Surfaces in the lightlike cone
Let \(E^{n+1}_q\) be \(n+1\)-dimensional pseudo-Riemannian space with pseudo-Riemannian metric \(\bar{G}\). It is well known that the pseudo-Riemannian light-like cone (or quadric cone) is defined by \(Q^{n}_{q}(c)=\{x \in E^{n+1}_q \mid \overline{G}(x-c, x-c)=0\}\). Since \(Q^{n}_{q}(c)\) is a degenerate hypersurface in \(E^{n+1}_q\), studying the geometry of such surfaces is harder than the other cases: space-like and time-like surfaces. In this interesting paper, the author obtains the structure equations and their integrability conditions for a surface with isothermal parameters in the lihgt-like cone \(Q^{n}\) (for \(c=0\) and \(q=1)\), and then classifies certain homogeneous surfaces in 3 and 4-dimensional cones \(Q^{3}\) and \(Q^{4}\).
scientific article; zbMATH DE number 5078883