Quasi-contact S-R metrics: Normal form in \(\mathbb{R}^{2n}\), wave front and caustic in \(\mathbb{R}^4\)
A quasi-contact structure on a \((2n+2)\)-manifold \(M\) is a sub-bundle \(D\) of the tangent bundle, of codimension 1, such that if \(D\) is the kernel of a 1-form \(\omega\), then the kernel of \(d\omega\) restricted to \(D\) has dimension 1. A quasi-constant sub-Riemannian metric on \(M\) is a couple \((D,g)\), where \(g\) is a nondegenerate Riemannian metric on \(D\). By analogy with [\textit{A. Agrachev, E. C. El Alaoui, J.-P. Gauthier} and \textit{I. Krupka}, C. R. Acad. Sci., Paris, Sér. I 322, 377-384 (1996; Zbl 0843.53025), \textit{A. Agrachev, G. Charlot} and \textit{V. Zakalyukin}, Dyn. Control Syst. 6, 365-395 (2000), \textit{A. Agrachev, G. Stefani} and \textit{P. Zezza}, Proc. Symp. Pure Math. 64, 1-11 (1999; Zbl 0937.49012) or \textit{A. Agrachev} and \textit{J.-P. Gauthier}, J. Math. Sci. 103, 639-663 (2001; Zbl 1008.53019)] which deal with any odd dimension, the present paper constructs the same type of normal coordinates, normal frame and classical invariant tensors in the quasi-contact case in any even dimension. Local singularities of the exponential map are studied in dimension~4.
scientific article; zbMATH DE number 1879603