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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)

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)

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 Riemmanian 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)

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)

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 $\mathbb{R}^{2n}\setminus\{0\}$:

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

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

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 Riemmanian 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)