codimension 1 distribution on a sphere you will obtain codimension 1 distribution. For example, consider $S^3$ embedded in $\mathbb{R}^4 = \mathbb{C}^2 = \mathbb{H}$. Then, the distribution $\text{Ker}(\omega(v, *))$ is also euclidean orthogonal to $I(v)$ due to the identity $\omega(v, x) = g(Iv, x)$. Now, it is actually generated by vector fields $Jv$ and $Kv$ ($I J K$ being quaternion imaginary units). Commutator of $J$ and $K$ is indeed $I$, so this distribution is non-integrable and generates the whole tangent space. $\endgroup$ May 20 '20 at 11:37

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