The contact structure on the real projective space












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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







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    0












    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 4 at 19:02









      BrianTBrianT

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