The contact structure on the real projective space
$begingroup$
I came across the following description of the standard contact structure on the real projective space:
let $mathbb{R}^{2n}$ be endowed with its standard symplectic structure. The real projectivization, $mathbb{R} P^{2n-1}$ has a standard contact structure, and a point $p$ in $mathbb{R} P^{2n-1}$ is a real line in $mathbb{R}^{2n}$; its orthogonal complement relatively to the symplectic form is a hyperplane containing this line. In $mathbb{R} P^{2n-1}$ we get a hyperplane through $p$, and the tangent space of this defines the element of the contact structure at $p$.
I have trouble understanding why the last sentence is true. Any help would be greatly appreciated. Thanks !
projective-space contact-topology
asked Jan 4 at 19:02
BrianTBrianT
1866
$endgroup$
add a comment |
$begingroup$
I came across the following description of the standard contact structure on the real projective space:
let $mathbb{R}^{2n}$ be endowed with its standard symplectic structure. The real projectivization, $mathbb{R} P^{2n-1}$ has a standard contact structure, and a point $p$ in $mathbb{R} P^{2n-1}$ is a real line in $mathbb{R}^{2n}$; its orthogonal complement relatively to the symplectic form is a hyperplane containing this line. In $mathbb{R} P^{2n-1}$ we get a hyperplane through $p$, and the tangent space of this defines the element of the contact structure at $p$.
I have trouble understanding why the last sentence is true. Any help would be greatly appreciated. Thanks !
projective-space contact-topology
asked Jan 4 at 19:02
BrianTBrianT
1866
$endgroup$
add a comment |
$begingroup$
I came across the following description of the standard contact structure on the real projective space:
let $mathbb{R}^{2n}$ be endowed with its standard symplectic structure. The real projectivization, $mathbb{R} P^{2n-1}$ has a standard contact structure, and a point $p$ in $mathbb{R} P^{2n-1}$ is a real line in $mathbb{R}^{2n}$; its orthogonal complement relatively to the symplectic form is a hyperplane containing this line. In $mathbb{R} P^{2n-1}$ we get a hyperplane through $p$, and the tangent space of this defines the element of the contact structure at $p$.
I have trouble understanding why the last sentence is true. Any help would be greatly appreciated. Thanks !
projective-space contact-topology
asked Jan 4 at 19:02
BrianTBrianT
1866
$endgroup$
I came across the following description of the standard contact structure on the real projective space:
let $mathbb{R}^{2n}$ be endowed with its standard symplectic structure. The real projectivization, $mathbb{R} P^{2n-1}$ has a standard contact structure, and a point $p$ in $mathbb{R} P^{2n-1}$ is a real line in $mathbb{R}^{2n}$; its orthogonal complement relatively to the symplectic form is a hyperplane containing this line. In $mathbb{R} P^{2n-1}$ we get a hyperplane through $p$, and the tangent space of this defines the element of the contact structure at $p$.
I have trouble understanding why the last sentence is true. Any help would be greatly appreciated. Thanks !
projective-space contact-topology
projective-space contact-topology
asked Jan 4 at 19:02
BrianTBrianT
1866
asked Jan 4 at 19:02
BrianTBrianT
1866
asked Jan 4 at 19:02
BrianTBrianT
1866
asked Jan 4 at 19:02
BrianTBrianT
1866
asked Jan 4 at 19:02
BrianTBrianT
1866
1866
add a comment |
add a comment |
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