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)



A metric naturally arise from the Euclidean symplectic structure?
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.


