A bilinear form associated to contact sub-conformal manifolds
Let \(M\) be a \((2n + 1)\)-dimensional manifold endowed with a sub-Riemannian structure \((M, D, g)\) where \(D\) is a contact distribution on \(M\) and \(g\) is a metric on \(D\). Denote by \((M D, \tilde{g})\) a sub-conformal structure on \(M\) such that \(\tilde{g}\) is a conformal class of sub-Riemannian metrics on \(D\). Consider the \(SO(2n)\) bundle \(E\) of coframes \((\theta, \theta^i)\) on \(M\) such that the \(\theta^i\) restricted to \(D\) are orthonormal, \(\theta\) is the positive contact form defined by \[ d\theta = h_{ij} \theta^i \wedge \theta^j,\quad \text{det}h_{ij} = 1. \] Denote by \(\wedge\) the canonical form of \(H = (h_{ij})\) and \(E_1\) the subset of points of \(E\) such that \(H = \wedge\). The authors have proved the following three results {\parindent=4mm \begin{itemize}\item[{\(\bullet\)}] Suppose \(H\) is constant on \(E_1\). Then there exists a torus bundle \(T\) over \(M\) and a conformal invariant bilinear form \(b\) on \(T\). \item[{\(\bullet\)}] If \(H^2 = -Id\) on \(E_1\), then the torus bundle \(T\) is a \(U(1)\) bundle and \(b\) is a conformal Lorentz metric. \item[{\(\bullet\)}] The bilinear form \(b\) descends to a Lorentz metric on the \(U(1)\) bundle \(N = T/ \exp g\) defined by another conformal invariant bilinear form. \end{itemize}}
scientific article; zbMATH DE number 5163861
scientific article