Heisenberg groups, Carnot groups and contact forms












2












$begingroup$


The horizontal distribution in the Heisenberg group is the kernel of the standard contact form:
$$
alpha = dt + 2 sum_{j=1}^n (x_j , dy_j - y_j , dx_j).
$$




Question. Can one describe horizontal distribution in any Carnot group in terms of kernels of some $1$-forms?




I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.






lie-groups sub-riemannian-geometry heisenberg-groups







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    The horizontal distribution in the Heisenberg group is the kernel of the standard contact form:
    $$
    alpha = dt + 2 sum_{j=1}^n (x_j , dy_j - y_j , dx_j).
    $$




    Question. Can one describe horizontal distribution in any Carnot group in terms of kernels of some $1$-forms?




    I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.






    lie-groups sub-riemannian-geometry heisenberg-groups







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      The horizontal distribution in the Heisenberg group is the kernel of the standard contact form:
      $$
      alpha = dt + 2 sum_{j=1}^n (x_j , dy_j - y_j , dx_j).
      $$




      Question. Can one describe horizontal distribution in any Carnot group in terms of kernels of some $1$-forms?




      I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.






      lie-groups sub-riemannian-geometry heisenberg-groups







      share|cite|improve this question









      $endgroup$




      The horizontal distribution in the Heisenberg group is the kernel of the standard contact form:
      $$
      alpha = dt + 2 sum_{j=1}^n (x_j , dy_j - y_j , dx_j).
      $$




      Question. Can one describe horizontal distribution in any Carnot group in terms of kernels of some $1$-forms?




      I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.






      lie-groups sub-riemannian-geometry heisenberg-groups




      lie-groups sub-riemannian-geometry heisenberg-groups






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 5 hours ago









      Piotr HajlaszPiotr Hajlasz

      8,57142867




      8,57142867






















          1 Answer
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          4












          $begingroup$

          The distribution on any Carnot group is left invariant, so pick any set of differential forms at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.



          On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank, you that was very simple.
            $endgroup$
            – Piotr Hajlasz
            4 hours ago










          • $begingroup$
            At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
            $endgroup$
            – YCor
            2 hours ago













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          $begingroup$

          The distribution on any Carnot group is left invariant, so pick any set of differential forms at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.



          On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank, you that was very simple.
            $endgroup$
            – Piotr Hajlasz
            4 hours ago










          • $begingroup$
            At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
            $endgroup$
            – YCor
            2 hours ago


















          4












          $begingroup$

          The distribution on any Carnot group is left invariant, so pick any set of differential forms at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.



          On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank, you that was very simple.
            $endgroup$
            – Piotr Hajlasz
            4 hours ago










          • $begingroup$
            At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
            $endgroup$
            – YCor
            2 hours ago
















          4












          4








          4





          $begingroup$

          The distribution on any Carnot group is left invariant, so pick any set of differential forms at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.



          On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.






          share|cite|improve this answer











          $endgroup$



          The distribution on any Carnot group is left invariant, so pick any set of differential forms at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.



          On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 4 hours ago

























          answered 5 hours ago









          RazielRaziel

          2,08411425




          2,08411425












          • $begingroup$
            Thank, you that was very simple.
            $endgroup$
            – Piotr Hajlasz
            4 hours ago










          • $begingroup$
            At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
            $endgroup$
            – YCor
            2 hours ago




















          • $begingroup$
            Thank, you that was very simple.
            $endgroup$
            – Piotr Hajlasz
            4 hours ago










          • $begingroup$
            At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
            $endgroup$
            – YCor
            2 hours ago


















          $begingroup$
          Thank, you that was very simple.
          $endgroup$
          – Piotr Hajlasz
          4 hours ago




          $begingroup$
          Thank, you that was very simple.
          $endgroup$
          – Piotr Hajlasz
          4 hours ago












          $begingroup$
          At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
          $endgroup$
          – YCor
          2 hours ago






          $begingroup$
          At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
          $endgroup$
          – YCor
          2 hours ago




















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