Mathematical Journal of Okayama University volume63 issue1
da Silva, Luiz C. B.
Department of Physics of Complex Systems, Weizmann Institute of Science
We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of isotropic rigid motions lead to seven types of invariant surfaces, which then generalizes the study of revolution and helicoidal surfaces in Euclidean and Lorentzian spaces to the context of singular metrics. After computing the two fundamental forms of these surfaces and their Gaussian and mean curvatures, we solve the corresponding problem of prescribed curvature for invariant surfaces whose generating curves lie on a plane containing the degenerated direction.
Simply isotropic space
prescribed Gaussian curvature
prescribed mean curvature
Mathematics Subject Classiﬁcation. Primary 53A35; Secondary 53A10; 53A40.