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

Abstract

In this work, we are interested in the differential geometry of surfaces in simply isotropic ${\mathbb {I}}^3$ and pseudo-isotropic ${\mathbb {I}}_{\mathrm {p}}^3$ spaces, which consists of the study of ${\mathbb {R}}^3$ equipped with a degenerate metric such as $\mathrm {d}s^2=\mathrm {d}x^2\pm \mathrm {d}y^2$. 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 ${\mathbb {I}}^3$ and ${\mathbb {I}}_{\mathrm {p}}^3$ does not vanish identically and is directly related to the relative Gaussian curvature. We also compute the Gauss and Codazzi–Mainardi equations for the r-connection and show that r-geodesics on spheres of parabolic type are obtained via intersections with planes passing through their center (focus). Finally, we show that admissible pseudo-isotropic surfaces are timelike and that their shape operator may fail to be diagonalizable, in analogy to Lorentzian geometry. We also prove that the only totally umbilical surfaces in ${\mathbb {I}}_{\mathrm {p}}^3$ 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.