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

Luiz C. B. da Silva

Published 2018-01-03Version 1

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 is basically the study of $\mathbb{R}^3$ equipped with a degenerate metric: $\mathrm{d}s^2=\mathrm{d}x^2\pm\mathrm{d}y^2$. Pushing further results concerning simply isotropic surfaces [B. Pavkovi\'c, Glas. Mat. Ser. III \textbf{15}, 149 (1980)], here we introduce a Gauss map in both $\mathbb{I}^3$ and $\mathbb{I}_{\mathrm{p}}^3$ taking values on a certain unit sphere of parabolic type, define a shape operator $L_q$ from it, and show that the relative Gaussian and Mean curvatures are respectively the determinant and trace of $L_q$. After proving that every (admissible) pseudo-isotropic surface is timelike, we show that in analogy to what happens in Lorentz-Minkowski geometry the pseudo-isotropic shape operator may fail to be diagonalizable. On the other hand, as also happens in simply isotropic space, we prove that the only totally umbilical surfaces in $\mathbb{I}_{\mathrm{p}}^3$ are spheres of parabolic type and that the curvature tensor associated with the Levi-Civita connection vanishes identically for any pseudo-isotropic surface. Later, based on the Gauss map, we introduce a new notion of connection, named \emph{relative connection} (or \emph{r-connection}, for short), whose curvature tensor does not vanish identically and is directly related to the relative Gaussian curvature. Finally, we compute the Gauss and Codazzi-Mainardi equations for the r-connection, define \emph{r-geodesics} as autoparallel curves according to the new connection, and illustrate the new concept by showing that r-geodesics on a sphere of parabolic type are obtained by intersecting it with planes passing through its center (focus).