The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces

In this work, we are interested in the differential geometry of surfaces in simply isotropic I^3 and pseudo-isotropic I_p^3 spaces, which consists of the study of R^3 equipped with a degenerate metric such as ds^2 = dx^2 \pm dy^2. The investigation is based on previous results in the simply isotropic space (Pavković in Glas Mat Ser III 15:149–152, 1980; Rad JAZU 450:129–137, 1990), which point to the possibility of introducing an isotropic Gauss map taking values on a unit sphere of parabolic type and of defining a shape operator from it, whose determinant and trace give the known relative Gaussian and mean curvatures, respectively. Based on the isotropic Gauss map, a new notion of connection is also introduced, the relative connection (r-connection, for short). We show that the new curvature tensor in both I^3 and I_p^3 does not vanish identically and is directly related to the relative Gaussian curvature. We also prove that the only totally umbilical surfaces in I_p^3 are planes and spheres of parabolic type and that, in contrast to the r-connection, the curvature tensor associated with the isotropic Levi-Civita connection vanishes identically for any pseudo-isotropic surface, as also happens in simply isotropic space.