Generalized translation hypersurfaces in conformally flat spaces

The graph of a real function f defined in some open set of the Euclidean space of dimension (p+q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p+q)$$\end{document} is said to be a generalized translation graph (GTG) if f may be expressed as the sum of two independent functions ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} and ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} defined in open sets of the Euclidean spaces of dimension p and q, respectively. In this work, we study the geometry of GTG immersed in Euclidean space equipped with a metric conformal to Euclidean metric and obtain results that characterize such hypersurfaces. Applying the characterization results, and using ODE solving techniques, we build examples of GTG satisfying geometric properties not valid in relation to the Euclidean metric.