Transversal Cartan chains in a real hyperquadric (Q2714199)

Let \(G\) be a subgroup of the special group \(SL(n+1,C)\) which leaves invariant \(Q\), the real hyperquadric of a Hermitian quadratic form \(\phi(\zeta)\) of index \((n,1)\) in the complex projective space \(P^nV\), and \(D\) the \(n-2\) dimensional distribution defined by the Cauchy-Riemann structure induced over \(Q\).NEWLINENEWLINENEWLINEIn the paper the induced \(G\) action on the manifold \(C^2Q\) of contact elements of order 2 and dimension 1 is studied. It is proved that the Cartan chains transversal to \(D\) are solutions of a differential system defined as a submanifold of \(C^2Q\). This yields a characterization of the transversal chains as curves whose second order contact elements are singular at all their points. It is also shown that the Cartan chains are orbits of order 1 induced by the action of a closed subgroup \(K\) of \(G\) on \(Q\).